On the methods of mechanical non-theorems (latest version)
On the methods of mechanical non-theorems (latest version)
On the methods of mechanical non-theorems (latest version)
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<strong>On</strong> <strong>the</strong> <strong>methods</strong> <strong>of</strong> <strong>mechanical</strong> <strong>non</strong>-<strong>the</strong>orems<br />
André Rognes<br />
April 12, 2013
Contents<br />
0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1<br />
0.2 Summary <strong>of</strong> <strong>the</strong> chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1<br />
0.2.1 Summary <strong>of</strong> chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2<br />
0.2.2 Summary <strong>of</strong> chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2<br />
0.2.3 Summary <strong>of</strong> chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 3<br />
0.3 What is known . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4<br />
0.3.1 Classification <strong>of</strong> <strong>the</strong> Entscheidungsproblem based on syntax . . . . . . . 4<br />
0.3.2 Beyond syntactically defined classes . . . . . . . . . . . . . . . . . . . . 6<br />
0.3.3 Algebraic logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6<br />
0.3.4 Automated reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . 8<br />
0.3.5 Personal perspective and acknowledgements . . . . . . . . . . . . . . . . 8<br />
1 Turning decision procedures into disprovers 11<br />
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12<br />
1.1.1 Sets and mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13<br />
1.1.2 First-order formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13<br />
1.1.3 Algebras, homomorphisms, and preservation . . . . . . . . . . . . . . . 14<br />
1.1.4 Algebras <strong>of</strong> boolean signature . . . . . . . . . . . . . . . . . . . . . . . 15<br />
1.2 Algebras <strong>of</strong> polyadic signature . . . . . . . . . . . . . . . . . . . . . . . . . . . 15<br />
1.2.1 Operations related to variable substitution . . . . . . . . . . . . . . . . 16<br />
1.2.2 Algebras <strong>of</strong> polyadic signature . . . . . . . . . . . . . . . . . . . . . . . 16<br />
1.2.3 L 3 as an algebra <strong>of</strong> polyadic signature . . . . . . . . . . . . . . . . . . . 17<br />
1.2.4 Polyadic set algebras <strong>of</strong> ternary relations . . . . . . . . . . . . . . . . . . 18<br />
1.2.5 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19<br />
1.2.6 Additivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20<br />
1.2.7 The Lindenbaum algebra <strong>of</strong> a <strong>the</strong>ory Γ . . . . . . . . . . . . . . . . . . 21<br />
1.2.8 Exhaustive search for satisfying interpretations in a Ps 3 . . . . . . . . . . 22<br />
1.3 Algebras <strong>of</strong> directed many-sorted polyadic signature . . . . . . . . . . . . . . . . 22<br />
1.3.1 Algebras <strong>of</strong> directed many-sorted polyadic signature . . . . . . . . . . . . 22<br />
1.3.2 A conservative reduction class with sub-formulae as an algebra . . . . . . 23<br />
1.3.3 Directed many-sorted polyadic set algebras . . . . . . . . . . . . . . . . 24<br />
1.3.4 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25<br />
1.3.5 The directed many-sorted polyadic closure . . . . . . . . . . . . . . . . 26<br />
1.3.6 The closure as an algorithm . . . . . . . . . . . . . . . . . . . . . . . . 28<br />
1.3.7 Exhaustive search for satisfying interpretations in a dMsPs 3 . . . . . . . 29<br />
iii
1.4 A construction <strong>of</strong> Ps 3 ’s and dMsPs 3 ’s . . . . . . . . . . . . . . . . . . . . . . 30<br />
1.4.1 A disprover deviced according to <strong>the</strong> method . . . . . . . . . . . . . . . 31<br />
1.5 Related constructions and algebras . . . . . . . . . . . . . . . . . . . . . . . . . 32<br />
1.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33<br />
2 Automata for mechanising consistency pro<strong>of</strong>s 35<br />
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36<br />
2.1.1 Outline <strong>of</strong> paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37<br />
2.1.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38<br />
2.1.3 p-automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38<br />
2.1.4 The Ehrenfeucht-Fraïssé method . . . . . . . . . . . . . . . . . . . . . 40<br />
2.1.5 Mappings from and to a finite set . . . . . . . . . . . . . . . . . . . . . 41<br />
2.1.6 Vector-spaces over finite fields . . . . . . . . . . . . . . . . . . . . . . . 42<br />
2.2 n-fold vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43<br />
2.2.1 3-fold vector spaces over <strong>the</strong> minimal field . . . . . . . . . . . . . . . . 44<br />
2.2.2 n-fold vector spaces over a given finite field . . . . . . . . . . . . . . . . 45<br />
2.3 p-automata with abstract alphabets . . . . . . . . . . . . . . . . . . . . . . . . 47<br />
2.3.1 n-tape p-automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47<br />
2.3.2 Infinite tapes with finite support . . . . . . . . . . . . . . . . . . . . . 48<br />
2.4 Transitive automata and reachability in general . . . . . . . . . . . . . . . . . . 49<br />
2.4.1 A not quite classical notion <strong>of</strong> equivalence . . . . . . . . . . . . . . . . . 49<br />
2.4.2 Transitive p-automata, <strong>the</strong> two-sorted case . . . . . . . . . . . . . . . . 50<br />
2.4.3 Transitive p-automata, <strong>the</strong> one-sorted case . . . . . . . . . . . . . . . . 51<br />
2.5 Reachability refined . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55<br />
2.5.1 Two-sorted multi-automata . . . . . . . . . . . . . . . . . . . . . . . . 55<br />
2.5.2 Definable linear transformations . . . . . . . . . . . . . . . . . . . . . 56<br />
2.5.3 Projective transitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 57<br />
2.5.4 Projection automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58<br />
2.5.5 Substitution automata . . . . . . . . . . . . . . . . . . . . . . . . . . . 60<br />
2.5.6 Properties <strong>of</strong> <strong>the</strong> two-sorted transition function . . . . . . . . . . . . . . 62<br />
2.6 Moving to <strong>the</strong> abstract and to one sort . . . . . . . . . . . . . . . . . . . . . . . 63<br />
2.6.1 <strong>On</strong>e-sorted multi-automata . . . . . . . . . . . . . . . . . . . . . . . . 64<br />
2.6.2 Properties <strong>of</strong> <strong>the</strong> one-sorted transition function . . . . . . . . . . . . . . 66<br />
2.7 The automata for a formula and its sub-formulae . . . . . . . . . . . . . . . . . 68<br />
2.8 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73<br />
3 Automata for <strong>the</strong> computation <strong>of</strong> finite representable polyadic algebras 75<br />
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76<br />
3.1.1 Outline <strong>of</strong> paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77<br />
3.1.2 Sets, relations and mappings . . . . . . . . . . . . . . . . . . . . . . . . 77<br />
3.1.3 Finite-dimensional (quasi) polyadic algebras . . . . . . . . . . . . . . . . 78<br />
3.1.4 Purely infinite spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 81<br />
3.2 The polyadic atom-structure and <strong>the</strong> h-complex algebra . . . . . . . . . . . . . . 82<br />
3.2.1 Atom-structures and n-homomorphisms . . . . . . . . . . . . . . . . . 82<br />
iv
3.2.2 The complex algebra tailored for many sorts . . . . . . . . . . . . . . . 84<br />
3.3 The h-complex algebra <strong>of</strong> a multi-automaton . . . . . . . . . . . . . . . . . . . 86<br />
3.3.1 Concrete PTPS-automata defined . . . . . . . . . . . . . . . . . . . . . 86<br />
3.3.2 Properly partitioned automata . . . . . . . . . . . . . . . . . . . . . . . 90<br />
3.3.3 Automata as atom-structures . . . . . . . . . . . . . . . . . . . . . . . 90<br />
3.3.4 The n-homomorphism induced by an automaton . . . . . . . . . . . . . 91<br />
3.3.5 Representability <strong>of</strong> <strong>the</strong> h-complex algebra <strong>of</strong> an automaton . . . . . . . . 92<br />
3.3.6 The main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95<br />
3.4 Diversity <strong>of</strong> <strong>the</strong> h-complex algebras <strong>of</strong> finite automata . . . . . . . . . . . . . . . 95<br />
3.4.1 dMsPs n ’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95<br />
3.4.2 Ps n ’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96<br />
3.4.3 MsPs n ’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97<br />
3.4.4 Polyadic equality algebras . . . . . . . . . . . . . . . . . . . . . . . . . 97<br />
3.5 Axiomatisation by a finite set <strong>of</strong> first order sentences . . . . . . . . . . . . . . . . 98<br />
3.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101<br />
v
0.1 Introduction<br />
The present <strong>the</strong>sis is on mechanisation <strong>of</strong> <strong>the</strong> part <strong>of</strong> ma<strong>the</strong>matical reasoning that amounts to<br />
establishing that a formal sentence is not a <strong>the</strong>orem, i.e. it is about dispro<strong>of</strong>. A ma<strong>the</strong>matician typically<br />
uses examples in <strong>the</strong> form <strong>of</strong> infinite structures to establish that a sentence is not a <strong>the</strong>orem.<br />
Infinite structures are however unsuitable as objects <strong>of</strong> <strong>mechanical</strong> computation by virtue <strong>of</strong> not<br />
being finite. The present <strong>the</strong>sis is about a recently discovered class <strong>of</strong> finite boolean algebras with<br />
operators that can be used in place <strong>of</strong> infinite structures for establishing that a formal sentence is<br />
a <strong>non</strong>-<strong>the</strong>orem. There is no <strong>mechanical</strong> procedure for constructing all <strong>of</strong> <strong>the</strong> finite algebras, but<br />
<strong>the</strong> algebras can be used to construct <strong>mechanical</strong> procedures with capabilities that for a ma<strong>the</strong>matician<br />
implies pondering structures that necessarily are infinite. Such capability is witnessed when a<br />
procedure establishes <strong>the</strong> consistency <strong>of</strong> a formal sentence that is true only <strong>of</strong> infinite structures. A<br />
formal sentence <strong>of</strong> this kind is called an infinity axiom.<br />
We work in <strong>the</strong> framework for formal <strong>mechanical</strong> procedure and formal <strong>the</strong>orem, as set forth<br />
and studied by, amongst o<strong>the</strong>rs <strong>the</strong> ma<strong>the</strong>maticians D. Hilbert, K. Gödel, A. Church and A. Turing<br />
in <strong>the</strong> 1930-ies. D. Hilbert posed <strong>the</strong> problem <strong>of</strong> whe<strong>the</strong>r it is possible to device a <strong>mechanical</strong> procedure<br />
for deciding whe<strong>the</strong>r given first-order formulae are (in)consistent. This problem is known<br />
as Hilberts Entscheidungsproblem. We shall mostly be concerned with <strong>the</strong> Entscheidungsproblem<br />
in its most classical form namely by considering sentences <strong>of</strong> pure predicate logic, i.e. <strong>the</strong> logic <strong>of</strong><br />
first-order formulae without equality or function symbols. Note that by combining results <strong>of</strong> J. von<br />
Neumann, P. Bernays and K. Gödel it is possible to axiomatise set-<strong>the</strong>ory with such a sentence, and<br />
<strong>the</strong>refore questions <strong>of</strong> derivability from set-<strong>the</strong>ory can be rephrased as questions <strong>of</strong> consistency <strong>of</strong><br />
sentences in pure predicate logic.<br />
For <strong>the</strong> purpose <strong>of</strong> experimentation several <strong>mechanical</strong> disproving procedures, based on <strong>the</strong><br />
newly discovered algebras, have been implemented by <strong>the</strong> present author. They were made publicly<br />
available and presented at <strong>the</strong> international workshop on disproving held as part <strong>of</strong> <strong>the</strong> Federated<br />
Logic Conference in Seattle in 2006. <strong>On</strong> grounds <strong>of</strong> <strong>the</strong> response <strong>the</strong>re, we believe that that <strong>the</strong>se<br />
disprovers were <strong>the</strong> first implemented procedures with disproving capability for pure predicate logic,<br />
that fully automatically and naturally recognise a class <strong>of</strong> consistent sentences including infinity<br />
axioms in reasonable time. Natural here means undoctored or <strong>non</strong>-ad-hoc, i.e., <strong>the</strong> capability <strong>of</strong><br />
recognising infinity axioms stems from <strong>the</strong> algebra on which <strong>the</strong> procedure is based alone. The use<br />
<strong>of</strong> abstract algebra was crucial for solving <strong>the</strong> practical problem <strong>of</strong> making <strong>the</strong> procedures terminate<br />
for at least some infinity axioms in reasonable time.<br />
0.2 Summary <strong>of</strong> <strong>the</strong> chapters<br />
Apart from <strong>the</strong> introduction, <strong>the</strong> present <strong>the</strong>sis consists <strong>of</strong> three chapters, <strong>the</strong> first <strong>of</strong> which is based<br />
on a preprint <strong>of</strong> a journal article, see [Rog09], and <strong>the</strong> remaining two are manuscripts submitted to<br />
journals for publication.<br />
Before some elaboration, we summarise <strong>the</strong> chapters in one sentence each as follows. The<br />
algebras and <strong>the</strong> <strong>the</strong> way <strong>of</strong> constructing new algebras from old ones, introduced in chapter 1<br />
provide a novel way <strong>of</strong> subdividing <strong>the</strong> consistent sentences into <strong>mechanical</strong>ly listable classes <strong>of</strong><br />
sentences. In chapter 2 we introduce a novel kind <strong>of</strong> automaton for <strong>the</strong> purpose <strong>of</strong> tackling a<br />
class <strong>of</strong> consistent sentences, which properly includes those tackled by <strong>the</strong> method <strong>of</strong> finite model<br />
1
search. In chapter 3 we show how to use <strong>the</strong> automata <strong>of</strong> chapter 2, to compute algebras <strong>of</strong> <strong>the</strong><br />
kind introduced in chapter 1, as well as <strong>the</strong> well known quasi-polyadic- and substitution-cylindric<br />
algebras, for <strong>the</strong> purpose <strong>of</strong> naturally and <strong>mechanical</strong>ly tackling infinity axioms in reasonable time.<br />
0.2.1 Summary <strong>of</strong> chapter 1<br />
In chapter 1 we work with <strong>the</strong> Entscheidungsproblem in an extremely reduced form, namely in <strong>the</strong><br />
form <strong>of</strong> <strong>the</strong> subclass <strong>of</strong> pure predicate logic consisting <strong>of</strong> finite conjunctions <strong>of</strong> prenex sentences<br />
using no more than three variables. The class is known to be a conservative reduction class, which<br />
is to say that in a coarse but precise sense <strong>of</strong> sameness, working with <strong>the</strong> Entscheidungsproblem<br />
for this subclass is <strong>the</strong> same as working on <strong>the</strong> Entscheidungsproblem for first-order language with<br />
an arbitrary supply <strong>of</strong> variables. D. Scott, in 1962, proved that pure predicate logic with less than<br />
three variables is not, in <strong>the</strong> above sense, <strong>the</strong> same as working with an arbitrary supply.<br />
The afore mentioned axiomatisation <strong>of</strong> set-<strong>the</strong>ory is not known to belong to <strong>the</strong> three variable<br />
class o<strong>the</strong>r than via a <strong>non</strong>-trivial conservative reduction. However, working with this class facilitates<br />
getting to <strong>the</strong> core <strong>of</strong> <strong>the</strong> Entscheidungsproblem. Restriction to three variables is also beneficial in<br />
that it allows us to compute some <strong>of</strong> <strong>the</strong> algebras <strong>of</strong> interest in a matter <strong>of</strong> days. <strong>On</strong>ce computed, an<br />
algebra can be used in place <strong>of</strong> finite structures in a variant <strong>of</strong> <strong>the</strong> well known method <strong>of</strong> establishing<br />
consistency by searching for a finite model for a sentence, i.e. to attempt to find a finite structure <strong>of</strong><br />
which <strong>the</strong> formal sentence is true. The algebras <strong>of</strong> interest are those where <strong>the</strong> modified method<br />
yields consistency <strong>of</strong> infinity axioms. In this case <strong>the</strong> method <strong>of</strong> finite model search is strictly<br />
generalised.<br />
Novelties presented in chapter 1 are <strong>the</strong> newly discovered class <strong>of</strong> boolean algebras with operators,<br />
called directed many-sorted polyadic set algebras toge<strong>the</strong>r with a result on <strong>the</strong> class stating that<br />
<strong>the</strong> finite members suffice for dispro<strong>of</strong> <strong>of</strong> any consistent sentence <strong>of</strong> <strong>the</strong> conservative reduction class.<br />
We have likened this to <strong>the</strong> downward Löwenheim-Skolem <strong>the</strong>orem because it states that countable<br />
models suffice for dispro<strong>of</strong> <strong>of</strong> any consistent sentence <strong>of</strong> first-order logic, which is a conservative<br />
reduction class trivially. A novel way <strong>of</strong> constructing new algebras <strong>of</strong> interest from a given one is introduced<br />
and we show how to use this construction to generalise <strong>the</strong> method <strong>of</strong> finite model search.<br />
The construction is related to <strong>the</strong> tensor product <strong>of</strong> polyadic algebras introduced by A. Daigneault<br />
in 1963, see [Dai63]. The fact that this construction on <strong>the</strong> well known quasi-polyadic algebras<br />
gives rise to a way <strong>of</strong> generalising finite model search is also covered in chapter 1 and is novel.<br />
0.2.2 Summary <strong>of</strong> chapter 2<br />
It is a consequence <strong>of</strong> <strong>the</strong> undecidability <strong>of</strong> <strong>the</strong> Entscheidungsproblem and <strong>of</strong> our analogue to <strong>the</strong><br />
downward Löwenheim-Skolem <strong>the</strong>orem that <strong>the</strong> finite directed many-sorted polyadic set algebras<br />
can not completely be listed by means <strong>of</strong> a <strong>mechanical</strong> procedure. Subclasses <strong>of</strong> <strong>the</strong> algebras may,<br />
however, be listed and <strong>the</strong> remaining two chapters are on such a subclass.<br />
In chapter 2 we prepare <strong>the</strong> ground for <strong>mechanical</strong>ly listing algebras, some <strong>of</strong> which yield<br />
disprovers that recognise classes with infinity axioms. The ground is prepared by introducing a<br />
variant <strong>of</strong> <strong>the</strong> finite automata used by J.R. Büchi in <strong>the</strong> 1960-ies in a procedure for deciding Presburger<br />
arithmetic, <strong>the</strong> first-order <strong>the</strong>ory <strong>of</strong> addition <strong>of</strong> natural numbers. This procedure uses finite<br />
automata to represent possibly infinite relations definable in Presburger arithmetic. Operations corresponding<br />
to union, negation and projection which are needed to interpret Presburger formulae,<br />
2
can be done on automata computationally. This implies that it is possible to search for models for<br />
given formal sentences amongst, <strong>the</strong> not necessarily finite, relations definable in Presburger arithmetic.<br />
The automata we introduce are <strong>the</strong> finite models <strong>of</strong> a first order <strong>the</strong>ory. Moreover we show<br />
that <strong>the</strong> operations corresponding to union, negation and projection are not only computable but<br />
definable using first-order sentences about <strong>the</strong> automata. This is relevant for computation, since<br />
a property that is definable by means <strong>of</strong> a first-order formula can be checked in polynomial time.<br />
Polynomial time is here with respect to <strong>the</strong> maximum <strong>of</strong> <strong>the</strong> size <strong>of</strong> <strong>the</strong> state-set and alphabet <strong>of</strong><br />
given automata.<br />
The above allows us to transform given sentences in pure predicate logic into equiconsistent<br />
first-order sentences about finite automata. Moreover consistency can be established by <strong>the</strong> familiar<br />
method <strong>of</strong> finite model search, provided <strong>the</strong> pure sentence has a model definable in Presburger<br />
arithmetic.<br />
Novelties <strong>of</strong> chapter 2 include a fairly general technique for tackling <strong>the</strong> problem that reachability<br />
is not a first-order property over finite automata. It is known that over finite structures this<br />
problem can be tackled by giving a first-order definition <strong>of</strong> some higher-order constructs and to<br />
define reachability by means <strong>of</strong> that. The novel technique <strong>of</strong> chapter 2 avoids defining higherorder<br />
constructs by allowing some flexibility in <strong>the</strong> alphabet <strong>of</strong> <strong>the</strong> automata and by postulating<br />
that states that are reachable in two steps also are reachable in one step. We introduce a novel<br />
class <strong>of</strong> automata, called PTPS-automata. The class is defined by means <strong>of</strong> a finite set <strong>of</strong> first-order<br />
formulae. The finite members <strong>of</strong> <strong>the</strong> class are shown to be <strong>of</strong> <strong>the</strong> same computational strength<br />
as <strong>the</strong> multi-track automata introduced by J.R. Büchi in that <strong>the</strong>y may replace one ano<strong>the</strong>r in his<br />
procedure for deciding Presburger arithmetic. The transformation <strong>of</strong> sentences <strong>of</strong> pure predicate<br />
logic into equiconsistent first-order statements about automata, without introducing higher-order<br />
constructs, is novel.<br />
0.2.3 Summary <strong>of</strong> chapter 3<br />
In chapter 3 we turn to <strong>the</strong> algebras introduced in chapter 1. When using an algebra in place <strong>of</strong><br />
a finite structure in <strong>the</strong> modified method <strong>of</strong> finite model search, <strong>the</strong> algebra has to be representable<br />
in order to terminate for consistent sentences only. It is a consequence <strong>of</strong> <strong>the</strong> undecidability <strong>of</strong><br />
<strong>the</strong> Entscheidungsproblem and our analogue to <strong>the</strong> downward Löwenheim-Skolem <strong>the</strong>orem that<br />
<strong>the</strong>re is no <strong>mechanical</strong> procedure for telling whe<strong>the</strong>r an algebra is representable. In chapter 3 we<br />
introduce a way <strong>of</strong> turning PTPS-automata, introduced in chapter 2, into directed many-sorted<br />
polyadic algebras that necessarily are representable. Any algebra finitely generated by relations<br />
definable in a slight expansion <strong>of</strong> Presburger arithmetic is shown to be embeddable in <strong>the</strong> algebra<br />
<strong>of</strong> a PTPS-automaton. This class <strong>of</strong> algebras results in disproving procedures that recognise infinity<br />
axioms. We show some results on criteria making <strong>the</strong> construction on PTPS-automata result in <strong>the</strong><br />
classical one-sorted polyadic algebras and polyadic equality algebras. The criteria and <strong>the</strong> PTPSautomata<br />
are shown to be definable using a finite set <strong>of</strong> first-order formulae.<br />
The construction <strong>of</strong> representable algebras from PTPS-automata goes via <strong>the</strong> well known atomstructure<br />
<strong>of</strong> a finite algebra <strong>of</strong> polyadic similarity type. The atom-structure plays <strong>the</strong> same role as<br />
a basis for a vector-space or a topology. Working with atom-structures is beneficial with respect<br />
to <strong>mechanical</strong> procedures since a polyadic algebra with 2 k elements has an atom-structure with k<br />
elements. The afore mentioned implemented disprovers are based on algebras stored in <strong>the</strong> form <strong>of</strong><br />
3
atom-structures.<br />
Novelties <strong>of</strong> <strong>of</strong> chapter 3 include <strong>the</strong> generalisation <strong>of</strong> <strong>the</strong> well known complex algebra <strong>of</strong> an<br />
atom-structure to fit many-sorted algebras and <strong>the</strong> sufficient conditions for such a complex algebra<br />
to be representable. The application <strong>of</strong> <strong>the</strong> generalised complex algebra to finite automata for <strong>the</strong><br />
purpose <strong>of</strong> computing representable algebras <strong>of</strong> <strong>the</strong> well known quasi polyadic and substitutioncylindric<br />
similarity types with and without diagonals is novel.<br />
0.3 What is known<br />
The present <strong>the</strong>sis is a contribution to <strong>the</strong> subjects <strong>of</strong> <strong>the</strong> Entscheidungsproblem and <strong>of</strong> representability<br />
in algebraic logic. We briefly go through <strong>the</strong> most relevant results as ga<strong>the</strong>red from <strong>the</strong><br />
two major books on <strong>the</strong> subjects, namely <strong>the</strong> book “The Classical Decision Problem” by E. Börger,<br />
E.Grädel and Y. Gurevich [BGG97] and “Cylindric Algebras part II” by J.D. Monk, L. Henkin and<br />
A. Tarski, [MHT85]. As <strong>the</strong> latter, i.e. [MHT85], may not be easily accessible, we mention that it<br />
is based on some lecture notes that may be available, namely “Cylindric set algebras”, [HMA + 81].<br />
For algebraic logic <strong>the</strong> survey articles <strong>of</strong> I. Németi [Ném97] and <strong>of</strong> H. Andréka, I. Sain and I.<br />
Németi [ANS01] give a somewhat more updated picture <strong>of</strong> what has been done. In algebraic logic<br />
<strong>the</strong> chapters on polyadic algebras are <strong>the</strong> relevant ones. We work with finite dimensional algebras<br />
and in <strong>the</strong> present setting polyadic algebras are in all relevant respects <strong>the</strong> same as quasi-polyadic<br />
algebras, pinter-algebras and substitution-cylindric algebras as encountered in <strong>the</strong> surveys.<br />
0.3.1 Classification <strong>of</strong> <strong>the</strong> Entscheidungsproblem based on syntax<br />
In 1920 <strong>the</strong> Norwegian ma<strong>the</strong>matician T. Skolem published a pro<strong>of</strong> <strong>of</strong> <strong>the</strong> fact that any sentence <strong>of</strong><br />
pure predicate logic can be brought to an equisatisfiable, i.e. equiconsistent, sentence <strong>of</strong> <strong>the</strong> form<br />
∀x 0 . . . ∀x n−1 ∃y 0 . . . ∃y m−1 φ<br />
where φ is pure and quantifier free. This reduction is by means <strong>of</strong> a <strong>mechanical</strong> procedure as are<br />
<strong>the</strong> rest <strong>of</strong> <strong>the</strong> reductions mentioned in this section.<br />
In 1933 K. Gödel published an improvement <strong>of</strong> T. Skolems result by showing that sentences<br />
can be brought to <strong>the</strong> form<br />
∀x 0 ∀x 1 ∀x 2 ∃y 0 . . . ∃y m−1 φ<br />
where φ is pure and quantifier free. This is to say that no more than 3 universal quantifiers need<br />
be considered when working with <strong>the</strong> Entscheidungsproblem. Following <strong>the</strong> book [BGG97] <strong>the</strong>se<br />
two classes are denoted [∀ ∗ ∃ ∗ ] and [∀ 3 ∃ ∗ ] respectively.<br />
Then in 1935 T. Skolem published a simpler pro<strong>of</strong> <strong>of</strong> K. Gödels result and in doing so he<br />
showed that any sentence can be brought to one <strong>of</strong> <strong>the</strong> form<br />
(∀x∃y 0 . . . ∃y n−1 φ) ∧ (∀x∀y∀zφ ′ ) ∧ ( ∧ i
Parallel to this ma<strong>the</strong>maticians worked on syntactic criteria for a decision procedure to exist.<br />
In doing so, sentences <strong>of</strong> <strong>the</strong> following two classes were shown to be consistent if and only if <strong>the</strong><br />
method <strong>of</strong> finite model search succeeds.<br />
[∃ ∗ ∀ ∗ ] by P. Bernays and M. Schönfinkel (1928)<br />
[∃ ∗ ∀ 2 ∃ ∗ ] proven independently by K. Gödel (1932), L. Kalmár (1933) and K. Schütte (1934).<br />
In 1936 and 1937 respectively, A. Church and A. Turing proved <strong>the</strong> undecidability <strong>of</strong> <strong>the</strong> Entscheidungsproblem.<br />
An analysis <strong>of</strong> A. Turings pro<strong>of</strong> yields <strong>the</strong> undecidability <strong>of</strong> <strong>the</strong> class [∀∃ ∧ ∃∀ 5 ].<br />
Gradual sharpening <strong>of</strong> A. Turings result led to pro<strong>of</strong>s that <strong>the</strong> following two prefix classes are undecidable.<br />
[∀ 3 ∃] and [∀ 3 ∧ ∀ 2 ∃] by J. Surányi (1943) improving a result <strong>of</strong> J. Pepis (1938)<br />
[∀∃∀] by A. Kahr, E. Moore and H. Wang (1962)<br />
Toge<strong>the</strong>r with <strong>the</strong> two decidable classes this yields a complete classification <strong>of</strong> pure predicate logic<br />
in terms <strong>of</strong> syntactic criteria on <strong>the</strong> quantifiers. In short; if we have a maximum <strong>of</strong> two universal<br />
quantifiers preceding some existential quantifier and <strong>the</strong> two are consecutive <strong>the</strong>n <strong>the</strong> method <strong>of</strong><br />
finite model search succeeds. O<strong>the</strong>rwise we have to look for fur<strong>the</strong>r criteria, or hope that <strong>the</strong><br />
sentence has a finite model anyway. Beyond that we may use some <strong>of</strong> <strong>the</strong> <strong>methods</strong> presented in <strong>the</strong><br />
present <strong>the</strong>sis and hope those succeed, which <strong>the</strong>y do when <strong>the</strong> sentence happens to have a finite<br />
model as well as in several cases where all <strong>the</strong> models are infinite. We note that <strong>the</strong> latter two classes<br />
remain undecidable when restricted to relation symbols <strong>of</strong> arity at most 3.<br />
The undecidable classes can fur<strong>the</strong>r be divided into decidable and undecidable classes in terms<br />
<strong>of</strong> syntactic criteria. We mention that <strong>the</strong> class [∀∃∀] restricted to formulae in which all disjunctions<br />
are binary at most, was proven to be decidable by S.O. Aanderaa in 1971/73, see [Aan71],[AL73].<br />
This class contains <strong>the</strong> possibly shortest infinity axiom known to mankind, namely<br />
∀x∃y∀z¬Rxx ∧ Rxy ∧ (Rzx → Rzy),<br />
which states that <strong>the</strong> relation R is irreflexive, moreover it has no maximal elements and is transitive<br />
on connected components. Based on experiments we conjecture that <strong>the</strong> published implementations<br />
<strong>of</strong> <strong>the</strong> diprovers described in <strong>the</strong> present <strong>the</strong>sis recongnises <strong>the</strong> consistent members <strong>of</strong> this<br />
class.<br />
O<strong>the</strong>r infinity axioms where <strong>the</strong> mentioned implementations, work well in experiments, belong<br />
to a class shown to be decidable by W. Ackermann (1936), namely sentences <strong>of</strong> <strong>the</strong> form<br />
∀x∃ySxy ∧ ∀x∀y∀zφ<br />
where S is a binary relation symbol and φ is quantifier-free and uses S only. An infinity axiom <strong>of</strong><br />
this class is<br />
∀x∃ySxy ∧ ∀x∀y∀z¬Sxx ∧ (Sxy ∧ Syz → Sxz).<br />
In 1983 S.O. Aanderaa presented a pro<strong>of</strong> sketch <strong>of</strong> <strong>the</strong> decidability <strong>of</strong> a generalisation <strong>of</strong> W. Ackermanns<br />
class [Aan83].<br />
5
0.3.2 Beyond syntactically defined classes<br />
By <strong>the</strong> completeness <strong>the</strong>orem <strong>of</strong> K. Gödel (1930) we may conclude that <strong>the</strong>re is a procedure for<br />
listing <strong>the</strong> inconsistent sentences. By <strong>the</strong> undecidability <strong>of</strong> <strong>the</strong> Entscheidungsproblem we fur<strong>the</strong>r<br />
conclude that <strong>the</strong>re is no procedure listing <strong>the</strong> consistent sentences. A sub-class <strong>of</strong> pure predicate<br />
logic, where ei<strong>the</strong>r <strong>the</strong> sub-class or its complement can not be listed by means <strong>of</strong> a <strong>mechanical</strong><br />
procedure is said to be <strong>non</strong>-recursive.<br />
In 1950 B. Trakhtenbrot proved that <strong>the</strong> sentences <strong>of</strong> pure predicate logic for which <strong>the</strong> method<br />
<strong>of</strong> finite model search succeeds is <strong>non</strong>-recursive. In <strong>the</strong> same year W. Craig independently announces<br />
a pro<strong>of</strong> <strong>of</strong> <strong>the</strong> same. In 1953 B. Trakhtenbrot improved his result by showing that any<br />
class <strong>of</strong> consistent sentences that contains <strong>the</strong> sentences for which <strong>the</strong> method <strong>of</strong> finite model search<br />
succeeds, necessarily is <strong>non</strong>-recursive.<br />
In 1962 J.R. Büchi showed that <strong>the</strong> situation is <strong>the</strong> same for <strong>the</strong> class [∃ ∧ ∀∃∀] by introducing<br />
<strong>the</strong> notion now known as a conservative reduction. This is to say that any sentence <strong>of</strong> pure predicate<br />
logic can, by means <strong>of</strong> a <strong>mechanical</strong> procedure, be brought to an equiconsistent sentence <strong>of</strong> <strong>the</strong><br />
class in such a way that <strong>the</strong> method <strong>of</strong> finite model search is ’equisuccessful’, i.e., it ei<strong>the</strong>r succeeds<br />
for a given sentence in both <strong>the</strong> reduced form and <strong>the</strong> unreduced form or for nei<strong>the</strong>r.<br />
Using a technique introduced by Y. Gurevich (1976), it was possible to prove that <strong>the</strong>re exist<br />
conservative reductions for both <strong>the</strong> undecidable class <strong>of</strong> J. Surányi and <strong>the</strong> class <strong>of</strong> A. Kahr, E.<br />
Moore and H. Wang. Pro<strong>of</strong>s are available in <strong>the</strong> book [BGG97].<br />
As a consequence <strong>of</strong> J.R. Büchis result and <strong>of</strong> Y. Gurevich’s results on <strong>the</strong> earlier reduction<br />
class <strong>of</strong> J. Surányi, <strong>the</strong> disprovers <strong>of</strong> <strong>the</strong> present <strong>the</strong>sis recognise a <strong>non</strong>-recursive class <strong>of</strong> sentences.<br />
Under <strong>the</strong> fair assumption that syntactic criteria are precise enough to be recursive, we may draw<br />
<strong>the</strong> following conclusion. The disprovers <strong>of</strong> <strong>the</strong> present <strong>the</strong>sis recognise a set <strong>of</strong> sentences that is<br />
not contained in any syntactically defined decidable class.<br />
0.3.3 Algebraic logic<br />
The class <strong>of</strong> finite algebras, proposed as a substitute for infinite structures in <strong>the</strong> present <strong>the</strong>sis, is<br />
due to <strong>the</strong> present author and are a variation <strong>of</strong> polyadic algebras introduced by P. Halmos’ around<br />
1954. Polyadic algebras are in turn are a variation <strong>of</strong> cylindric algebras, introduced by A. Tarski<br />
around 1952. The study <strong>of</strong> such algebras belongs to <strong>the</strong> branch <strong>of</strong> ma<strong>the</strong>matics called algebraic<br />
logic. Algebraic logic and indeed ma<strong>the</strong>matical logic as a whole, was pioneered by G. Boole and A.<br />
DeMorgan in <strong>the</strong> mid eighteen hundreds. What A. Tarski and P. Halmos did was to equip boolean<br />
algebras with extra operators so as to describe how families <strong>of</strong> n-ary relations behave under <strong>the</strong><br />
boolean operations as well as <strong>the</strong> o<strong>the</strong>r operations needed to interpret first-order language. <strong>On</strong> A.<br />
Tarskis part, this was generalisation <strong>of</strong> work he had done on relation algebras which are models <strong>of</strong> a<br />
formalism on binary relations under boolean operations, composition and reciprocation.<br />
A. Tarskis cylindric algebras have extra-boolean operations consisting <strong>of</strong> cylindrifications and<br />
diagonals which are sufficient to interpret first-order language with equality. P. Halmos’ polyadic<br />
algebras have cylindrifications and operations for substituting variables for variables, which is sufficient<br />
for interpreting formulae <strong>of</strong> pure predicate logic. Since <strong>the</strong> present <strong>the</strong>sis is about pure<br />
predicate logic, variants <strong>of</strong> polyadic algebras are our main concern. In part 3 <strong>of</strong> <strong>the</strong> present <strong>the</strong>sis<br />
we however touch upon polyadic equality algebras which are a common expansion <strong>of</strong> both cylindric<br />
and polyadic algebras.<br />
6
For an algebra to soundly replace a finite structure in <strong>the</strong> modified method <strong>of</strong> finite model<br />
search, it has to be finite and representable. We are <strong>the</strong>refore interested in computing, or <strong>mechanical</strong>ly<br />
listing, finite and representable algebras. <strong>On</strong>e method <strong>of</strong> listing finite representable algebras<br />
is to use n-ary relations over finite sets. Algebras produced thusly do however not give rise to disprovers<br />
that recognise infinity axioms. The algebras <strong>of</strong> interest are finite algebras which give rise to<br />
disprovers that recognise only consistent sentences and some infinity axioms.<br />
If it were possible to finitely axiomatise <strong>the</strong> representable algebras, we would have a necessary<br />
and sufficient condition by which we could <strong>mechanical</strong>ly list finite representable algebras. In chapter<br />
1 we show that finite axiomatisation is not possible in any reasonable language for <strong>the</strong> variant<br />
<strong>of</strong> polyadic algebras introduced in [Rog09]. For <strong>the</strong> classical variants <strong>of</strong> cylindric and polyadic algebras<br />
<strong>the</strong> situation is as follows. J.S. Johnson in 1969 proved that representable polyadic algebras<br />
<strong>of</strong> dimension n > 2 can not be axiomatised with a finite set <strong>of</strong> first-order axioms, see [Joh69]. J.S.<br />
Johnson extended <strong>the</strong> <strong>non</strong>-axiomatisability results <strong>of</strong> J.D. Monk on cylindric algebras from 1965<br />
and 1969, see [Mon69]. In 1969 J.D. Monk also considered infinite dimensional cylindric algebras.<br />
But for <strong>the</strong> modified method <strong>of</strong> finite model search to work, <strong>the</strong> representable algebras have<br />
to be finite and <strong>the</strong>refore finite dimensional.<br />
What remains is to find sufficient and finite conditions for an algebra to be representable. The<br />
book by L. Henkin, J.D. Monk and A. Tarski, [MHT85], has several sufficient conditions for<br />
representability <strong>of</strong> finite dimensional polyadic algebras and polyadic equality algebras. These are<br />
<strong>the</strong>orem 5.4.34 on rich algebras, <strong>the</strong>orem 5.4.36 and 5.4.28 on algebras whose atoms are rectangular<br />
and <strong>the</strong>orem 5.4.39 by L. Henkin on algebras whose characteristic is finite. It is in general<br />
hard to tell from a representation <strong>the</strong>orem whe<strong>the</strong>r <strong>the</strong> finite algebras, that satisfy a condition, are<br />
representable over finite sets and <strong>the</strong>refore only <strong>of</strong> interest in regards to infinite algebras. In <strong>the</strong> case<br />
<strong>of</strong> <strong>the</strong> condition that <strong>the</strong> atoms be rectangular, <strong>the</strong>orem 5.4.35 sets up a correspondence between<br />
rectangular atoms and singleton sets which implies that finite algebras with rectangular atoms are<br />
representable over finite sets. Thus using finite algebras that satisfy <strong>the</strong> condition that atoms be<br />
rectangular gives nothing more than <strong>the</strong> good old method <strong>of</strong> finite model search.<br />
As for more recent developments we mention <strong>the</strong> following. H. Andréka et. al. [AGM + 98]<br />
introduce a generalised notion <strong>of</strong> rectangular atoms and a modified <strong>version</strong> <strong>of</strong> richness which imply<br />
representability. Also M. Ferenczi and G. Sági [FS06] survey recent developments on two approaches<br />
to representability <strong>of</strong> cylindric algebras. <strong>On</strong>e approach is on relativised set algebras, which<br />
is <strong>of</strong> interest in regards to pro<strong>of</strong>, but not to dispro<strong>of</strong> in any obvious way. The o<strong>the</strong>r approach is<br />
on proving representability using a <strong>non</strong>-standard set <strong>the</strong>ory whose consistency follows from <strong>the</strong> set<br />
<strong>the</strong>ory known as ZFC. The latter approach may be <strong>of</strong> interest in regards to dispro<strong>of</strong>, but <strong>the</strong>re are<br />
serious complexity issues in regards to <strong>the</strong> use <strong>of</strong> axiomatisations <strong>of</strong> set <strong>the</strong>ory as part <strong>of</strong> sufficient<br />
conditions for representability .<br />
Chapter 3 <strong>of</strong> <strong>the</strong> present <strong>the</strong>sis provides sufficient conditions for an algebra to be representable<br />
provided it is finite. The conditions are in <strong>the</strong> form <strong>of</strong> a finite set <strong>of</strong> first-order sentences about<br />
<strong>the</strong> atom-structure. Moreover <strong>the</strong>re are finite algebras that satisfy <strong>the</strong> conditions, which are not<br />
representable over finite sets. Disprovers based on finite algebras that satisfy <strong>the</strong> conditions, but<br />
that are not representable over finite sets, are capable <strong>of</strong> recognising infinity axioms.<br />
7
0.3.4 Automated reasoning<br />
In regards to dispro<strong>of</strong> and implementations <strong>the</strong> branch <strong>of</strong> computer science called automated reasoning<br />
is relevant. In particular <strong>the</strong> part <strong>of</strong> automated reasoning called model generation or model<br />
building. In <strong>the</strong> “Handbook <strong>of</strong> Automated Reasoning” <strong>the</strong>re is a chapter written by C.G. Fermüller,<br />
A. Leitsch, U. Hustadt and T. Tammet, [FLHT01] which contains a survey <strong>of</strong> model generation<br />
<strong>methods</strong> up until 2001. Put very briefly, <strong>the</strong>re have been two basic approaches. <strong>On</strong>e approach is<br />
that <strong>of</strong> pruning <strong>the</strong> search tree in regards to <strong>the</strong> method <strong>of</strong> finite model search. The o<strong>the</strong>r approach<br />
is that <strong>of</strong> producing a description <strong>of</strong> a model from <strong>the</strong> deduced formulae <strong>of</strong> a complete refutation<br />
procedure, that has terminated without finding an inconsistency. Although exhibiting a model implies<br />
consistency, we note that model building is more than dispro<strong>of</strong> in that a model provides more<br />
information than <strong>the</strong> confirmation we get from a procedure that merely terminates on consistency.<br />
The models resulting from refutation procedures are typically finite descriptions <strong>of</strong> infinite Herbrand<br />
models. However no examples <strong>of</strong> infinity axioms for which a model can be described is given<br />
in <strong>the</strong> model generation survey in [FLHT01].<br />
Fur<strong>the</strong>r developments are reported in <strong>the</strong> 2004 book “Automated model building” by R. Caferra,<br />
A. Leitsch and N. Peltier, which <strong>the</strong> present author has not had <strong>the</strong> opportunity to read.<br />
However in 1997 N. Peltier, [Pel97a], [Pel97b], reports on <strong>methods</strong> capable <strong>of</strong> recognising infinity<br />
axioms. These appear to be unimplemented <strong>methods</strong>. In 2000 R. Caferra and N. Peltier, [CP00]<br />
report on a method which has been implemented. The implementation is reported to have succeed<br />
on an example <strong>of</strong> an infinity axiom discovered by W. Goldfarb [Gol84]. The implementation<br />
needed human interaction to succeed, but this is perhaps because <strong>of</strong> <strong>the</strong> complexity <strong>of</strong> <strong>the</strong> particular<br />
infinity axiom chosen.<br />
In 2009 N. Peltier published on transforming first-order sentences into equiconsistent firstorder<br />
descriptions <strong>of</strong> tree-automata, toge<strong>the</strong>r with an example <strong>of</strong> an infinity axiom <strong>of</strong> pure predicate<br />
logic for which <strong>the</strong> description <strong>of</strong> said automaton has a finite model, see [Pel09]. Chapter 2 <strong>of</strong> <strong>the</strong><br />
present <strong>the</strong>sis is about a similar but distinct transformation. Nei<strong>the</strong>r <strong>of</strong> <strong>the</strong> two transformations<br />
are known to result in infinity axioms recognisable in reasonable time. We do however in chapter<br />
3 propose to use first-order descriptions <strong>of</strong> automata to compute atom-structures <strong>of</strong> polyadic set<br />
algebras. The present author has used polyadic atom-structures, when implementing some <strong>of</strong> <strong>the</strong><br />
<strong>methods</strong> <strong>of</strong> <strong>the</strong> present <strong>the</strong>sis. The implementations are available on <strong>the</strong> present authors web-page<br />
and terminate in reasonable time for several infinity axioms <strong>of</strong> pure predicate logic without human<br />
interaction.<br />
0.3.5 Personal perspective and acknowledgements<br />
For my Cand. Scient. degree I worked on strategies for <strong>the</strong> method <strong>of</strong> finite model search under<br />
<strong>the</strong> supervision <strong>of</strong> Pr<strong>of</strong>essor S.O. Aanderaa. After obtaining <strong>the</strong> Cand. Scient. degree in 1993, I<br />
attempted to enter a Dr. Scient program with S.O. Aanderaa as my supervisor.<br />
I was not accepted for such a program, but had begun work on generalising a decidability result<br />
due to S.O Aanderaa, see [Aan83]. I ended up with <strong>the</strong> generalisations <strong>of</strong> <strong>the</strong> method <strong>of</strong> finite<br />
model search described in <strong>the</strong> present <strong>the</strong>sis. S.O. Aanderaa also found a serious flaw in an early<br />
<strong>version</strong> <strong>of</strong> chapter 2 <strong>of</strong> <strong>the</strong> present <strong>the</strong>sis. Early means some time before 1999 in this setting.<br />
As I was not accepted for <strong>the</strong> Dr. Scient. program I started working full time outside <strong>of</strong><br />
academia. In <strong>the</strong> year 2000 however, I took a part time job for <strong>the</strong> reason that I wanted go for a<br />
8
Dr. Philos. degree. As life is not always predictable I have had several breaks in my project and my<br />
<strong>non</strong>-academic career. After one such break I had <strong>the</strong> opportunity to join an automated reasoning<br />
project with Pr<strong>of</strong>essor A. Waaler at <strong>the</strong> Department <strong>of</strong> Informatics in Oslo for half a year in 2007.<br />
I have also had <strong>the</strong> opportunity to follow <strong>the</strong> weekly ma<strong>the</strong>matical logic seminar at <strong>the</strong> University<br />
<strong>of</strong> Oslo, where besides Pr<strong>of</strong>essor S.O. Aanderaa, <strong>the</strong> Pr<strong>of</strong>essors D. Normann and H.R. Jervell<br />
have been able to follow my progress and to give advice. I hereby thank all <strong>of</strong> <strong>the</strong> above for <strong>the</strong> said<br />
opportunities. I also thank my personal friend Dr. F.A. Dahl for helpful comments.<br />
9
Chapter 1<br />
Turning decision procedures into disprovers<br />
11
1.1 Introduction<br />
In this setting a disprover is a procedure that terminates on input <strong>of</strong> satisfiable first-order sentences<br />
only. The purpose being to use <strong>the</strong> disprover in conjunction with a procedure that terminates on<br />
input <strong>of</strong> inconsistent first-order sentences only, in attempts at deciding whe<strong>the</strong>r given sentences are<br />
satisfiable by means <strong>of</strong> a computer.<br />
The present paper describes a method for devising disprovers, some <strong>of</strong> which have been implemented.<br />
Focus is on why disprovers deviced according to <strong>the</strong> method work in principle and to what<br />
extent. We rely on established results on <strong>the</strong> Entscheidungsproblem and use techniques common<br />
in algebraic logic for pro<strong>of</strong>s. A particular class <strong>of</strong> many-sorted polyadic set algebras suitable for <strong>the</strong><br />
task is introduced.<br />
<strong>On</strong>e needs a decision procedure for some first-order <strong>the</strong>ory to devise a disprover according to <strong>the</strong><br />
method. By means <strong>of</strong> <strong>the</strong> decision procedure a finite many-sorted polyadic set algebra is computed.<br />
Such an algebra forms <strong>the</strong> basis <strong>of</strong> one disprover, which on input <strong>of</strong> a first-order sentence works<br />
by exhaustive search for satisfying interpretations for <strong>the</strong> sentence in <strong>the</strong> algebra and in successively<br />
refined <strong>version</strong>s <strong>of</strong> <strong>the</strong> algebra.<br />
Depending on <strong>the</strong> decision procedure, resulting disprovers can be made to recognise satisfiable<br />
sentences that are not finitely satisfiable. Such sentences are called infinity axioms, and here is an<br />
example.<br />
∀x∃yRxy∧ ∀x¬Rxx∧ ∀x∀y∀zRxy ∧ Ryz → Rxz<br />
The set <strong>of</strong> sentences recognised by each disprover, turns out to be <strong>non</strong>-recursive. This is a<br />
consequence <strong>of</strong> a result <strong>of</strong> Büchi, sharpening Trakhtenbrots <strong>the</strong>orem, toge<strong>the</strong>r with <strong>the</strong> ability to<br />
recognise any finitely satisfiable sentence <strong>of</strong> a substantial fragment <strong>of</strong> first-order language [Büc62].<br />
This implies that no decidable class <strong>of</strong> sentences covers <strong>the</strong> set <strong>of</strong> sentences recognised by any one<br />
<strong>of</strong> <strong>the</strong> disprovers described here. Since finite unions <strong>of</strong> decidable sentence classes are decidable, no<br />
finite union <strong>of</strong> decidable sentence classes will cover any <strong>of</strong> <strong>the</strong> recognised sets ei<strong>the</strong>r.<br />
As any disprover can be repaired in a most ad hoc way so as to recognise any given infinity axiom,<br />
<strong>the</strong> following naturalness property is shown. The set <strong>of</strong> sentences recognised by each disprover is<br />
closed under logical equivalence, within <strong>the</strong> aforementioned fragment. This property is not shared<br />
with procedures that work by first checking whe<strong>the</strong>r <strong>the</strong> input is syntactically equal to one <strong>of</strong> a<br />
finite set <strong>of</strong> satisfiable sentences and <strong>the</strong>n go on with, say, search for satisfying interpretations over<br />
finite sets.<br />
Procedures that do search for satisfying interpretations over finite sets are said to do finite model<br />
search. Each <strong>of</strong> <strong>the</strong> disprovers described here externally share <strong>the</strong> ability to recognise any finitely<br />
satisfiable sentence <strong>of</strong> a substantial fragment <strong>of</strong> first-order language and <strong>the</strong> naturalness property<br />
with finite model search procedures. Also internally <strong>the</strong>re is some resemblance. We <strong>the</strong>refore use<br />
<strong>the</strong> term generalised finite model search, for describing how <strong>the</strong> presented disprovers work. Those<br />
<strong>of</strong> <strong>the</strong> disprovers that recognise infinity axioms represent a strict generalisation <strong>of</strong> finite model<br />
search.<br />
The model search generalisation and <strong>the</strong> results about it, work for conjunctions <strong>of</strong> purely relational<br />
prenex sentences whose length <strong>of</strong> quantifier prefix is limited by a constant. For <strong>the</strong> following<br />
reasons that constant is fixed at 3 throughout. It has made implementation feasible. To some extent<br />
it makes geometric inspection feasible. It economises notation. There is no loss <strong>of</strong> generality, in<br />
12
terms <strong>of</strong> computability, satisfiability and finite satisfiability as conjunctions <strong>of</strong> 3-variable prenex sentences<br />
are known to form a conservative reduction class. This is to say; <strong>the</strong> existence <strong>of</strong> algorithms<br />
are known that transform any first-order sentence to an equi-satisfiable and equi-finitely-satisfiable<br />
sentence <strong>of</strong> <strong>the</strong> required form. The earliest to show <strong>the</strong> existence <strong>of</strong> such a transformation was<br />
Büchi [Büc62], and several are known. These transformations are called conservative reductions,<br />
where conservativeness relates to finite satisfiability. A well known example <strong>of</strong> a conservative reduction<br />
is skolemisation, and it is conservative because a model for <strong>the</strong> transformed sentence can<br />
be defined by expanding a model for <strong>the</strong> original one. A class <strong>of</strong> 3-variable prenex sentences that<br />
form a reduction class on <strong>the</strong>ir own is <strong>the</strong> ∀∃∀ class <strong>of</strong> Kahr Moore and Wang [KMW62]. This<br />
class turned out to be conservative by a result <strong>of</strong> Gurevich and Koriakov[GK72] utilising a result <strong>of</strong><br />
Berger [Ber66]. Pro<strong>of</strong>s <strong>of</strong> this and related results are accessible in <strong>the</strong> book [BGG97].<br />
That which is needed about <strong>the</strong> well known <strong>the</strong>ory <strong>of</strong> polyadic set algebras, for statement and<br />
pro<strong>of</strong> <strong>of</strong> <strong>the</strong> results <strong>of</strong> <strong>the</strong> present paper is given in section 1.2. In Section 1.3 a particular class <strong>of</strong><br />
many-sorted polyadic set algebras, that is believed to be new, is introduced. They are called directed<br />
many-sorted polyadic set algebras <strong>of</strong> dimension 3 (dMsPs 3 ). A downward Löwenheim-Skolem<br />
type <strong>of</strong> result is shown, in that any satisfiable sentence <strong>of</strong> <strong>the</strong> aforementioned form is satisfiable in<br />
a finite such algebra, even if <strong>the</strong> sentence is an infinity axiom. In Section 1.4 a construction <strong>of</strong><br />
polyadic set algebras, many-sorted or not, is defined. It is for building more refined algebras from<br />
a given one. In particular one can build algebras refined enough to allow a satisfying interpretation<br />
for any finitely satisfiable (in <strong>the</strong> usual sense) sentence.<br />
1.1.1 Sets and mappings<br />
The numeral 3 denotes {0, 1, 2} and ω <strong>the</strong> natural numbers. The set <strong>of</strong> mappings from 3 to 2 is<br />
written both as 2 3 and as 3 → 2. Formally a mapping f in 3 → 2 is <strong>the</strong> set <strong>of</strong> pairs {(u, f(u)) :<br />
u ∈ 3}. The f-image <strong>of</strong> 2 is <strong>the</strong> set {f(u) : u ∈ 2}. The restriction <strong>of</strong> f to <strong>the</strong> set 2 is <strong>the</strong> set <strong>of</strong><br />
pairs {(u, f(u)) : u ∈ 2}. If f and g are mappings <strong>the</strong>n f ◦ g denotes <strong>the</strong>ir composition, applying<br />
g first <strong>the</strong>n f. If f ⊆ g <strong>the</strong>n f is said to be a restriction <strong>of</strong> g and g an extension <strong>of</strong> f. Mappings<br />
in 3 → 3 are written as triples <strong>of</strong> elements <strong>of</strong> 3, where <strong>the</strong> value <strong>of</strong> 0 is leftmost. So <strong>the</strong> identity<br />
mapping for instance is written 012 and <strong>the</strong> mapping that subtracts one modulo 3 is written 201.<br />
1.1.2 First-order formulae<br />
First-order formulae in <strong>the</strong> present paper are taken from pure relation calculus with three variables.<br />
Pure here means being without distinguished equality-, function- and constant-symbols. Relationsymbols<br />
are fur<strong>the</strong>r assumed to be <strong>of</strong> arity 3. See chapter 6.5 <strong>of</strong> [DG79] for how to conservatively<br />
reduce ∀∃∀-sentences with relation-symbols <strong>of</strong> varying arities to those <strong>of</strong> arity 2. For arities lower<br />
than 3 one can replace each occurrence <strong>of</strong> Rx for instance, with Rxxx and Rxy with Rxyy. The<br />
obtained language fragment is denoted by L 3 .<br />
L 3 is assumed to have a countable unlimited supply <strong>of</strong> distinct ternary relation-symbols. Ternary<br />
relation-symbols are also <strong>the</strong> only kind <strong>of</strong> symbol taken from an infinite set. Rxy and xRy are used<br />
as abbreviations for Rxyy. The notation L ′ 3 will be used for L 3 fragments that have a limited finite<br />
number <strong>of</strong> relation-symbols. A formula is a sentence if every variable is bound by a quantifier.<br />
Theories are sets <strong>of</strong> L ′ 3 sentences, and <strong>the</strong> symbol Γ is used for <strong>the</strong>ories. A <strong>the</strong>ory Γ is complete if<br />
for every sentence φ ∈ L ′ 3 it is <strong>the</strong> case that φ ∈ Γ or ¬φ ∈ Γ. If it is <strong>the</strong> case that for all φ ∈ L ′ 3<br />
13
ei<strong>the</strong>r φ ∈ Γ or ¬φ ∈ Γ, and not both, <strong>the</strong>n Γ is consistent and complete. The set <strong>of</strong> L ′ 3 sentences<br />
true under a fixed interpretation is a complete and consistent <strong>the</strong>ory. A complete <strong>the</strong>ory is said to<br />
be decidable if it is recursive. Two L ′ 3 formulae φ and ψ are said to be Γ-equivalent if <strong>the</strong> sentence<br />
∀x∀y∀z(φ ↔ ψ) is an element <strong>of</strong> Γ. Note that <strong>the</strong> variables x, y and z each may occur free, bound<br />
<strong>of</strong> both in φ and ψ.<br />
For each relation-symbol R and each mapping σ in 3 → 3, Rσ is an atomic formula. An<br />
example <strong>of</strong> an atomic formula is R012 which is, or denotes, a formula more commonly written as<br />
Rxyz. It is to be understood that <strong>the</strong> variable x has index 0, y has index 1 and z index 2. The<br />
word atom is used for minimal <strong>non</strong>zero elements <strong>of</strong> boolean algebras and not for atomic formulae.<br />
The literals are <strong>the</strong> set <strong>of</strong> formulae generated by <strong>the</strong> atomic formulae under negation (¬). The open<br />
formulae are <strong>the</strong> set <strong>of</strong> formulae generated by <strong>the</strong> atomic formulae under negation and disjunction<br />
(∨). L 3 is <strong>the</strong> set <strong>of</strong> formulae generated by <strong>the</strong> atomic formulae under negation, disjunction and<br />
<strong>the</strong> existential quantifiers (∃ 0 , ∃ 1 , ∃ 2 ). Again ∃ 0 φ is, or denotes ∃xφ, etc. The following are used<br />
as abbreviations; ⊥ for some contradiction depending on <strong>the</strong> language fragment in question, ⊤ for<br />
¬(⊥), φ ∧ ψ for ¬(¬φ ∨ ¬ψ), φ ↔ ψ for (¬φ ∨ ψ) ∧ (¬ψ ∨ φ) and ∀ i φ for ¬∃ i (¬φ). A formula<br />
is said to be a prenex sentence if it is <strong>of</strong> <strong>the</strong> form Q 0 Q 1 Q 2 φ where φ is open and each Q i is one<br />
<strong>of</strong> ∃ i or ∀ i . Since <strong>the</strong> languages considered here are equality-free <strong>the</strong> upward Löwenheim-Skolem<br />
<strong>the</strong>orem also holds in <strong>the</strong> finite. This is to say that if φ has a model <strong>of</strong> cardinality n ∈ ω <strong>the</strong>n φ<br />
also has a model <strong>of</strong> cardinality n + 1 [Men87](p.73).<br />
1.1.3 Algebras, homomorphisms, and preservation<br />
In <strong>the</strong> present paper algebras and homomorphisms <strong>of</strong> several kinds are used. Here we summarise<br />
some properties <strong>the</strong> kinds have in common. An algebra is a set toge<strong>the</strong>r with a family <strong>of</strong> operations<br />
on that set. The given set is called <strong>the</strong> carrier-set. Operations are mappings under which <strong>the</strong><br />
algebra is closed, which is to say that <strong>the</strong> image <strong>of</strong> <strong>the</strong> carrier-set under an operation is a subset <strong>of</strong><br />
<strong>the</strong> carrier-set. Algebras have signatures, <strong>the</strong>se provide information on <strong>the</strong> arities <strong>of</strong> <strong>the</strong> operations,<br />
moreover signatures determine a notion <strong>of</strong> correspondence between operations in pairs <strong>of</strong> algebras<br />
<strong>of</strong> similar kind.<br />
Definition Let A and B be algebras, let f be an n-ary operation <strong>of</strong> A and g <strong>the</strong> corresponding<br />
n-ary operation <strong>of</strong> B. A mapping h from a subset X <strong>of</strong> <strong>the</strong> carrier-set <strong>of</strong> A to <strong>the</strong> carrier-set <strong>of</strong> B,<br />
is said to preserve f if for every x 0 , . . . , x n ∈ X it is <strong>the</strong> case that<br />
f(x 0 , . . . , x n−1 ) = x n implies g(h(x 0 ), . . . , h(x n−1 )) = h(x n ).<br />
A homomorphism from A to B is a mapping from <strong>the</strong> carrier-set <strong>of</strong> A to <strong>the</strong> carrier-set <strong>of</strong> B that<br />
preserves <strong>the</strong> operations <strong>of</strong> A. Embeddings are homomorphisms that are one to one. Isomorphisms<br />
are embeddings that are onto.<br />
The following properties follow by <strong>the</strong> definition above. Homomorphisms are closed under<br />
composition which is to say that if h 0 and h 1 are homomorphisms <strong>the</strong>n h 1 ◦ h 0 is also a homomorphism.<br />
If h is a homomorphism from A and if X is a subset <strong>of</strong> <strong>the</strong> carrier-set <strong>of</strong> A, <strong>the</strong>n <strong>the</strong><br />
restriction <strong>of</strong> h to X also preserves <strong>the</strong> operations <strong>of</strong> A. If X is closed under <strong>the</strong> operations <strong>of</strong> A<br />
<strong>the</strong>n <strong>the</strong> restriction <strong>of</strong> h to X is a homomorphism from X.<br />
14
1.1.4 Algebras <strong>of</strong> boolean signature<br />
Algebras <strong>of</strong> boolean signature are algebras with signature B = (B, ∨, ¬, ⊥). ∨, ¬, ⊥ are called join,<br />
negation, and bottom respectively. An algebra <strong>of</strong> boolean signature A is a B-sub-algebra if it is <strong>of</strong><br />
<strong>the</strong> form (X, ∨|, ¬|, ⊥|) where X is a subset <strong>of</strong> B and ∨|, ¬|, ⊥| are <strong>the</strong> restrictions <strong>of</strong> ∨, ¬, ⊥ to<br />
X, moreover X must be such that ∨| ∈ X × X → X and ¬|, ⊥| ∈ X → X. The last requirements<br />
here are to exclude X’s that are not closed under ∨|, ¬|, ⊥|. A boolean homomorphism is<br />
a mapping from an algebra <strong>of</strong> boolean signature to an algebra <strong>of</strong> boolean signature that preserves<br />
∨, ¬, ⊥.<br />
The algebras that are <strong>of</strong> interest in <strong>the</strong> present paper are each expansions <strong>of</strong> algebras <strong>of</strong> boolean<br />
signature. This is to say that <strong>the</strong>y are algebras <strong>of</strong> boolean signature with extra operations on <strong>the</strong>m.<br />
Those <strong>of</strong> <strong>the</strong> algebras that are expansions <strong>of</strong> boolean algebras turn out to be very hard to axiomatise.<br />
We <strong>the</strong>refore dispose <strong>of</strong> with <strong>the</strong> usual axioms all toge<strong>the</strong>r. By <strong>the</strong> representation <strong>the</strong>orem <strong>of</strong> Stone<br />
<strong>the</strong> following is way <strong>of</strong> defining boolean algebras.<br />
Definition An algebra <strong>of</strong> boolean signature is a boolean algebra if it is embeddable into<br />
(P(U), ∪, −, ∅), where U is some set,P(U) is <strong>the</strong> power-set <strong>of</strong> U, ∪ is union, − is (absolute)<br />
complement and ∅ is <strong>the</strong> empty set.<br />
A boolean algebra is finitely generated if it is generated by a finite subset <strong>of</strong> <strong>the</strong> carrier-set under <strong>the</strong><br />
boolean operations. The following property <strong>of</strong> boolean algebras is central in <strong>the</strong> present paper.<br />
Lemma 1.1.1 Finitely generated boolean algebras are finite.<br />
Now if U is infinite <strong>the</strong>n a finitely generated sub algebra <strong>of</strong> (P(U), ∪, −, ∅) is finite but its<br />
elements may be infinite sets. As we seek to make computers work with boolean algebras <strong>the</strong><br />
following is useful.<br />
Lemma 1.1.2 Let A = ({⊥, ⊤}, ∨, ¬, ⊥) be <strong>the</strong> usual two element boolean algebra. Every finite<br />
boolean algebra is isomorphic to an algebra with carrier-set n → {⊥, ⊤} where n ∈ ω and where <strong>the</strong><br />
operations are defined component-wise.<br />
To say that <strong>the</strong> operations on a set <strong>of</strong> mappings are defined component-wise is to say that<br />
<strong>the</strong> join <strong>of</strong> f and g for instance, which for now we denote by (f∨g), is <strong>the</strong> uniquely determined<br />
operation with <strong>the</strong> following property. For each i ∈ n it is <strong>the</strong> case that (f∨g)(i) = f(i) ∨ g(i).<br />
For <strong>the</strong> particular case, where <strong>the</strong> mappings are to a two element set, <strong>the</strong> term bit-wise is sometimes<br />
used.<br />
1.2 Algebras <strong>of</strong> polyadic signature<br />
Inspired by Tarskis cylindric algebras, Halmos introduced and studied polyadic algebras. Some<br />
papers on <strong>the</strong>se studies are collected in [Hal62]. A particular class <strong>of</strong> polyadic algebras are polyadic<br />
set algebras <strong>of</strong> dimension 3, denoted Ps 3 (see definition below). In <strong>the</strong> present paper <strong>the</strong>se serve<br />
as abstract or generalised models for L 3 sentences. The well known Lindenbaum algebra, L ′ 3/Γ,<br />
<strong>of</strong> a consistent and complete L ′ 3 <strong>the</strong>ory Γ can be equipped with extra operations related to variable<br />
substitution and existential quantifiers to make it a Ps 3 . It is possible to define interpretation and<br />
15
satisfaction in Ps 3 ’s in such a way that finding a satisfying interpretation for an L 3 -sentence implies<br />
satisfiability in a usual sense.<br />
Observe that <strong>the</strong> number <strong>of</strong> equivalence classes <strong>of</strong> L ′ 3/Γ may be finite also when Γ does not have<br />
finite models. The <strong>the</strong>ory <strong>of</strong> a dense order without endpoints is an example <strong>of</strong> such a Γ. To see this,<br />
note that this Γ can be stated in a language, L ′ 3, with one relation-symbol (
Definition Let A = (B, r, p, s, c 0 , c 1 , c 2 ) and C be algebras <strong>of</strong> polyadic signature. h is a polyadic<br />
homomorphism from A to C if h is a boolean homomorphism from B to <strong>the</strong> boolean reduct <strong>of</strong> C<br />
that preserves each <strong>of</strong> r, p, s, c 0 , c 1 , c 2 .<br />
1.2.3 L 3 as an algebra <strong>of</strong> polyadic signature<br />
Here <strong>the</strong> language L 3 , seen as an algebra with operations for boolean connectives, is expanded to<br />
make an algebra <strong>of</strong> polyadic signature. Interpretations <strong>of</strong> L 3 turn out to be <strong>the</strong> polyadic homomorphisms<br />
from this algebra to o<strong>the</strong>r algebras <strong>of</strong> polyadic signature. An interpretation is satisfying for<br />
a sentence if <strong>the</strong> sentence is interpreted as ⊤. The following defines some syntactic operations on<br />
L 3 . Semantics for <strong>the</strong>se operations are to be found fur<strong>the</strong>r on.<br />
Definition for i ∈ 3 and σ ∈ 3 → 3 define r ∗ , p ∗ , s ∗ ∈ L 3 → L 3 by<br />
r ∗ (Rσ) = R(r ◦ σ)<br />
p ∗ (Rσ) = R(p ◦ σ)<br />
s ∗ (Rσ) = R(s ◦ σ)<br />
r ∗ (¬φ) = ¬r ∗ (φ)<br />
p ∗ (¬φ) = ¬p ∗ (φ)<br />
s ∗ (¬φ) = ¬s ∗ (φ)<br />
r ∗ (φ ∨ ψ) = r ∗ (φ) ∨ r ∗ (ψ)<br />
p ∗ (φ ∨ ψ) = p ∗ (φ) ∨ p ∗ (ψ)<br />
s ∗ (φ ∨ ψ) = s ∗ (φ) ∨ s ∗ (ψ)<br />
r ∗ (∃ i φ) = ∃ r(i) r ∗ (φ)<br />
p ∗ (∃ i φ) = ∃ p(i) p ∗ (φ)<br />
s ∗ (∃ 0 φ) = ∃ 0 φ<br />
s ∗ (∃ 1 φ) = ∃ 0 p ∗ (φ)<br />
s ∗ (∃ 2 φ) = ∃ 2 s ∗ (φ)<br />
Note that r ∗ , p ∗ and s ∗ may be “moved inwards” relative to each <strong>of</strong> <strong>the</strong> connectives <strong>of</strong> L 3 . So L 3 , <strong>the</strong><br />
open and <strong>the</strong> atomic formulae are each closed under <strong>the</strong>se operations. They don’t quite commute<br />
as s ∗ involves renaming only free occurrences <strong>of</strong> a certain variable, while r ∗ and p ∗ rename all<br />
occurrences <strong>of</strong> involved variables. Compare this definition to <strong>the</strong> axiomatisation <strong>of</strong> quasi polyadic<br />
algebras in [AGM + 98] Section 4. We now expand L 3 to make it an algebra <strong>of</strong> polyadic signature.<br />
Definition L 3 = (L 3 , ∨, ¬, ⊥, r ∗ , p ∗ , s ∗ , ∃ 0 , ∃ 1 , ∃ 2 )<br />
17
1.2.4 Polyadic set algebras <strong>of</strong> ternary relations<br />
We have seen how to turn a fragment <strong>of</strong> first-order language into an algebra <strong>of</strong> polyadic signature.<br />
Here <strong>the</strong> ternary relations over given sets are are turned into algebras <strong>of</strong> polyadic signature. A<br />
mapping from <strong>the</strong> variables <strong>of</strong> L 3 to some set U is called a sequence and an interpretation is an<br />
assignment <strong>of</strong> each formula <strong>of</strong> L 3 to a set <strong>of</strong> sequences. This form <strong>of</strong> interpretation is due to<br />
Tarski. In contrast to o<strong>the</strong>r, quite viable, forms <strong>of</strong> interpretation tarskian interpretation makes<br />
Γ-equivalence coincide with equality <strong>of</strong> interpretations in given models for Γ.<br />
Since we use 3 variables, a set <strong>of</strong> sequences is a ternary relation. We refer to <strong>the</strong> relation assigned<br />
to φ by an interpretation as <strong>the</strong> relation defined by φ. By <strong>the</strong> classical Löwenheim-Skolem-Tarski<br />
<strong>the</strong>orems and <strong>the</strong> absence <strong>of</strong> a symbol for equality, satisfiable sentences each have a model over<br />
<strong>the</strong> reals, R. In such models L 3 -formulae define subsets <strong>of</strong> R 3 . This gives rise to a geometric<br />
interpretation <strong>of</strong> elements <strong>of</strong> Ps 3 ’s that will be appealed to later, instead <strong>of</strong> very detailed symbolic<br />
pro<strong>of</strong>, see fig.1.1 and fig. 1.2.<br />
Definition Let U be some set. The full Ps 3 over U is an algebra<br />
(P(U 3 ), ∪, −, ∅, r U , p U , s U , c U 0 , c U 1 , c U 2 ) where (P(U 3 ), ∪, −, ∅) is <strong>the</strong> boolean algebra <strong>of</strong> sets <strong>of</strong><br />
triples <strong>of</strong> elements <strong>of</strong> U. Operations related to variable substitutions are for V ⊆ U<br />
r U (V ) = {u ∈ U 3 : u ◦ r ∈ V } = {abc ∈ U 3 : cab ∈ V }<br />
p U (V ) = {u ∈ U 3 : u ◦ p ∈ V } = {abc ∈ U 3 : bac ∈ V }<br />
s U (V ) = {u ∈ U 3 : u ◦ s ∈ V } = {abc ∈ U 3 : bbc ∈ V }<br />
Operations for <strong>the</strong> existential quantifiers are<br />
c U 0 (V ) = {abc ∈ U 3 : <strong>the</strong>re is a u ∈ U such that ubc ∈ V }<br />
c U 1 (V ) = {abc ∈ U 3 : <strong>the</strong>re is a u ∈ U such that auc ∈ V }<br />
c U 2 (V ) = {abc ∈ U 3 : <strong>the</strong>re is a u ∈ U such that abu ∈ V }<br />
An algebra is said to be a full Ps 3 if it is <strong>the</strong> full Ps 3 over some set U.<br />
By abuse <strong>of</strong> notation let P(U 3 ) denote <strong>the</strong> full Ps 3 over U.<br />
Figure 1.1: Cylindrification along x-axis, z-axis is orthogonal to <strong>the</strong> present paper and <strong>the</strong> dotted<br />
line marks <strong>the</strong> xy-diagonal plane which is also orthogonal to <strong>the</strong> present paper.<br />
18
Figure 1.2: Substitution, cylindrifies that which meets <strong>the</strong> xy-diagonal plane. Rotation and permutation<br />
amount to suitably renaming <strong>the</strong> axis, <strong>the</strong>n rotating or mirroring <strong>the</strong> figures respectively.<br />
Definition A Ps 3 is an algebra <strong>of</strong> polyadic signature that is embeddable into a full Ps 3 .<br />
The next property relates <strong>the</strong> defining equations for r ∗ , s ∗ , p ∗ to those <strong>of</strong> r U , s U , p U . For this<br />
<strong>the</strong> author finds geometric interpretation helpful, see fig. 1.1 and fig. 1.2.<br />
Lemma 1.2.2 For each i ∈ 3 and V, W ⊆ U<br />
r U (¬V ) = ¬(r U (V )).<br />
p U (¬V ) = ¬(p U (V )).<br />
s U (¬V ) = ¬(s U (V )).<br />
r U (V ∪ W ) = r U (V ) ∪ r U (W ).<br />
p U (V ∪ W ) = p U (V ) ∪ p U (W ).<br />
s U (V ∪ W ) = s U (V ) ∪ s U (W ).<br />
r U (c U i (V )) = c U r(i) V .<br />
p U (c U i (V )) = p U p(i) V .<br />
s U (c U 0 (V )) = c U 0 (V ).<br />
s U (c U 1 (V )) = c U 0 (p U (V ).<br />
s U (c U 2 (V )) = c U 2 (s U (V ).<br />
1.2.5 Interpretation<br />
Here we go some length to see that polyadic homomorphisms from L 3 to Ps 3 ’s are essentially<br />
<strong>the</strong> same as interpretations <strong>of</strong> L 3 formulae as ternary relations over some set. The statement is<br />
broken down into a few extension properties, which are <strong>of</strong> use later in <strong>the</strong> present paper. Extension<br />
properties are also known as universal properties.<br />
The first extension property says that polyadic homomorphisms from L 3 to Ps 3 ’s are like<br />
interpretations in that if we interpret <strong>the</strong> relation-symbols <strong>of</strong> L 3 <strong>the</strong>n an interpretation <strong>of</strong> each<br />
atomic formula <strong>of</strong> L 3 is determined. We use <strong>the</strong> notation {R, ...} × {012} for <strong>the</strong> set <strong>of</strong> atomic<br />
formulae <strong>of</strong> L 3 in which all variables <strong>of</strong> L 3 occur and in which <strong>the</strong>y occur in <strong>the</strong> order specified by<br />
012. The following is a consequence <strong>of</strong> lemma 1.2.1.<br />
19
Lemma 1.2.3 If {R, ...} are <strong>the</strong> relation-symbols <strong>of</strong> L 3 and if A is a Ps 3 , <strong>the</strong>n each mapping f ∈<br />
{R, ...} × {012} → A extends to a unique r ∗ , s ∗ , p ∗ -preserving mapping from <strong>the</strong> atomic formulae <strong>of</strong><br />
L 3 to A.<br />
The second extension property has as a consequence that polyadic homomorphisms are like<br />
interpretations in that if an interpretation for each atomic formula <strong>of</strong> L 3 is determined <strong>the</strong>n an<br />
interpretation for each formula <strong>of</strong> L 3 is uniquely determined.<br />
Definition A set <strong>of</strong> formulae X ⊆ L 3 is said to be sub-formula-closed if each atomic formula <strong>of</strong><br />
L 3 is an element <strong>of</strong> X and if for each φ ∈ X it is <strong>the</strong> case that every sub-formula <strong>of</strong> φ is also in X.<br />
Lemma 1.2.4 Let X ⊆ Y ⊆ L 3 be such that X is sub-formula-closed. Let A be a Ps 3 . Then each<br />
∨, ¬, ⊥, ∃ 0 , ∃ 1 , ∃ 2 -preserving mapping f from X to A extends to a unique<br />
∨, ¬, ⊥, ∃ 0 , ∃ 1 , ∃ 2 -preserving mapping f ∗ from Y to A.<br />
The above two extension properties can now be combined with lemma 1.2.2 which provides semantics<br />
for r ∗ , s ∗ , p ∗ .<br />
Lemma 1.2.5 If {R, ...} are <strong>the</strong> relation-symbols <strong>of</strong> L 3 and if A is a Ps 3 , <strong>the</strong>n each mapping f ∈<br />
{R 0 , ...} × {012} → A extends to a unique polyadic homomorphism f ∗ from L 3 to A<br />
The following can now be seen by comparing <strong>the</strong> uniquely determined homomorphisms with<br />
interpretation and satisfaction as found in for example [Men87].<br />
Proposition 1.2.6 A tarskian interpretation <strong>of</strong> L 3 as relations over some set U, is a polyadic homomorphism<br />
from L 3 to <strong>the</strong> full Ps 3 over U. Also each homomorphism from L 3 to some full Ps 3 is a<br />
tarskian interpretation <strong>of</strong> L 3 -formulae.<br />
1.2.6 Additivity<br />
The following lemma is not central for understanding why <strong>the</strong> presented disprovers do what <strong>the</strong>y<br />
are supposed to do. It does however contribute to actually fitting <strong>the</strong> disprovers into a computers<br />
memory as finite Ps 3 ’s mostly are <strong>of</strong> considerable size. Additivity later turns out to provide a way<br />
<strong>of</strong> representing finite Ps 3 ’s in quite a compact form.<br />
The carrier-set <strong>of</strong> a Ps 3 toge<strong>the</strong>r with <strong>the</strong> boolean operations ∨, ¬, ⊥ are by definition a<br />
boolean algebra, called <strong>the</strong> boolean reduct <strong>of</strong> <strong>the</strong> Ps 3 .<br />
Definition An operation f on a boolean algebra is called additive if f(⊥) = ⊥ and f(x ∨ y) =<br />
f(x) ∨ f(y).<br />
The following property can to some extent be inspected geometrically, see fig. 1.1 and fig. 1.2.<br />
Lemma 1.2.7 The boolean reduct <strong>of</strong> a Ps 3 is a boolean algebra and each <strong>of</strong> r, p, s, c 0 , c 1 , c 2 is additive.<br />
20
1.2.7 The Lindenbaum algebra <strong>of</strong> a <strong>the</strong>ory Γ<br />
A full Ps 3 over U is finite if U is finite, and exhaustive search for satisfying interpretations can<br />
readily be done. If U is infinite however, <strong>the</strong> full Ps 3 over U is uncountable and unsuitable for<br />
exhaustive search as such. It turns out that <strong>the</strong> well known Lindenbaum algebra <strong>of</strong> a <strong>the</strong>ory Γ<br />
sometimes provides a way <strong>of</strong> computing and representing finite Ps 3 ’s that can not be embedded<br />
into a full Ps 3 over any finite U. In <strong>the</strong>se algebras, satisfying interpretations for infinity axioms<br />
can be found. The question <strong>of</strong> whe<strong>the</strong>r a satisfying interpretation in a finite Ps 3 exists, can even<br />
be decided by exhaustive search.<br />
We identify <strong>the</strong> Lindenbaum algebra <strong>of</strong> a <strong>the</strong>ory Γ with a particular way <strong>of</strong> representing it. This<br />
representation is chosen so as to be in <strong>the</strong> form <strong>of</strong> a finite set <strong>of</strong> finite objects (formulae), toge<strong>the</strong>r<br />
with a finite set <strong>of</strong> tables (finite sets <strong>of</strong> pairs or triples <strong>of</strong> formulae), when <strong>the</strong> Lindenbaum algebra<br />
has a finite number <strong>of</strong> equivalence classes.<br />
Throughout this paper; fixate a well-ordering ≼ on L 3 based on a well-ordering <strong>of</strong> <strong>the</strong> symbols<br />
<strong>of</strong> L 3 . We assume that <strong>the</strong> ordering, ≼, is such that if φ is a sub-formula <strong>of</strong> ψ <strong>the</strong>n φ ≼ ψ. We also<br />
assume that (L 3 , ≼) is order-isomorphic to (ω, ≤). Well-orderedness ensures that <strong>the</strong> following<br />
defines a mapping.<br />
Definition µ Γ ∈ L ′ 3 → L ′ 3 is defined by µ Γ (φ) = ψ where ψ is <strong>the</strong> ≼-minimal L ′ 3 formula such<br />
that φ and ψ are Γ-equivalent.<br />
As long as Γ is given we write [φ] in stead <strong>of</strong> µ Γ (φ). We even refer to [φ] as <strong>the</strong> equivalence class<br />
<strong>of</strong> φ, and say that ψ is in [φ] when φ and ψ are Γ-equivalent.<br />
Definition L ′ 3/Γ = ([L ′ 3], ∨ ′ , ¬ ′ , ⊥ ′ , r ′ , p ′ , s ′ , ∃ ′ 0, ∃ ′ 1, ∃ ′ 2), where for i ∈ 3<br />
[L ′ 3] is <strong>the</strong> µ Γ -image <strong>of</strong> L ′ 3<br />
[φ] ∨ ′ [ψ] = [φ ∨ ψ]<br />
¬ ′ [φ] = [¬φ]<br />
r ′ [φ] = [r ∗ φ]<br />
p ′ [φ] = [p ∗ φ]<br />
s ′ [φ] = [s ∗ φ]<br />
∃ ′ i[φ] = [∃ i φ]<br />
Note that by this definition, µ Γ is a polyadic homomorphism from L ′ 3 to L ′ 3/Γ. Moreover it is<br />
a satisfying interpretation for every logical consequence <strong>of</strong> Γ.<br />
The following property <strong>of</strong> consistent and complete <strong>the</strong>ories follows by <strong>the</strong> completeness <strong>the</strong>orem<br />
<strong>of</strong> Gödel. Consistency provides a model with universe U, interpretation <strong>of</strong> L ′ 3-formulae as<br />
relations over U, provides a homomorphism h from L ′ 3 to <strong>the</strong> full Ps 3 over U. The restriction <strong>of</strong><br />
h to <strong>the</strong> carrier-set <strong>of</strong> L ′ 3/Γ is a homomorphism from L ′ 3/Γ to <strong>the</strong> full Ps 3 over U. Completeness<br />
ensures that this homomorphism is an embedding.<br />
Proposition 1.2.8 If Γ is consistent and complete <strong>the</strong>n L ′ 3/Γ is a Ps 3<br />
21
1.2.8 Exhaustive search for satisfying interpretations in a Ps 3<br />
We finish this section with a summary <strong>of</strong> <strong>the</strong> results so far and how <strong>the</strong>y may be applied in disproving.<br />
As noted; if L ′ 3 is a language with a finite number <strong>of</strong> relation-symbols and if Γ is a complete<br />
and consistent <strong>the</strong>ory that has quantifier-elimination within L ′ 3, <strong>the</strong>n L ′ 3/Γ has a finite number <strong>of</strong><br />
equivalence classes. By <strong>the</strong> particular representation chosen, this L ′ 3/Γ is a finite object. Given a<br />
decision procedure for Γ, L ′ 3/Γ can be computed and put in <strong>the</strong> form <strong>of</strong> tables for <strong>the</strong> operations<br />
on <strong>the</strong> carrier-set <strong>of</strong> L ′ 3/Γ. With such tables one can by exhaustive search decide whe<strong>the</strong>r <strong>the</strong>re<br />
exists a satisfying interpretation from L 3 to L ′ 3/Γ for any given L 3 sentence φ. This is because one<br />
only needs to enumerate mappings f in {R 0 , ...R m } × 012 → L ′ 3/Γ, where {R 0 , ...R m } are <strong>the</strong><br />
relation-symbols <strong>of</strong> φ, to decide whe<strong>the</strong>r <strong>the</strong>re exists an interpretation f ∗ from L 3 to L ′ 3/Γ that is<br />
satisfying for φ.<br />
If a satisfying interpretation f ∗ for φ in L ′ 3/Γ is found <strong>the</strong>n a satisfying interpretation for φ in<br />
<strong>the</strong> tarskian sense is obtained by h ◦ f ∗ . Here h is an embedding from L ′ 3/Γ to P(U 3 ) provided<br />
by proposition 1.2.8. Since polyadic homomorphisms and interpretations are <strong>the</strong> same, h ◦ f ∗ is<br />
an interpretation as polyadic homomorphisms are closed under composition. It is satisfying for φ<br />
as h, like any o<strong>the</strong>r polyadic homomorphism, preserves ⊤.<br />
1.3 Algebras <strong>of</strong> directed many-sorted polyadic signature<br />
We have seen how to do exhaustive search for satisfying interpretations in an L ′ 3/Γ when Γ has<br />
quantifier-elimination within L ′ 3. Theories do not in general have quantifier-elimination like this<br />
however. By an argument found at <strong>the</strong> end <strong>of</strong> section 1.5, <strong>the</strong> Lindenbaum algebra <strong>of</strong> <strong>the</strong> <strong>the</strong>ory<br />
<strong>of</strong> <strong>the</strong> usual strict order (
In <strong>the</strong> following definition boolean homomorphisms are used. Note that our definition <strong>of</strong> a<br />
boolean homomorphisms does not require that <strong>the</strong> domain is a boolean algebra, only an algebra <strong>of</strong><br />
boolean signature.<br />
Definition Let A = (B 3 , B 2 , B 1 , B 0 , r, p, s, c 0 , c 1 , c 2 ) be as above and let C be ei<strong>the</strong>r an algebra <strong>of</strong><br />
polyadic signature or an algebra <strong>of</strong> directed many-sorted polyadic signature. A directed many-sorted<br />
polyadic homomorphism from A to C is a quadruple <strong>of</strong> boolean homomorphisms<br />
h 3 from B 3 to C<br />
h 2 from B 2 to C<br />
h 1 from B 1 to C<br />
h 0 from B 0 to C<br />
such that each <strong>of</strong> r, s, p, c 0 , c 1 , c 2 are preserved.<br />
We define what it means for a many-sorted algebra to be a sub-algebra <strong>of</strong> a one-sorted algebra.<br />
Definition Let C = (B, r, s, p, c 0 , c 1 , c 2 ) be an algebra <strong>of</strong> polyadic signature. A C-sub-algebra <strong>of</strong><br />
directed many-sorted polyadic signature is an A = (B 3 , B 2 , B 1 , B 0 , r|, s|, p|, c 0 |, c 1 |, c 2 |) with sorts<br />
B 3 , B 2 , B 1 , B 0 where<br />
B 3 , B 2 , B 1 , B 0 are B-sub-algebras <strong>of</strong> boolean signature,<br />
r|, s|, p| are <strong>the</strong> restrictions <strong>of</strong> r, s, p to B 3 ,<br />
c 0 |, c 1 |, c 2 | are <strong>the</strong> restrictions <strong>of</strong> c 0 , c 1 , c 2 to B 1 , B 2 , B 3 respectively,<br />
r|, p|, s| ∈ B 3 → B 3 ,<br />
c 2 | ∈ B 3 → B 2 ,<br />
c 1 | ∈ B 2 → B 1 ,<br />
c 0 | ∈ B 1 → B 0 .<br />
1.3.2 A conservative reduction class with sub-formulae as an algebra<br />
Here <strong>the</strong> set <strong>of</strong> sub-formulae <strong>of</strong> every sentence <strong>of</strong> a conservative reduction class is expanded to make<br />
an algebra <strong>of</strong> directed many-sorted polyadic signature. What sort a formula is, depends on which<br />
quantifiers occur in it.<br />
Definition The four algebras L 33 , L 32 , L 31 , L 30 <strong>of</strong> boolean signature, with respective carrier-sets<br />
L 33 , L 32 , L 31 , L 30 are defined as follows ...<br />
L 33 = <strong>the</strong> L 3 -sub-algebra <strong>of</strong> boolean signature generated by <strong>the</strong> atomic formulae <strong>of</strong> L 3 , making<br />
L 33 <strong>the</strong> set <strong>of</strong> open formulae.<br />
L 32 = <strong>the</strong> L 3 -sub-algebra <strong>of</strong> boolean signature generated by {∃ 2 (φ) : φ ∈ L 33 }<br />
L 31 = <strong>the</strong> L 3 -sub-algebra <strong>of</strong> boolean signature generated by {∃ 1 (φ) : φ ∈ L 32 }<br />
L 30 = <strong>the</strong> L 3 -sub-algebra <strong>of</strong> boolean signature generated by {∃ 0 (φ) : φ ∈ L 31 }<br />
23
Lemma 1.3.1 L 33 ∪ L 32 ∪ L 31 ∪ L 30 is a sub-formula-closed subset <strong>of</strong> L 3 .<br />
Pro<strong>of</strong>: L 33 ∪ L 32 ∪ L 31 ∪ L 30 is built by means <strong>of</strong> logical connectives beginning with <strong>the</strong> atomic<br />
formulae.<br />
qed<br />
The above four algebras <strong>of</strong> boolean signature are now interconnected with operations related to<br />
variable substitution and existential quantifiers.<br />
Definition L crc<br />
3 = (L 33 , L 32 , L 31 , L 30 , r ∗ |, p ∗ |, s ∗ |, ∃ 0 |, ∃ 1 |, ∃ 2 |) where<br />
r ∗ |, p ∗ |, s ∗ | are <strong>the</strong> restrictions <strong>of</strong> r ∗ , p ∗ , s ∗ to L 33 .<br />
∃ 2 | is <strong>the</strong> restriction <strong>of</strong> ∃ 2 to L 33 , making it an element <strong>of</strong> L 33 → L 32<br />
∃ 1 | is <strong>the</strong> restriction <strong>of</strong> ∃ 1 to L 32 , making it an element <strong>of</strong> L 32 → L 31<br />
∃ 0 | is <strong>the</strong> restriction <strong>of</strong> ∃ 0 to L 31 , making it an element <strong>of</strong> L 31 → L 30<br />
Proposition 1.3.2 The sort L 30 <strong>of</strong> L crc<br />
3 is a conservative reduction class.<br />
Pro<strong>of</strong>: We prove that L 30 contains <strong>the</strong> sentences <strong>of</strong> <strong>the</strong> form ∀ 0 ∃ 1 ∀ 2 φ where φ is open. This class<br />
<strong>of</strong> sentences contains <strong>the</strong> reduction class <strong>of</strong> Kahr, More and Wang, which turned out to be conservative<br />
by results <strong>of</strong> Berger, Gurevich and Koriakov. For ∀ 0 ∃ 1 ∀ 2 φ is by definition ¬∃ 0 ¬∃ 1 ¬∃ 2 ¬φ.<br />
As long as φ is open ¬φ ∈ L 33 ,<br />
<strong>the</strong>n ¬∃ 2 ¬φ ∈ L 32 ,<br />
<strong>the</strong>n ¬∃ 1 ¬∃ 2 ¬φ ∈ L 31 ,<br />
<strong>the</strong>n ¬∃ 0 ¬∃ 1 ¬∃ 2 ¬φ ∈ L 30 .<br />
The following is virtually <strong>the</strong> same as above but works for <strong>the</strong> conservative reduction class <strong>of</strong> Büchi.<br />
Corollary 1.3.3 The sort L 30 <strong>of</strong> L crc<br />
3 has every conjunction <strong>of</strong> prenex sentences in L 3 as an element.<br />
Pro<strong>of</strong>: It can be proven as above that every sentence Q 0 Q 1 Q 2 φ where φ is open is in L 30 . The<br />
corollary <strong>the</strong>n follows since L 30 has boolean signature and thus has conjunctions.<br />
qed<br />
1.3.3 Directed many-sorted polyadic set algebras<br />
We have seen how to turn a substantial fragment <strong>of</strong> first-order language into an algebra <strong>of</strong> directed<br />
many-sorted polyadic signature. Here we define <strong>the</strong> algebras that are <strong>the</strong> core <strong>of</strong> <strong>the</strong> present paper<br />
and <strong>the</strong> basis <strong>of</strong> each disprover deviced. These algebras turn out to be finite if <strong>the</strong>y are finitely<br />
generated.<br />
Definition An algebra A is a dMsPs 3 (directed many-sorted polyadic set algebra <strong>of</strong> dimension 3)<br />
if it is a C-sub algebra <strong>of</strong> directed many-sorted polyadic signature where C is a Ps 3 .<br />
qed<br />
24
1.3.4 Interpretation<br />
We show that one can interpret conjunctions <strong>of</strong> prenex L 3 sentences in dMsPs 3 ’s much as in<br />
Ps 3 ’s, and that each such interpretation can be used to represent a tarskian interpretation.<br />
Lemma 1.3.4 Let h = (h 3 , h 2 , h 1 , h 0 ) be a directed many-sorted polyadic homomorphism from L crc<br />
3<br />
to an A in Ps 3 . Then h 3 ∪ h 2 ∪ h 1 ∪ h 0 is a mapping from a subset <strong>of</strong> L 3 to A that extends to a<br />
unique polyadic homomorphism h ∗ from L 3 to A.<br />
Pro<strong>of</strong>: Recall that mappings formally are represented by sets <strong>of</strong> pairs. The boolean homomorphisms<br />
h 3 , h 2 , h 1 , h 0 are defined on disjoint sets, distinguished by what quantifiers occur in <strong>the</strong> elements.<br />
Thus h 3 ∪ h 2 ∪ h 1 ∪ h 0 is a mapping from L 33 ∪ L 32 ∪ L 31 ∪ L 30 . Which is easily seen to be a<br />
sub-formula-closed subset <strong>of</strong> L 3 . By lemma 1.2.4, letting X denote L 33 ∪ L 32 ∪ L 31 ∪ L 30 and<br />
Y denote L 3 , <strong>the</strong> mapping h 3 ∪ h 2 ∪ h 1 ∪ h 0 uniquely extends to a ∨, ¬, ⊥, ∃ 0 , ∃ 1 , ∃ 2 -preserving<br />
mapping h ∗ from L 3 . It remains to show that r ∗ , s ∗ , p ∗ are preserved by h ∗ to show that h ∗ is a<br />
polyadic homomorphism. Let f denote <strong>the</strong> restriction <strong>of</strong> h ∗ to <strong>the</strong> atomic formulae <strong>of</strong> <strong>the</strong> form<br />
{R, ...} × {012}. By definition, h ∗ extends f and must be <strong>the</strong> uniquely determined polyadic<br />
homomorphism f ∗ <strong>of</strong> lemma 1.2.5.<br />
qed<br />
The following says that homomorphisms are like interpretations in <strong>the</strong> usual sense in that if an interpretation<br />
<strong>of</strong> <strong>the</strong> relation-symbols is given <strong>the</strong>n an interpretation <strong>of</strong> each formula in our fragment<br />
is determined.<br />
Lemma 1.3.5 If {R, ...} are <strong>the</strong> relation-symbols <strong>of</strong> L 3 and if A is a dMsPs 3 and B 3 <strong>of</strong> A has<br />
carrier-set B 3 , <strong>the</strong>n each mapping f ∈ {R, ...} × {012} → B 3 extends to a unique directed manysorted<br />
polyadic homomorphism (f ∗ 3 , f ∗ 2 , f ∗ 1 , f ∗ 0 ) from L crc<br />
3 to A.<br />
Pro<strong>of</strong>: We display boolean homomorphisms f3 ∗ , f2 ∗ , f1 ∗ , f0 ∗ from L 33 , L 32 , L 31 , L 30 to <strong>the</strong> boolean<br />
reducts B 3 , B 2 , B 1 , B 0 <strong>of</strong> A respectively, we show that <strong>the</strong> extra-boolean operations <strong>of</strong> L crc<br />
3 are preserved<br />
and that this is a unique directed many-sorted polyadic homomorphism. By <strong>the</strong> definition<br />
<strong>of</strong> dMsPs 3 <strong>the</strong>re is a C in Ps 3 such that A is a C-sub-algebra <strong>of</strong> directed many-sorted signature.<br />
Lemma 1.2.5 provides a unique polyadic homomorphism f ∗ from L 3 to C that extends f. Define<br />
f3 ∗ , f2 ∗ , f1 ∗ , f0 ∗ as <strong>the</strong> restrictions <strong>of</strong> f ∗ to L 33 , L 32 , L 31 , L 30 respectively. These preserve <strong>the</strong> required<br />
operations as <strong>the</strong>y are restrictions <strong>of</strong> f ∗ , which preserves <strong>the</strong>m.<br />
Let (h 3 , h 2 , h 1 , h 0 ) be an arbitrary directed many-sorted polyadic homomorphism <strong>of</strong> <strong>the</strong> required<br />
form. To show uniqueness we show that h 3 ∪ h 2 ∪ h 1 ∪ h 0 and f3 ∗ ∪ f2 ∗ ∪ f1 ∗ ∪ f0 ∗ are<br />
<strong>the</strong> same. By lemma 1.2.3, <strong>the</strong> two are <strong>the</strong> same on atomic formulae. Letting X be <strong>the</strong> atomic<br />
formulae and Y = L 33 ∪ L 32 ∪ L 31 ∪ L 30 , in lemma 1.2.4 we get <strong>the</strong> desired result. Note that f3<br />
∗<br />
for instance trivially preserves ∃ 0 , ∃ 1 , ∃ 2 , as <strong>the</strong>re is no pair <strong>of</strong> open formulae φ and ψ such that <strong>the</strong><br />
syntactic equality ∃ i φ = ψ holds.<br />
qed<br />
As before a homomorphism is satisfying for a sentence if it is mapped to ⊤.<br />
Definition Let L 30 be <strong>the</strong> sort <strong>of</strong> L crc<br />
3 consisting <strong>of</strong> sentences. A directed many-sorted polyadic<br />
homomorphism (h 3 , h 2 , h 1 , h 0 ) from L crc<br />
3 to an A in dMsPs 3 is satisfying for a sentence φ ∈ L 30<br />
if h 0 (φ) = ⊤.<br />
25
The following allows us to use homomorphisms as representatives <strong>of</strong> tarskian interpretations.<br />
Proposition 1.3.6 Each homomorphism from L crc<br />
3 to an A in dMsPs 3 corresponds to a tarskian<br />
interpretation <strong>of</strong> L 3 as relations over some set U. Moreover a homomorphism is satisfying for a sentence<br />
φ if and only if <strong>the</strong> corresponding interpretation is satisfying for φ.<br />
Pro<strong>of</strong>: The correspondence is set up in two steps. First we compose <strong>the</strong> given homomorphism<br />
h = (h 3 , h 2 , h 1 , h 0 ) from L crc<br />
3 to A with an embedding f from A to a full Ps 3 obtaining<br />
f ◦ h = (f ◦ h 3 , f ◦ h 2 , f ◦ h 1 , f ◦ h 0 ). The definitions <strong>of</strong> dMsPs 3 and Ps 3 provide such an f.<br />
Secondly we use lemma 1.3.4 and extend f ◦ h 3 ∪ f ◦ h 2 ∪ f ◦ h 1 ∪ f ◦ h 0 to a homomorphism<br />
(f ◦ h) ∗ from L 3 to <strong>the</strong> full Ps 3 . By proposition 1.2.6 this is an interpretation in <strong>the</strong> usual sense.<br />
If h 0 (φ) = ⊤ <strong>the</strong>n (f ◦h) ∗ is a satisfying interpretation since f ◦h 0 is a boolean homomorphism<br />
which preserves ⊤ and since (f ◦ h) ∗ extends f ◦ h 3 ∪ f ◦ h 2 ∪ f ◦ h 1 ∪ f ◦ h 0 . For <strong>the</strong> o<strong>the</strong>r<br />
direction: if φ ∈ L 30 and if (f ◦ h) ∗ is satisfying for φ <strong>the</strong>n f ◦ h is satisfying for φ. Finally h is<br />
satisfying for φ since f is an embedding.<br />
qed<br />
By <strong>the</strong> above we may refer to homomorphisms from L crc<br />
3 to dMsPs 3 ’s as interpretations. A<br />
computer can decide whe<strong>the</strong>r <strong>the</strong>re exist satisfying interpretation for given prenex sentences in a<br />
finite dMsPs 3 by means <strong>of</strong> exhaustive search. The following proposition, provides a naturalness<br />
property for disprovers based on such exhaustive search. The property does in particular say that if<br />
one has two logically equivalent conjunctions <strong>of</strong> prenex sentences whose status regarding satisfiability<br />
one wishes to decide, running a disprover once, for any one <strong>of</strong> <strong>the</strong> two sentences suffices. This<br />
property is not shared with procedures that compare <strong>the</strong> input with sentences from a given finite<br />
set <strong>of</strong> sentences, as to each sentence <strong>the</strong>re is an infinite number <strong>of</strong> logically equivalent sentences.<br />
Proposition 1.3.7 Let A be a (finite) dMsPs 3 , let φ be a sentence <strong>of</strong> L crc<br />
3 and h an interpretation<br />
in A such that h is satisfying for φ. If ψ is a sentence <strong>of</strong> L crc<br />
3 such that φ and ψ are logically equivalent<br />
<strong>the</strong>n h is a satisfying interpretation for ψ<br />
Pro<strong>of</strong>: To say that φ and ψ are logically equivalent is to say that h(φ) = h(ψ). To say that h is<br />
satisfying for φ is to say that h(φ) = h(⊤). These two equations yield h(ψ) = h(⊤), which is to<br />
say that h is satisfying for ψ.<br />
qed<br />
1.3.5 The directed many-sorted polyadic closure<br />
Here we show that any satisfiable finite conjunction <strong>of</strong> prenex sentences is satisfied in a finite<br />
dMsPs 3 , even if such a conjunction is an infinity axiom.<br />
Let O ′ 3 denote <strong>the</strong> open formulae <strong>of</strong> L ′ 3, and O ′ 3/Γ <strong>the</strong> appropriate sub-boolean algebra <strong>of</strong><br />
L ′ 3/Γ. Recall that O ′ 3 is closed under <strong>the</strong> operations r ∗ , p ∗ , s ∗ .<br />
Definition This defines <strong>the</strong> directed many-sorted polyadic closure <strong>of</strong> <strong>the</strong> atomic formulae <strong>of</strong> L ′ 3<br />
in L ′ 3/Γ. The closure defines boolean algebras B 3 , B 2 , B 1 , B 0 , with carrier-sets B 3 , B 2 , B 1 , B 0 and<br />
and operations r, p, s, c 0 , c 1 , c 2 , which constitute a dMsPs 3 .<br />
B 3 = <strong>the</strong> boolean algebra O ′ 3/Γ<br />
B 2 = <strong>the</strong> sub-boolean-algebra <strong>of</strong> L ′ 3/Γ generated by <strong>the</strong> ∃ ′ 2-image <strong>of</strong> B 3<br />
26
B 1 = <strong>the</strong> sub-boolean-algebra <strong>of</strong> L ′ 3/Γ generated by <strong>the</strong> ∃ ′ 1-image <strong>of</strong> B 2<br />
B 0 = <strong>the</strong> sub-boolean-algebra <strong>of</strong> L ′ 3/Γ generated by <strong>the</strong> ∃ ′ 0-image <strong>of</strong> B 1<br />
r = {(φ, ψ) ∈ B 3 × B 3 : ∀ 0 ∀ 1 ∀ 2 (r ∗ (φ) ↔ ψ) ∈ Γ}<br />
p = {(φ, ψ) ∈ B 3 × B 3 : ∀ 0 ∀ 1 ∀ 2 (p ∗ (φ) ↔ ψ) ∈ Γ}<br />
s = {(φ, ψ) ∈ B 3 × B 3 : ∀ 0 ∀ 1 ∀ 2 (s ∗ (φ) ↔ ψ) ∈ Γ}<br />
c 2 = {(φ, ψ) ∈ B 3 × B 2 : ∀ 0 ∀ 1 (∃ 2 (φ) ↔ ψ) ∈ Γ}<br />
c 1 = {(φ, ψ) ∈ B 2 × B 1 : ∀ 0 (∃ 1 (φ) ↔ ψ) ∈ Γ}<br />
c 0 = {(φ, ψ) ∈ B 1 × B 0 : ∃ 0 (φ) ↔ ψ ∈ Γ}<br />
With <strong>the</strong> above definition <strong>of</strong> closure finitely generated dMsPs 3 ’s are finite.<br />
Proposition 1.3.8 Let Γ be a consistent and complete L ′ 3 <strong>the</strong>ory. As long as <strong>the</strong> number <strong>of</strong> atomic<br />
formulae <strong>of</strong> L ′ 3 is finite, <strong>the</strong>ir directed many-sorted polyadic closure in L ′ 3/Γ is finite.<br />
Pro<strong>of</strong>: The initial boolean algebra B 3 is O 3/Γ, ′ <strong>the</strong> boolean algebra generated by <strong>the</strong> equivalence<br />
classes that have atomic formulae in <strong>the</strong>m. It is finite since it is a finitely generated boolean algebra.<br />
The subsequent boolean algebras are generated by images <strong>of</strong> finite ones.<br />
qed<br />
Note that in <strong>the</strong> above proposition, Γ needs only be complete enough to contain formulae needed<br />
for <strong>the</strong> closure. Since <strong>the</strong> closure is finite <strong>the</strong>re is to each satisfiable L ′ 3 sentence a finite Γ sufficient<br />
for <strong>the</strong> closure to be defined. There is no general way <strong>of</strong> computing consistent Γ sufficient for <strong>the</strong><br />
closure to be defined but, as it turns out, sufficiency can be established computationally. In <strong>the</strong><br />
following we appeal to Lindenbaums lemma which says that any consistent first-order <strong>the</strong>ory has a<br />
completion.<br />
Corollary 1.3.9 A finite conjunction <strong>of</strong> prenex L 3 sentences, is satisfiable iff it is satisfiable in a finite<br />
dMsPs 3 .<br />
Pro<strong>of</strong>: Let L ′ 3 denote <strong>the</strong> language whose relation-symbols are those <strong>of</strong> <strong>the</strong> given conjunction.<br />
View such a conjunction as an L ′ 3 <strong>the</strong>ory by itself, and let Γ denote one <strong>of</strong> its consistent completions.<br />
The sentence is <strong>the</strong>n satisfied in L ′ 3/Γ by <strong>the</strong> homomorphism that maps each formula to<br />
it’s equivalence class. The sentence is also satisfied in <strong>the</strong> dMsPs 3 that is <strong>the</strong> closure <strong>of</strong> its atoms<br />
in L ′ 3/Γ. For <strong>the</strong> o<strong>the</strong>r direction; a satisfying interpretation in a finite dMsPs 3 does by lemma<br />
1.3.4 extend to a satisfying interpretation in <strong>the</strong> Ps 3 <strong>of</strong> which <strong>the</strong> finite dMsPs 3 is a directed<br />
many-sorted sub-algebra.<br />
qed<br />
The following sets things straight with fundamental results <strong>of</strong> Church and Turing. It also excludes<br />
<strong>the</strong> possibility <strong>of</strong> defining <strong>the</strong> class <strong>of</strong> dMsPs 3 by means <strong>of</strong> a finite number <strong>of</strong> axioms in a language<br />
where one can check whe<strong>the</strong>r an axiom holds in a finite algebra recursively.<br />
In <strong>the</strong> following <strong>the</strong> letter R stands for representable, a notion we have not defined in <strong>the</strong><br />
present paper but see [Ném91] or [AGM + 98]. Letting R denote <strong>the</strong> finite dMsPs 3 ’s and |= <strong>the</strong><br />
notion <strong>of</strong> satisfaction between finite algebras <strong>of</strong> suitable signature and L crc<br />
3 -sentences used in <strong>the</strong><br />
present paper, <strong>the</strong> corrolary states that <strong>the</strong> finite dMsPs 3 ’s are not recursively enumerable.<br />
27
Corollary 1.3.10 Let |= be a recursively enumerable binary relation between conjunctions <strong>of</strong> prenex L 3<br />
sentences and algebras <strong>of</strong> directed many-sorted polyadic signature. Let R be a class <strong>of</strong> algebras <strong>of</strong> directed<br />
many-sorted polyadic signature, containing <strong>the</strong> finite dMsPs 3 ’s. Let R and |= have <strong>the</strong> following two<br />
properties for each finite conjunction <strong>of</strong> prenex L 3 sentences φ:<br />
if A is a finite dMsPs 3 and <strong>the</strong>re is a satisfying homomorphism for φ in A <strong>the</strong>n A |= φ.<br />
if A is in R and A |= φ <strong>the</strong>n φ is satisfiable in <strong>the</strong> tarskian sense.<br />
Then R is not recursively enumerable.<br />
Pro<strong>of</strong>: For <strong>the</strong> purpose <strong>of</strong> arriving at a contradiction assume that <strong>the</strong>re is a way <strong>of</strong> recursively<br />
enumerating <strong>the</strong> class R <strong>On</strong>e could <strong>the</strong>n write a procedure that recognised only satisfiable and all<br />
satisfiable first-order sentences working as follows. This procedure would first transform an input<br />
to an equi-satisfiable conjunction <strong>of</strong> prenex L 3 sentences φ by <strong>the</strong> reduction <strong>of</strong> Büchi. Then <strong>the</strong><br />
procedure would enumerate algebras A in R, and whilst doing so seek to find if A |= φ. If A |= φ<br />
were found hold for some A <strong>the</strong> procedure would terminate.<br />
Since every satisfiable finite conjunction <strong>of</strong> prenex sentences has a satisfying interpretation in<br />
a finite A in dMsPs 3 , (corollary 1.3.9), and since this implies A |= φ this procedure would<br />
terminate for every satisfiable sentence. It would only terminate for satisfiable sentences as A |= φ<br />
implies that φ is satisfiable in <strong>the</strong> tarskian sense.<br />
Combining this hypo<strong>the</strong>tical procedure with a known to exist complete procedure for recognising<br />
inconsistent sentences, results in a procedure for <strong>the</strong> Entscheidungsproblem. Assuming <strong>the</strong><br />
existence <strong>of</strong> such, was shown to lead to a contradiction by Church and Turing.<br />
qed<br />
1.3.6 The closure as an algorithm<br />
Assume that Γ is a complete and consistent <strong>the</strong>ory in a language L ′ 3, with a finite number <strong>of</strong><br />
relation-symbols. Also assume that <strong>the</strong>re is a decision procedure for Γ. The directed many-sorted<br />
polyadic closure <strong>of</strong> <strong>the</strong> atomic formulae <strong>of</strong> L ′ 3 in L ′ 3/Γ can <strong>the</strong>n be seen as an algorithm. First<br />
<strong>of</strong> all, <strong>the</strong> definition <strong>of</strong> <strong>the</strong> closure has <strong>the</strong> overall structure <strong>of</strong> a procedure. Moreover B 3 for<br />
instance, can be computed by starting with Γ-distinct (not Γ-equivalent) atomic formulae and<br />
generating new formulae by <strong>the</strong> boolean operations. As new formulae φ are generated <strong>the</strong>se are kept<br />
or discarded, depending on whe<strong>the</strong>r <strong>the</strong>y are Γ-equivalent to some already generated formula ψ.<br />
That is; depending on what <strong>the</strong> decision procedure has to say about <strong>the</strong> sentence ∀ 0 ∀ 1 ∀ 2 (φ ↔ ψ).<br />
The kept formulae serve as indices for distinct elements <strong>of</strong> O ′ 3/Γ, and <strong>the</strong> process terminates because<br />
<strong>the</strong> number <strong>of</strong> equivalence classes <strong>of</strong> O ′ 3/Γ is finite. Thus <strong>the</strong> procedure so far is an algorithm. <strong>On</strong>e<br />
can also show that all <strong>of</strong> O ′ 3/Γ is generated like this by induction on <strong>the</strong> well-ordering used in <strong>the</strong><br />
definition <strong>of</strong> L ′ 3/Γ. The rest <strong>of</strong> <strong>the</strong> closure is seen to be algorithmic in a similar fashion.<br />
The outlined, straight forward approach, which is presenting <strong>the</strong> sub-boolean algebras <strong>of</strong> L ′ 3/Γ<br />
with one formula for each equivalence class and corresponding tables for <strong>the</strong> operations (finite<br />
sets <strong>of</strong> triples or pairs <strong>of</strong> formulae), is less than optimal with respect to size <strong>of</strong> <strong>the</strong> tables. By <strong>the</strong><br />
additivity <strong>of</strong> Ps 3 ’s (lemma 1.2.7), all one needs to store, is information about how r, p, s, c 0 , c 1<br />
and c 2 behave on <strong>the</strong> atoms <strong>of</strong> <strong>the</strong> sub-boolean algebras. Let, for example, [φ] and [ψ] be atoms<br />
<strong>of</strong> O ′ 3/Γ. Note that [φ] and [ψ] need not contain atomic formulae. Then r ′ ([φ ∨ ψ]) for instance<br />
is determined by r ′ ([φ]) ∨ ′ r ′ ([ψ]). By lemma 1.1.2 one may represent O ′ 3/Γ by an algebra with<br />
28
carrier-set n → {⊥, ⊤}. This can be done in such a way that <strong>the</strong> atoms are represented by <strong>the</strong><br />
mappings that equal ⊥ in all but one component. The representative <strong>of</strong> r ′ ([φ]) ∨ ′ r ′ ([ψ]) is <strong>the</strong>n<br />
determined by <strong>the</strong> bit-wise join <strong>of</strong> <strong>the</strong> representatives <strong>of</strong> r ′ ([φ]) and r ′ ([ψ]). This allows for a<br />
logarithmic reduction <strong>of</strong> <strong>the</strong> size <strong>of</strong> <strong>the</strong> tables for <strong>the</strong> corresponding algebra. For instance, O ′ 3/Γ <strong>of</strong><br />
<strong>the</strong> <strong>the</strong>ory <strong>of</strong> a dense order without endpoints turns out to have 2 13 elements and 13 atoms.<br />
It can be quite enlightening actually defining <strong>the</strong> mentioned 13 atoms by means <strong>of</strong> a knife,<br />
in some physical medium such as a reasonably cubic and homogenous piece <strong>of</strong> fruit. The cube<br />
represents R 3 . <strong>On</strong>e should make 3 slices, one for each <strong>of</strong> <strong>the</strong> planes defined by x = y, x = z<br />
and y = z, see fig. 1.3. <strong>On</strong>e ought <strong>the</strong>n end up with 6 pieces that are 3 dimensional, and thus<br />
visible. Wedged between <strong>the</strong>se pieces <strong>the</strong>re are ano<strong>the</strong>r 6 triangular pieces, and finally <strong>the</strong>re is a<br />
13th, 1-dimensional piece where <strong>the</strong> 3 defining planes meet. These 13 pieces correspond to subsets<br />
<strong>of</strong> R 3 which in turn correspond to <strong>the</strong> atoms <strong>of</strong> O ′ 3/Γ by <strong>the</strong> embedding obtained by restricting<br />
<strong>the</strong> usual interpretation <strong>of</strong> O ′ 3, as ternary relations over <strong>the</strong> dense order (R,
<strong>the</strong> full Ps 3 over 3 has 27 atoms. So in terms <strong>of</strong> size <strong>of</strong> search-space, doing exhaustive search for<br />
satisfying interpretations in an algebra whose B 3 has 13 atoms, is a lesser task than that <strong>of</strong> searching<br />
for satisfying interpretations over a 3 element set. Exhaustive search in such a space is well within<br />
reach for present day computers, for sentences <strong>of</strong> some length.<br />
1.4 A construction <strong>of</strong> Ps 3 ’s and dMsPs 3 ’s<br />
In <strong>the</strong> previous section effort was put into keeping only finite parts <strong>of</strong> polyadic set algebras, as this<br />
makes <strong>the</strong>m suitable as components <strong>of</strong> disprovers. The resulting algebras are however somewhat<br />
coarse. For example; to each such algebra <strong>the</strong>re exists a finitely satisfiable sentence not satisfiable in<br />
that algebra. Such a sentence may be constructed in a language built out <strong>of</strong> more relation-symbols<br />
than <strong>the</strong>re are elements in <strong>the</strong> algebra, by stating that <strong>the</strong> named relations are pairwise different.<br />
This section describes a way <strong>of</strong> constructing a new and more refined finite Ps 3 or dMsPs 3<br />
from a given one, by finite means. The construction is such that any finitely satisfiable sentence<br />
is satisfiable in an iterate <strong>of</strong> <strong>the</strong> construction. This can be used as part <strong>of</strong> a disprover to ensure<br />
termination in case <strong>of</strong> finite satisfiability. Here it is defined for Ps 3 ’s only. The construction is<br />
analogous for directed many-sorted ones.<br />
Definition Let A = (B, r, p, s, c 0 , c 1 , c 2 ) be a Ps 3 . Equip <strong>the</strong> set <strong>of</strong> mappings in 2 3 → A with<br />
polyadic structure as follows. Boolean operations are defined component-wise. For t ∈ 2 3 → A<br />
define r, p, s, c 0 , c 1 , c 2 ∈ (2 3 → A) → (2 3 → A) by:<br />
r(t)(abc) = r(t(abc ◦ r)),<br />
p(t)(abc) = p(t(abc ◦ p)),<br />
s(t)(abc) = s(t(abc ◦ s)),<br />
c 0 (t)(abc) = ∨ u∈2 c 0 (t(ubc)),<br />
c 1 (t)(abc) = ∨ u∈2 c 1 (t(auc)),<br />
c 2 (t)(abc) = ∨ u∈2 c 2 (t(abu)).<br />
By abuse <strong>of</strong> notation let 2 3<br />
construction follow.<br />
→ A denote <strong>the</strong> algebra just defined. Two propositions about <strong>the</strong><br />
Proposition 1.4.1 If A is a Ps 3 <strong>the</strong>n 2 3 → A is a Ps 3 .<br />
Pro<strong>of</strong>: Since A is a Ps 3 <strong>the</strong>re is a polyadic embedding h from A to a full Ps 3 over some set U.<br />
Using <strong>the</strong> fact that <strong>the</strong>re is a correspondence between sets <strong>of</strong> elements <strong>of</strong> U 3 and <strong>the</strong>ir characteristic<br />
functions (U 3 → 2), (2 3 → A) is embeddable into 2 3 → (U 3 → 2) by <strong>the</strong> mapping t ↦→ h ◦ t.<br />
Now <strong>the</strong>re is a natural isomorphism from 2 3 → (U 3 → 2) to (2 × U) 3 → 2 which is <strong>the</strong> set<br />
<strong>of</strong> characteristic functions <strong>of</strong> <strong>the</strong> elements <strong>of</strong> a full Ps 3 . Regard 3-dimensional figures with one<br />
quadrant, or “octant” ra<strong>the</strong>r, for each element <strong>of</strong> 2 3 for details <strong>of</strong> this, fig. 1.4.<br />
qed<br />
Proposition 1.4.2 Any finitely satisfiable sentence is satisfied in an iterate <strong>of</strong> <strong>the</strong> above construction<br />
beginning with an arbitrary Ps 3 .<br />
30
Figure 1.4: Cylindrification along x-axis <strong>of</strong> an element <strong>of</strong> 2 3 → P([0, 1〉 3 ). Here [0, 1〉 is a halfopen<br />
interval <strong>of</strong> <strong>the</strong> reals and <strong>the</strong> element represents a relation in [0, 2〉 3 visualised by suitably<br />
stacking 8 copies <strong>of</strong> [0, 1〉 3 .<br />
Pro<strong>of</strong>: Given an A in Ps 3 , iterates are <strong>of</strong> <strong>the</strong> form 2 3 → (. . . → (2 3 → A)) which are naturally<br />
isomorphic to those <strong>of</strong> <strong>the</strong> form (2 × . . . × 2) 3 → A. Since <strong>the</strong> top and bottom <strong>of</strong> A can serve<br />
as image <strong>of</strong> a characteristic function, <strong>the</strong> latter algebra contains (2 × . . . × 2) 3 → {⊥, ⊤}, which<br />
is isomorphic to <strong>the</strong> full polyadic set algebra over some set with 2 n elements for a suitable n ∈ ω.<br />
The proposition follows since any finitely satisfiable equality-free sentence is satisfiable over a set<br />
with 2 n elements, for some n ∈ ω.<br />
qed<br />
1.4.1 A disprover deviced according to <strong>the</strong> method<br />
To device a disprover according to <strong>the</strong> presented method, one needs a decision procedure for a<br />
<strong>the</strong>ory Γ, or alternatively a, known to be satisfiable, finite set <strong>of</strong> sentences sufficiently complete<br />
for a directed many-sorted polyadic closure to be defined. For implementation <strong>the</strong> present author<br />
has used a publicly available decision procedure for Presburger arithmetic by Karlund, Møller and<br />
Schwartzbach [KMS02]. For instance a computer can, over night, produce <strong>the</strong> closure <strong>of</strong> <strong>the</strong> atomic<br />
formulae <strong>of</strong> L ′ 3 in L ′ 3/Γ, where L ′ 3 has exactly one relation-symbol,
set. In <strong>the</strong> example, <strong>the</strong> initial sort B 3 <strong>of</strong> A, turned out to have 13 atoms, which can be verified<br />
by a geometric argument similar to <strong>the</strong> one depicted in fig. 1.3, when viewing ω as a subset <strong>of</strong><br />
R. B 3 <strong>of</strong> 2 3 → A has 8 times that number. Multiplication with 8 proceeds, so after i iterations<br />
search is done in an algebra with 13 · 8 i atoms. By lemma 1.1.2 and <strong>the</strong> possibility <strong>of</strong> storing <strong>the</strong><br />
extra-boolean operations <strong>of</strong> A in <strong>the</strong> form <strong>of</strong> tables (finite sets <strong>of</strong> pairs), <strong>the</strong> question <strong>of</strong> whe<strong>the</strong>r<br />
<strong>the</strong>re exists a satisfying interpretation for a sentence with m relation-symbols, in an algebra with<br />
13 · 8 i atoms, can be phrased as a constraint satisfaction problem with m · 13 · 8 i variables ranging<br />
over {⊥, ⊤}.<br />
If <strong>the</strong> disprover is based on <strong>the</strong> full Ps 3 over a finite set <strong>the</strong>n it is a finite model search procedure,<br />
which at each step doubles <strong>the</strong> size <strong>of</strong> <strong>the</strong> set over which satisfying interpretations are sought.<br />
Various disprovers have been implemented, based on <strong>the</strong> presented method. They are publicly<br />
available with source code included. 1 .<br />
1.5 Related constructions and algebras<br />
The construction <strong>of</strong> definition 1.4 is ra<strong>the</strong>r similar to what was called <strong>the</strong> cardinal multiple <strong>of</strong> <strong>the</strong>ories<br />
by Feferman and Vaught [FV59] (Section 4.7), when bearing in mind that polyadic set algebras<br />
correspond to complete and consistent <strong>the</strong>ories. The constructions <strong>of</strong> [FV59] were carried over<br />
to <strong>the</strong> setting <strong>of</strong> polyadic algebras and generalised by Daigneault [Dai63]. The main construction<br />
is called <strong>the</strong> tensor product <strong>of</strong> polyadic algebras. Daigneaults paper is mainly about infinite, not<br />
necessarily atomic nor complete, polyadic algebras so <strong>the</strong> definition goes via Stone-spaces. Finite<br />
polyadic algebras are all atomic and complete, so it is worth noting that this tensor product can also<br />
be constructed by finite means. It turns out to be <strong>the</strong> polyadic <strong>version</strong> <strong>of</strong> <strong>the</strong> Kronecker product,<br />
⊗, <strong>of</strong> finite-dimensional vector spaces. This can be seen by regarding elements <strong>of</strong> a finite (directed<br />
many-sorted) polyadic set algebra as vectors <strong>of</strong> zeros and ones and using boolean operations instead<br />
<strong>of</strong> ring-operations. In this view 2 3 → A is isomorphic to P(2 3 ) ⊗ A. Under one such isomorphism<br />
<strong>the</strong> function values <strong>of</strong> an element <strong>of</strong> 2 3 → A appear as rows <strong>of</strong> <strong>the</strong> corresponding matrix in<br />
P(2 3 ) ⊗ A. This product works for arbitrary pairs <strong>of</strong> set algebras and provides a way <strong>of</strong> combining<br />
any two disprovers devised as presented.<br />
An essential property <strong>of</strong> <strong>the</strong> construction, is that <strong>the</strong> polyadic set algebras are closed under<br />
it, and that one gets more than what one had to begin with. The direct product <strong>of</strong> polyadic<br />
algebras is not such a construction, as a sentence that is satisfiable in a product, by projection,<br />
already is satisfiable in one <strong>of</strong> <strong>the</strong> factors. There are no projections <strong>of</strong> this kind for Ps 3 ’s. They are<br />
simple. By an argument involving lemma 1.3.4, dMsPs 3 ’s are also simple. The direct product does<br />
however give rise to <strong>the</strong> extensively studied representable polyadic algebras, which toge<strong>the</strong>r with<br />
representable relation algebras and cylindric algebras, provide a potential source for dMsPs 3 ’s,<br />
besides decision procedures, and sufficiently complete finite sets <strong>of</strong> sentences.<br />
Regarding dMsPs 3 ’s, <strong>the</strong> idea <strong>of</strong> using partial and many-sorted variants <strong>of</strong> algebras for interpretation<br />
<strong>of</strong> languages with quantifiers is far from new. An early one is by Bernays [Ber59], more are<br />
mentioned by Németi [Ném91]. Connections to category <strong>the</strong>ory approaches are also mentioned<br />
<strong>the</strong>re. The approaches <strong>the</strong> present author is aware <strong>of</strong>, are each different from dMsPs 3 ’s in at least<br />
one <strong>of</strong> <strong>the</strong> following two respects. Firstly, classes <strong>of</strong> partial or many-sorted algebras are not gener-<br />
1<br />
http://flipper.berlios.de<br />
32
ally such that each arity respecting mapping <strong>of</strong> <strong>non</strong>-logical symbols into an A extends naturally to<br />
an interpretation in A <strong>of</strong> each sentence <strong>of</strong> an entire reduction class as in lemma 1.3.5. Secondly,<br />
known many-sorted variants, which obviously can be chopped <strong>of</strong>f at some finite dimension, typically<br />
allow <strong>the</strong> definition <strong>of</strong> cylindrification within each sort, which in <strong>the</strong> present vocabulary is to<br />
say that <strong>the</strong>y are not directed. Undirectedness can ruin <strong>the</strong> property that finitely generated algebras<br />
are finite. As an example <strong>of</strong> how little it takes for undirected closures to be infinite, consider P(ω 2 ),<br />
<strong>the</strong> full Ps 2 over <strong>the</strong> natural numbers. A signature for this algebra is (B, s, p, c x , c y ). Both c x and<br />
s may even be left out for <strong>the</strong> following. Let y < x denote <strong>the</strong> set <strong>of</strong> pairs <strong>of</strong> numbers whose<br />
second component is less than <strong>the</strong> first. Moreover let for each natural a, a < x denote <strong>the</strong> set<br />
<strong>of</strong> pairs whose first component is greater than a, and a < y denote <strong>the</strong> set <strong>of</strong> pairs whose second<br />
component is greater than a. The claim is that for each natural a, a < x lies in <strong>the</strong> closure <strong>of</strong><br />
y < x. Now 0 < x lies in <strong>the</strong> closure as c y (y < x) = 0 < x. Proceed by assuming that a < x lies<br />
in <strong>the</strong> closure. Consider c y (p(a < x) ∩ y < x), here p(a < x) = a < y so <strong>the</strong> term is true if <strong>the</strong>re<br />
is a y such that x is greater than y and y is greater than a, which is when a + 1 < x.<br />
1.6 Concluding remarks<br />
The author believes that <strong>the</strong> adaption <strong>of</strong> <strong>the</strong> polyadic algebras <strong>of</strong> Tarski and Halmos to <strong>the</strong> conservative<br />
reduction class <strong>of</strong> Büchi so as to form <strong>the</strong> class dMsPs 3 is a contribution to algebraic logic.<br />
Some potential in automated reasoning has also been made likely.<br />
Distinguishing features <strong>of</strong> <strong>the</strong> dMsPs 3 ’s are <strong>the</strong> naturalness property <strong>of</strong> lemma 1.3.5, <strong>the</strong>ir<br />
having a finite dimension higher than two and <strong>the</strong>ir being directed. Directedness enables <strong>the</strong> fur<strong>the</strong>r<br />
downward Löwenheim-Skolem property (corollary 1.3.9), for each sentence <strong>of</strong> <strong>the</strong> conservative<br />
reduction class. This property makes it possible to do exhaustive search for, and to sometimes find,<br />
satisfying interpretations for infinity axioms in finite dMsPs 3 ’s. Various finite dMsPs 3 ’s can be<br />
computed by means <strong>of</strong> given decision procedures for first-order <strong>the</strong>ories. The above makes finite<br />
dMsPs 3 ’s quite suitable as components <strong>of</strong> disprovers.<br />
Each finite dMsPs 3 can toge<strong>the</strong>r with <strong>the</strong> construction <strong>of</strong> definition 1.4 be used to devise<br />
a disprover that behaves naturally and that works by a generalisation <strong>of</strong> finite model search for a<br />
substantial fragment <strong>of</strong> first-order language. The fragment is substantial since it is a conservative<br />
reduction class. The disprover behaves naturally by proposition 1.3.7. The disprover works by<br />
a generalisation <strong>of</strong> finite model search since, by proposition 2.5.3, it does search through a set <strong>of</strong><br />
interpretations containing a representative for every interpretation over any finite set. As long as <strong>the</strong><br />
given dMsPs 3 allows a satisfying interpretation for an infinity axiom, <strong>the</strong> generalisation is strict.<br />
33
Chapter 2<br />
Automata for mechanising consistency<br />
pro<strong>of</strong>s<br />
35
2.1 Introduction<br />
This is <strong>the</strong> first <strong>of</strong> two papers on a kind <strong>of</strong> automaton that is suitable for consistency pro<strong>of</strong>s and for<br />
<strong>the</strong> computation <strong>of</strong> atom-structures <strong>of</strong> finite representable polyadic algebras, including some which<br />
have purely infinite spectrum. Finite representable algebras with infinite spectrum are <strong>of</strong> interest in<br />
regards to Hilberts Entscheidungsproblem as <strong>the</strong> algebras can be used to construct semi-decision<br />
procedures that recognise not only finitely satisfiable sentences as consistent but also some infinity<br />
axioms, see A. Rognes [Rog09]. Infinity axioms are consistent first-order sentences that have infinite<br />
models only. Devising reasonably natural semi-decision procedures that terminate on input <strong>of</strong> even<br />
a single infinity axiom is a challenge. Note, for instance, that an eventually periodic infinite branch<br />
<strong>of</strong> semantic tree implies finite satisfiability. Similar effects occur with finitely presented Herbrand<br />
models.<br />
Polyadic algebras were introduced by P. Halmos who was inspired by A.Tarskis cylindric algebras.<br />
Finite and representable polyadic algebras are ma<strong>the</strong>matical objects that can be used much like<br />
structures and models when recognising given first-order sentences as consistent. As with structures<br />
we may interpret relational sentences in <strong>the</strong>se algebras and only consistent sentences have satisfying<br />
interpretations in representable algebras. Curiously some <strong>of</strong> <strong>the</strong> finite and representable algebras<br />
have satisfying interpretations for infinity axioms. In computations involving infinity axioms explicitly<br />
presented models are generally unsuitable as arguments to computable functions by virtue<br />
<strong>of</strong> being infinite. Finite representable algebras with satisfying interpretations for infinity axioms<br />
however, work well by virtue <strong>of</strong> being finite. The atom-structures <strong>of</strong> finite algebras are even more<br />
suitable as <strong>the</strong>y are considerably smaller that <strong>the</strong> algebra it self. An atom-structure plays <strong>the</strong> same<br />
role for a finite polyadic algebra as does a basis for a vector space or a topology.<br />
The present paper, i.e. part one <strong>of</strong> two papers, can be read independently <strong>of</strong> <strong>the</strong> subsequent<br />
part and makes no use <strong>of</strong> algebraic logic. It introduces <strong>the</strong> automata mentioned in <strong>the</strong> title and<br />
shows how <strong>the</strong>se can be used to construct semi-decision procedures that tackle not only finitely<br />
satisfiable sentences but also some infinity axioms. The paper considers several classes <strong>of</strong> automata<br />
but culminates in a class that is basic elementary, i.e. <strong>the</strong> class is defined by a finite set <strong>of</strong> first-order<br />
sentences. The significance <strong>of</strong> a class being basic elementary is that this provides a sense in which<br />
<strong>the</strong> class is natural, moreover it is useful when computing, i.e., recursively enumerating, those <strong>of</strong><br />
<strong>the</strong> automata that are finite.<br />
A recurring <strong>the</strong>me in <strong>the</strong> two papers is <strong>the</strong> need to express reachability in <strong>the</strong> automata at<br />
hand, whilst keeping things basic elementary. It is known from model-<strong>the</strong>ory that notions which<br />
are naturally defined by means <strong>of</strong> transitive closure, are not definable in first-order language over<br />
structures in general. By <strong>the</strong> Ehrenfeucht-Fraïssé method one can prove that this remains true over<br />
finite structures. Reachability in finite automata is an example <strong>of</strong> such a notion.<br />
As we shall see in <strong>the</strong> present paper, a reason for <strong>the</strong> impossibility <strong>of</strong> defining reachability is <strong>the</strong><br />
fact that one considers classes <strong>of</strong> automata all <strong>of</strong> which have <strong>the</strong> same fixed alphabet. By allowing<br />
some flexibility in <strong>the</strong> alphabet however, we can show <strong>the</strong> existence <strong>of</strong> classes <strong>of</strong> automata with <strong>the</strong><br />
following properties.<br />
1. The class <strong>of</strong> automata is definable with a finite set <strong>of</strong> first-order axioms.<br />
2. There is in <strong>the</strong> language that defines <strong>the</strong> class <strong>of</strong> automata a first-order formula that defines<br />
<strong>the</strong> reachability relation in each automaton <strong>of</strong> <strong>the</strong> class.<br />
36
3. The finite automata <strong>of</strong> <strong>the</strong> class are as versatile as finite automata with fixed alphabets, at least<br />
in regards to deciding <strong>the</strong> <strong>the</strong>ory <strong>of</strong> automatic structures such as Presburger arithmetic.<br />
The first two properties are needed to obtain a basic elementary class <strong>of</strong> automata in which reachability<br />
is first-order. The third property is sufficient for a semi-decision procedure, based on an<br />
automaton, to recognise some infinity axiom. The latter is because Presburger arithmetic contains<br />
infinity axioms. To fulfil <strong>the</strong> second property we will simply postulate that <strong>the</strong> relations <strong>of</strong> which we<br />
need to take <strong>the</strong> transitive closure are transitive. This makes expressing reachability trivial. What<br />
remains <strong>the</strong>n is to make sure that we have not lost computational power, i.e., we need to show<br />
that those <strong>of</strong> <strong>the</strong> automata that fulfil <strong>the</strong> transitivity postulates suffice for deciding <strong>the</strong> <strong>the</strong>ories <strong>of</strong><br />
automatic structures.<br />
We introduce and show some results on two classes with <strong>the</strong> three properties. Firstly <strong>the</strong> class<br />
<strong>of</strong> transitive automata, made to be a simple and transparent example <strong>of</strong> such a class. Automata <strong>of</strong><br />
<strong>the</strong> second class, called PTPS automata, consist <strong>of</strong> transitive automata where one is able to express<br />
various refinements <strong>of</strong> <strong>the</strong> reachability relation in first-order language. We are in particular able<br />
to give first-order definitions <strong>of</strong> constructions on automata, that correspond to logical connectives,<br />
quantifiers and substitutions, internally in a finite PTPS automaton.<br />
2.1.1 Outline <strong>of</strong> paper<br />
In section 1 we repeat well known definitions and results to make <strong>the</strong> present paper fairly self<br />
contained. In section 2 we introduce n-fold vector-spaces. These are vector-spaces with two extra<br />
operators. Whe<strong>the</strong>r this particular subclass <strong>of</strong> groups with operators has been described before is<br />
unknown to <strong>the</strong> present author. The n-fold vector-spaces serve as alphabets for all <strong>the</strong> automata<br />
introduced in <strong>the</strong> present paper.<br />
In section 3, n-tape p-automata which have n-fold vector-spaces as alphabets are introduced.<br />
These are believed to be new. The rest <strong>of</strong> <strong>the</strong> automata described in <strong>the</strong> present paper are variations<br />
over <strong>the</strong>se. We define what it means for an automaton, with a possibly abstract n-fold vector-space<br />
as alphabet, to recognise an n-ary relation on natural numbers.<br />
In section 4 a subclass <strong>of</strong> n-tape p-automata is introduced. They are called transitive automata.<br />
We provide a finite set <strong>of</strong> first-order axioms for transitive automata, and show that in finite transitive<br />
automata <strong>the</strong> reachability relation is first-order definable, quite trivially. Less trivial is <strong>the</strong> fact that<br />
finite transitive n-tape p-automata are as versatile as finite automata in general when it comes to<br />
<strong>the</strong>ir use as parts <strong>of</strong> decision procedures for <strong>the</strong>ories <strong>of</strong> automatic structures.<br />
In section 5 <strong>the</strong> definitions and pro<strong>of</strong> techniques <strong>of</strong> section 4 are elaborated upon and we<br />
introduce PTPS automata. They are also definable by a finite set <strong>of</strong> first-order formula. In PTPS<br />
automata we are able to express, in first-order language, various forms <strong>of</strong> reachability between states.<br />
For example we are able to express that one state is reachable from ano<strong>the</strong>r using a tape whose second<br />
track represents <strong>the</strong> number 0.<br />
In section 6 we look at how PTPS automata, used to decide a given sentence about an automatic<br />
structure, are in relationship to one ano<strong>the</strong>r. We show that <strong>the</strong> relationship between an automaton<br />
for a formula and <strong>the</strong> automata for its sub-formulae is first-order definable over finite structures.<br />
Finally we see how this can be used for proving consistency computationally.<br />
37
2.1.2 Notation<br />
We assume familiarity with automata <strong>the</strong>ory. N denotes <strong>the</strong> natural numbers. A one-sorted structure,<br />
or just a structure, is a tuple (X, R 0 , . . . , R k−1 , f 0 , . . . , f l−1 ) where X is a set and where<br />
R 0 , . . . , R k−1 are relations and f 0 , . . . , f l−1 are functions on X <strong>of</strong> fixed finite arities. A two-sorted<br />
structure is a tuple<br />
(X, Y, R 0 , . . . , R k−1 , f 0 , . . . , f l−1 ) where X and Y are sets and where <strong>the</strong> relations and functions<br />
are on X or Y or both. Two structures are said to be <strong>of</strong> <strong>the</strong> same similarity-type if <strong>the</strong>y are both<br />
one-sorted or both two-sorted and if <strong>the</strong>y have <strong>the</strong> same number <strong>of</strong> relations and functions and<br />
<strong>the</strong>se correspond in arities.<br />
Moreover a matrix is a rectangular array <strong>of</strong> elements from a field. If X = (X, . . .) and X ′ =<br />
(X ′ , . . .) are structures <strong>the</strong>n <strong>the</strong>ir product, denoted X ×X ′ , is <strong>the</strong> structure with carrier-set X ×X ′<br />
and where <strong>the</strong> functions are defined component-wise, and where a tuple is in a relation if both<br />
projections are in <strong>the</strong> corresponding relations. The k-th power <strong>of</strong> X , denoted X k , is <strong>the</strong> product <strong>of</strong><br />
X with it self k times.<br />
The definition <strong>of</strong> a first-order language suitable for a class <strong>of</strong> structures is assumed to be known.<br />
If φ is a sentence (formula without free variables), and A a suitable structure we write A |= φ for φ<br />
is true in A, (alternatively A satisfies φ).<br />
For mappings f we use f k to denote f composed with it self k times. We use p to denote prime<br />
powers. The definition <strong>of</strong> groups, rings and fields are assumed to be known. For each prime power<br />
p <strong>the</strong>re is a unique field, F p , <strong>of</strong> that order, see a text on algebra such as Herstein [Her75].<br />
2.1.3 p-automata<br />
We use <strong>the</strong> word effective to mean computable and not necessarily fast. J.R. Büchi [B¨60] observed<br />
that effective constructions on finite synchronous automata could be used to decide Presburger<br />
arithmetic, <strong>the</strong> set <strong>of</strong> sentences true <strong>of</strong> <strong>the</strong> structure (N, +). Central in J.R. Büchis observation<br />
is to view strings <strong>of</strong> tuples <strong>of</strong> <strong>the</strong> form {0, . . . , p − 1} n as elements <strong>of</strong> N n written in base p, and<br />
considering automata that have alphabets <strong>of</strong> <strong>the</strong> form {0, . . . , p − 1} n .<br />
Example This is an example <strong>of</strong> a 3-tape with entries from {0, 1} which can be viewed as <strong>the</strong><br />
binary representation <strong>of</strong> <strong>the</strong> tuple (7, 2, 9).<br />
⎡<br />
⎢<br />
⎣<br />
1<br />
0<br />
1<br />
⎤ ⎡<br />
⎥ ⎢<br />
⎦ ⎣<br />
1<br />
1<br />
0<br />
⎤ ⎡<br />
⎥ ⎢<br />
⎦ ⎣<br />
1<br />
0<br />
0<br />
⎤ ⎡<br />
⎥ ⎢<br />
⎦ ⎣<br />
0<br />
0<br />
1<br />
⎤ ⎡<br />
⎥ ⎢<br />
⎦ ⎣<br />
0<br />
0<br />
0<br />
⎤ ⎡<br />
⎥ ⎢<br />
⎦ ⎣<br />
0<br />
0<br />
0<br />
⎤<br />
⎥<br />
⎦<br />
Here <strong>the</strong> binary expansion <strong>of</strong> 7 is at <strong>the</strong> top row and <strong>the</strong> most significant end is to <strong>the</strong> right.<br />
Observe that one may add any number <strong>of</strong> zero-vectors at <strong>the</strong> most significant end <strong>of</strong> this tape,<br />
without changing <strong>the</strong> tuple <strong>of</strong> numbers it represents. See one <strong>of</strong> <strong>the</strong> survey articles <strong>of</strong> V.Bruyère<br />
et al. [BHMV94] or W.Thomas [Tho96] for more on this encoding <strong>of</strong> tuples <strong>of</strong> numbers for use<br />
with automata.<br />
In accordance with <strong>the</strong> article <strong>of</strong> V.Bruyère et al. [BHMV94], automata that work in base p are<br />
referred to as p-automata. The relations on natural numbers recognised by p-automata are called<br />
p-recognisable. We shall in <strong>the</strong> next subsection recall <strong>the</strong> Ehrenfeucht-Fraïssé method and discuss<br />
first-order definable properties <strong>of</strong> finite p-automata, for a fixed alphabet {0, . . . , p − 1} n . When<br />
being specific about n ∈ N here, we will call <strong>the</strong>se n-tape p-automata. By <strong>the</strong> Büchi-Bruyère<br />
38
<strong>the</strong>orem [BHMV94], <strong>the</strong> p-recognisable relations are exactly those relations we can define in firstorder<br />
language over <strong>the</strong> structure (N, +, | p ). Here x| p y is true when x is a power <strong>of</strong> p dividing y.<br />
We define automatic structure as a structure which can be interpreted in <strong>the</strong> structure (N, +, | p ),<br />
see A. Blumensath E. Grädel [BG04] <strong>the</strong>orem 4.5. An automatic model for a set <strong>of</strong> sentences is an<br />
automatic structure that is also a model for <strong>the</strong> sentences.<br />
Büchis procedure to decide a sentence in <strong>the</strong> <strong>the</strong>ory <strong>of</strong> (N, +, | p ), and thus Presburger arithmetic<br />
and <strong>the</strong> <strong>the</strong>ory <strong>of</strong> <strong>the</strong> o<strong>the</strong>r automatic structures, can be outlined as follows. It is an easy<br />
exercise to construct two automata; one 3-tape automaton for <strong>the</strong> summation relation + and one<br />
2-tape automaton for <strong>the</strong> | p relation. Using <strong>the</strong>se two automata one can, by manipulating rows in<br />
<strong>the</strong> alphabet, effectively construct automata that recognise <strong>the</strong> interpretation <strong>of</strong> each atomic formula<br />
<strong>of</strong> <strong>the</strong> <strong>the</strong>ory <strong>of</strong> (N, +, | p ). Similarity <strong>the</strong>re are constructions on automata for each <strong>of</strong> <strong>the</strong><br />
logical connectives so that we get an association between formulae and automata. The association is<br />
such that an automaton that corresponds to a formula recognises <strong>the</strong> interpretation <strong>of</strong> <strong>the</strong> formula<br />
in (N, +, | p ). To check whe<strong>the</strong>r a sentence is true in (N, +, | p ) one checks whe<strong>the</strong>r every reachable<br />
state in an associated automaton is a terminal state.<br />
It is well known that regular languages are closed under reversal <strong>of</strong> direction, so when working<br />
with automata for automatic structures one has to decide which end one reads tapes from. We chose<br />
to read tapes from <strong>the</strong> least significant end. Likewise one can chose to work with deterministic or<br />
<strong>non</strong>-deterministic automata. We chose to use deterministic automata.<br />
There is a number <strong>of</strong> ways for defining <strong>the</strong> class <strong>of</strong> automata, over a fixed alphabet, so as to<br />
make it a class <strong>of</strong> structures suitable for first-order language. We shall consider two ways.<br />
Firstly we provide a definition that is perhaps <strong>the</strong> most standard for deterministic automata.<br />
These automata are two-sorted. The equivalence that forms <strong>the</strong> sole proper axiom <strong>of</strong> <strong>the</strong> definition<br />
relates to <strong>the</strong> convention that words are fed to our automata with <strong>the</strong> least significant end first, and<br />
that adding zeros at <strong>the</strong> most significant end does not change <strong>the</strong> represented numbers.<br />
Definition Let n, p ∈ N. A two-sorted n-tape p-automaton with classical alphabet is a structure<br />
({0, . . . , p − 1} n , K, δ, ι, T ) with carrier-sets {0, . . . , p − 1} n and K where<br />
K is a set whose elements are called states,<br />
{0, . . . , p − 1} n is called <strong>the</strong> alphabet,<br />
δ : K × {0, . . . , p − 1} n → K is called <strong>the</strong> transition function,<br />
ι ∈ K is called <strong>the</strong> initial state,<br />
T ⊆ K is called <strong>the</strong> set <strong>of</strong> terminal states.<br />
Moreover T is invariant under adding zero-vectors at <strong>the</strong> most significant end, i.e.<br />
⎡ ⎤<br />
0<br />
0<br />
q ∈ T iff δ(q,<br />
⎢<br />
⎣ . ⎥<br />
) ∈ T .<br />
⎦<br />
0<br />
Secondly we provide a definition that is intended to be <strong>the</strong> same as above in every relevant respect<br />
except for <strong>the</strong> number <strong>of</strong> sorts. These automata are one-sorted and which means we can apply<br />
<strong>methods</strong> <strong>of</strong> finite model-<strong>the</strong>ory to <strong>the</strong>m directly.<br />
39
Definition Let n, p ∈ N and let I = {0, . . . , p − 1} n . A one-sorted n-tape p-automaton with<br />
classical alphabet is a structure<br />
(K, {δ i } i∈I , ι, T ) with carrier-set K where<br />
K is a set whose elements are called states,<br />
For each symbol i <strong>of</strong> <strong>the</strong> alphabet {0, . . . , p − 1} n <strong>the</strong> mapping<br />
δ i : K → K is called <strong>the</strong> i-transition.<br />
ι ∈ K is called <strong>the</strong> initial state,<br />
T ⊆ K is called <strong>the</strong> set <strong>of</strong> terminal states.<br />
Moreover <strong>the</strong> following axiom holds, with <strong>the</strong> zero-vector transposed for typographical reasons,<br />
q ∈ T iff δ (0,0,...,0) (q) ∈ T .<br />
We now define reachability, <strong>the</strong> central relation <strong>of</strong> <strong>the</strong> present paper.<br />
Definition Let (K, {δ i } i∈I , ι, T ) be an automaton. The reachability relation is <strong>the</strong> ⊆-minimal<br />
relation, R ⊆ K × K, such that:<br />
If <strong>the</strong>re is an i ∈ I such that δ i (q) = q ′ <strong>the</strong>n (q, q ′ ) ∈ R.<br />
If (q, q ′ ) ∈ R and i ∈ I <strong>the</strong>n (q, δ i (q ′ )) ∈ R.<br />
2.1.4 The Ehrenfeucht-Fraïssé method<br />
The Ehrenfeucht-Fraïssé method <strong>of</strong> finite model-<strong>the</strong>ory is based on an application <strong>of</strong> one <strong>of</strong> two<br />
closely related <strong>the</strong>orems, one due to Fraïssé and ano<strong>the</strong>r to Ehrenfeucht. Which <strong>of</strong> <strong>the</strong>se <strong>the</strong>orems<br />
one uses is a mater <strong>of</strong> taste, in <strong>the</strong> setting <strong>of</strong> first-order logic.<br />
We use Fraïssés <strong>the</strong>orem, and we briefly recall <strong>the</strong> Ehrenfeucht-Fraïssé method to show that<br />
<strong>the</strong> reachability relation is not first-order definable over finite automata in general. This is for<br />
clarification as <strong>the</strong> present paper is about a class <strong>of</strong> automata where <strong>the</strong> reachability relation is<br />
indeed first-order definable.<br />
We shall, for this section alone, use <strong>the</strong> terms quantifier rank and m-isomorphism. For precise<br />
definitions <strong>of</strong> <strong>the</strong>se see a text on finite model-<strong>the</strong>ory such as Ebbinghaus and Flum [EF06]. The<br />
quantifier rank <strong>of</strong> a formula is a natural number that serves as a measurement <strong>of</strong> <strong>the</strong> complexity <strong>of</strong><br />
<strong>the</strong> formula with respect to occurrences <strong>of</strong> quantifiers. The relation <strong>of</strong> m-isomorphism is a weak<br />
form <strong>of</strong> isomorphism between pairs <strong>of</strong> structures. We write ∼ = m for m-isomorphism. The existence<br />
<strong>of</strong> an m-isomorphism is a sufficient condition for a pair <strong>of</strong> structures to satisfy <strong>the</strong> same sentences<br />
<strong>of</strong> quantifier rank m. If <strong>the</strong> language and structures in question are purely relational it is also a<br />
necessary condition. See e.g. Ebbinghaus and Flum [EF06] definition 2.3.1 and corollary 2.3.4.<br />
We now state Fraïssé’s <strong>the</strong>orem in <strong>the</strong> sufficiency direction. It is usually proven for purely<br />
relational sentences, but in <strong>the</strong> sufficiency direction it is easily generalised to full first-order language.<br />
Theorem 2.1.1 (Fraïssé) If φ is a first-order sentence, with or without constant- and function-symbols,<br />
<strong>the</strong>n <strong>the</strong>re exists a k ∈ N, namely <strong>the</strong> quantifier rank <strong>of</strong> φ, such that for all structures A, B we have; if<br />
both A |= φ and A ∼ = k B <strong>the</strong>n B |= φ.<br />
40
Using this <strong>the</strong>orem contrapositively we can, by for each k ∈ N showing <strong>the</strong> existence <strong>of</strong> a pair<br />
<strong>of</strong> structures A ∼ = k B where A has a given property that B does not, conclude that this is not a<br />
first-order property. This remains a valid way <strong>of</strong> reasoning when we restrict attention to a sub-class<br />
<strong>of</strong> all structures, such as <strong>the</strong> class <strong>of</strong> finite structures.<br />
Proposition 2.1.2 The reachability relation in one-sorted n-tape p-automata with classical alphabet<br />
is not first-order definable.<br />
Pro<strong>of</strong>: We prove this in three steps. In <strong>the</strong> first two steps we show that connectedness is not<br />
definable. This is done pretty much as with graphs, see a finite model-<strong>the</strong>ory text such as [EF06]<br />
for details.<br />
Firstly we show that <strong>the</strong> property “having an even number <strong>of</strong> states” is not first-order definable.<br />
Call an automaton trivial if each δ i is <strong>the</strong> identity mapping and T = ∅. For each k ∈ N let A k be<br />
<strong>the</strong> trivial automaton with 2k states and let B k be <strong>the</strong> trivial automaton with 2k +1 states. For each<br />
k we now have that A k has and even number <strong>of</strong> states and that A k<br />
∼ =k B k but B k does not have an<br />
even number <strong>of</strong> states. By Fraïssés (or Ehrenfeuchts) <strong>the</strong>orem we conclude that <strong>the</strong> automata with<br />
an even number <strong>of</strong> states is not first-order definable (over finite structures).<br />
Secondly one assumes that connectedness is first-order definable and shows that having an even<br />
number <strong>of</strong> states <strong>the</strong>n becomes first-order definable. This is in contradiction with <strong>the</strong> first step so<br />
connectedness is not first-order definable.<br />
Thirdly assume for <strong>the</strong> purpose <strong>of</strong> arriving at a contradiction that <strong>the</strong> reachability relation, R,<br />
is first-order definable. We can <strong>the</strong>n define connectedness as follows ∀q∀q ′ R(q, q ′ ). This is in<br />
contradiction to <strong>the</strong> second step, <strong>the</strong>refore reachability is not first-order definable over <strong>the</strong> finite<br />
automata.<br />
qed<br />
2.1.5 Mappings from and to a finite set<br />
Lemma 2.1.4, soon to follow, is central in various pr<strong>of</strong>s in <strong>the</strong> present paper. The lemma is on<br />
mappings from and to a finite set. The collection <strong>of</strong> such mappings is well known to form a<br />
semigroup under composition. As one may expect <strong>the</strong>n, <strong>the</strong> lemma follows fairly immediately<br />
from a result known to semigroup-<strong>the</strong>orists. See J-E.Pin [Pin97] <strong>the</strong> section on idempotents. The<br />
role <strong>of</strong> <strong>the</strong> lemma in <strong>the</strong> present paper justifies a pro<strong>of</strong>.<br />
Lemma 2.1.3 If X is a finite set with a point α ∈ X and a function f : X → X. Then <strong>the</strong>re exist<br />
numbers a > 0 and b > 0 s.t. for all l, m ∈ N we have f m+b (α) = f a·l+m+b (α).<br />
Pro<strong>of</strong>: Intuitively <strong>the</strong> successive application <strong>of</strong> f will bend in on it self at a point we refer to as<br />
f b (α). By bending in we mean that <strong>the</strong>re is an a > 0 such that f a+b (α) = f b (α). The number<br />
a is here <strong>the</strong> length <strong>of</strong> a cycle we may repeat. We may <strong>the</strong>refore without loss <strong>of</strong> generality assume<br />
that 0 < b. By using <strong>the</strong> cycle we get <strong>the</strong> property that for all l ∈ N we have f a·l+b (α) = f b (α).<br />
We may also jump into <strong>the</strong> cycle any number m <strong>of</strong> successions and do a-cycles from <strong>the</strong>re so, for<br />
all l and m we have f a·l+b+m (α) = f b+m (α).<br />
qed<br />
Lemma 2.1.4 If X is a finite set with an α ∈ X and a function f : X → X, <strong>the</strong>n <strong>the</strong>re exists a<br />
k ∈ N s.t. 0 < k and f k (α) = f 2·k (α).<br />
41
Pro<strong>of</strong>: Use a and b from <strong>the</strong> lemma above. Let l = b and m = a · b − b and k = a · b. Note that<br />
from <strong>the</strong> conditions that 0 < a and 0 < b we get 0 < k. Moreover:<br />
f k (α) = f a·b (α) by definition <strong>of</strong> k,<br />
= f a·b−b+b (α) = f m+b (α) by basic arithmetic,<br />
= f m+b (α) = f a·l+m+b (α) by <strong>the</strong> lemma,<br />
= f a·b+(a·b−b)+b (α) by definition <strong>of</strong> m and n,<br />
= f a·b+a·b (α) = f 2·k by basic arithmetic and <strong>the</strong> definition <strong>of</strong> k.<br />
2.1.6 Vector-spaces over finite fields<br />
The definition <strong>of</strong> a vector-space over a finite field is a ready made, finite and first-order definition<br />
<strong>of</strong> <strong>the</strong> notion <strong>of</strong> a set <strong>of</strong> tuples with entries from a finite set. When <strong>the</strong> vector-space has dimension<br />
m ∈ N, it is necessarily <strong>of</strong> <strong>the</strong> same form as <strong>the</strong> set <strong>of</strong> m-tuples with entries from <strong>the</strong> underlying<br />
finite field. These vector-spaces can accordingly be assembled to n-tapes with entries from <strong>the</strong><br />
underlying field, for a chosen n.<br />
A one-sorted definition <strong>of</strong> vector space over <strong>the</strong> field <strong>of</strong> integers modulo two follows.<br />
Definition A vector space V over <strong>the</strong> minimal field is a tuple (V, +, −, 0, 0, 1) where V is a set, +<br />
is a binary operation, −, 0, 1 are unary operations and where 0 is a constant. Moreover V is<br />
1. an abelian group, i.e.<br />
V |= ∀xyz[(x + y) + z = x + (y + z)],<br />
V |= ∀x[x + 0 = x],<br />
V |= ∀x[x + −(x) = 0],<br />
V |= ∀xy[x + y = y + x],<br />
2. left distributive, i.e.<br />
V |= ∀xy[0(x + y) = 0(x) + 0(y)],<br />
V |= ∀xy[1(x + y) = 1(x) + 1(y)],<br />
3. right distributive, i.e.<br />
V |= ∀x[0(x) = 0(x) + 0(x)],<br />
V |= ∀x[0(x) = 1(x) + 1(x)],<br />
V |= ∀x[1(x) = 0(x) + 1(x)],<br />
V |= ∀x[1(x) = 1(x) + 0(x)],<br />
qed<br />
42
4. associative, i.e.<br />
V |= ∀x[0(0(x)) = 0(x)],<br />
V |= ∀x[0(1(x)) = 0(x)],<br />
V |= ∀x[1(0(x)) = 0(x)],<br />
V |= ∀x[1(1(x)) = 1(x)],<br />
5. in possession <strong>of</strong> a unit, i.e.<br />
V |= ∀x[1(x) = x],<br />
V |= ∀x[0(x) = 0]. (*)<br />
The axiom with <strong>the</strong> (∗) is dependent i.e. it follows from <strong>the</strong> rest <strong>of</strong> <strong>the</strong> definition, in particular <strong>the</strong><br />
axiom 0(x) = 0(x) + 0(x). Mappings between vector-spaces over <strong>the</strong> same field that preserve <strong>the</strong><br />
vector-space operations are called linear transformations. If <strong>the</strong>re is a bijective linear transformation<br />
between two vector-spaces <strong>the</strong>y are said to be <strong>of</strong> <strong>the</strong> same form, or isomorphic. It is known that<br />
<strong>the</strong>re is a minimal field. It has two elements. Also all minimal fields are isomorphic, <strong>the</strong>refore we<br />
say <strong>the</strong> minimal field and write F 2 .<br />
Lemma 2.1.5 Every finite vector space over F 2 is <strong>of</strong> <strong>the</strong> form F 2 × · · · × F 2<br />
Pro<strong>of</strong>: It is known that finite dimensional vector-spaces in general are <strong>of</strong> this form. See an algebra<br />
text such as Herstein [Her75].<br />
qed<br />
We write h(V) for <strong>the</strong> image <strong>of</strong> a vector-space V under a linear transformation h.<br />
Lemma 2.1.6 If V and h are as as above <strong>the</strong>n h(V) is a vector-space.<br />
Pro<strong>of</strong>: see [Her75].<br />
qed<br />
2.2 n-fold vector spaces<br />
We introduce vector spaces with two additional operations. The operations and <strong>the</strong>ir accompanying<br />
axioms allow us to view <strong>the</strong> elements <strong>of</strong> <strong>the</strong> vector-spaces as n × m matrices with entries from<br />
<strong>the</strong> underlying field. These matrices will serve as symbols <strong>of</strong> alphabets for synchronous n-tape<br />
automata.<br />
43
2.2.1 3-fold vector spaces over <strong>the</strong> minimal field<br />
We begin with an axiomatisation <strong>of</strong> vector-spaces which can be assembled to 3-tapes with entries<br />
from {0, 1}. Later we do this for n-tapes with entries from larger sets.<br />
Definition A 3-fold vector space over F 2 is a tuple V = (V ′ , π, r) where<br />
V ′ = (V, +, −, 0, 0, 1) is a vector space over F 2<br />
π : V → V called <strong>the</strong> projection,<br />
r : V → V called <strong>the</strong> rotation.<br />
Moreover<br />
1. π is a linear transformation, i.e.<br />
V |= [π(0) = 0], (*)<br />
V |= ∀x[π(−(x)) = −(π(x))], (*)<br />
V |= ∀xy[π(x + y) = π(x) + π(y)],<br />
V |= ∀x[π(0(x)) = 0(π(x))],<br />
V |= ∀x[π(1(x)) = 1(π(x)))],<br />
2. r is a linear transformation, i.e.<br />
V |= ∀x[r(0) = 0], (*)<br />
V |= ∀x[r(−(x)) = −(r(x))], (*)<br />
V |= ∀xy[r(x + y) = r(x) + r(y)],<br />
V |= ∀x[r(0(x)) = 0(r(x))],<br />
V |= ∀x[r(1(x)) = 1(r(x))],<br />
3. r 3 is <strong>the</strong> identity mapping, i.e.<br />
V |= ∀x[r(r(r(x))) = x],<br />
4. π is idempotent, i.e.<br />
V |= ∀x[π(π(x)) = π(x)],<br />
5. V is <strong>the</strong> sum <strong>of</strong> 3 isomorphic copies <strong>of</strong> π(V), i.e.<br />
V |= ∀x[π(r(r(r(x)))) + r(π(r(r(x)))) + r(r(π(r(x)))) = x].<br />
44
Again <strong>the</strong> axioms with a (*) behind <strong>the</strong>m are dependent, which means <strong>the</strong>y follow from those that<br />
don’t have a (*), see [Her75] lemma 2.7.2.<br />
Example Consider <strong>the</strong> set <strong>of</strong> 3 × 5 matrices whose entries are from F 2 . Following convention<br />
this means an array <strong>of</strong> 3 rows and 5 columns. Letting 0 and 1 denote <strong>the</strong> elements <strong>of</strong> F 2 , <strong>the</strong> entries<br />
<strong>of</strong> <strong>the</strong> matrix are 0’s and 1’s. The top row and leftmost column have lowest index. It is clearly a<br />
vector-space when <strong>the</strong> vector-space operations are defined entry-wise. It is a 3-fold vector-space<br />
when r is <strong>the</strong> operation that moves <strong>the</strong> top row to <strong>the</strong> bottom <strong>of</strong> a matrix, and when π preserves<br />
<strong>the</strong> top row and replaces every entry <strong>of</strong>f <strong>the</strong> top row with a 0.<br />
2.2.2 n-fold vector spaces over a given finite field<br />
Using <strong>the</strong> fact that for each prime power p <strong>the</strong>re exists a unique field, F p , <strong>of</strong> that order, <strong>the</strong> axiomatisation<br />
<strong>of</strong> 3-fold vector spaces over <strong>the</strong> minimal field ,F 2 , can clearly be generalised to 3-fold<br />
vector spaces over each F p . Likewise we may generalise from 3-fold to n-fold as follows.<br />
Definition Let n ∈ N. An n-fold vector-space over F p is a tuple V = (V ′ , π, r) where<br />
V ′ = (V, +, −, 0, 0, 1, . . .) is a vector space over F p ,<br />
π : V → V is <strong>the</strong> projection,<br />
r : V → V is <strong>the</strong> rotation.<br />
Moreover<br />
1. π is a linear transformation,<br />
2. r is a linear transformation,<br />
3. r n is <strong>the</strong> identity mapping, i.e<br />
V |= ∀x[r n (x) = x]<br />
4. π is idempotent, i.e.<br />
V |= ∀x[π(π(x)) = π(x)]<br />
5. V is <strong>the</strong> sum <strong>of</strong> n isomorphic copies <strong>of</strong> π(V), i.e.<br />
V |= ∀x[π(r n (x)) + r(π(r n−1 (x))) + · · · + r n−1 (π(r(x))) = x]<br />
We recall a result on vector-spaces, n-fold or not.<br />
Lemma 2.2.1 Every finite vector-space over F p is <strong>of</strong> <strong>the</strong> form F p × · · · × F p .<br />
Pro<strong>of</strong>: See an algebra text such as Herstein [Her75].<br />
qed<br />
Definition A linear transformation between two n-fold vector-spaces is said to be an n-fold linear<br />
transformation if in addition to <strong>the</strong> vector-space operations it preserves π and r.<br />
45
As usual we will say that two n-fold vector-spaces are isomorphic (as n-fold vector-spaces) if<br />
<strong>the</strong>re is a bijection between <strong>the</strong>m that is also an n-fold linear transformation.<br />
Definition We write 0 n×m for <strong>the</strong> n × m matrix whose entries are all zero. Also write O n×m for<br />
<strong>the</strong> unique n-fold vector-space whose sole element is 0 n×m<br />
Definition M p (n, m) is <strong>the</strong> n-fold vector-space <strong>of</strong> n × m matrices where <strong>the</strong> vector-space operations<br />
are defined entry-wise and where for rows A i in F m p we have<br />
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤<br />
A 0 A 0<br />
A 0 A 1<br />
A 1<br />
π(<br />
⎢<br />
⎣ . ⎥<br />
) =<br />
0 1×m<br />
A 1<br />
and r(<br />
⎢<br />
⎦ ⎣ . ⎥ ⎢<br />
⎦ ⎣ . ⎥<br />
) =<br />
.<br />
⎢<br />
⎦ ⎣ A ⎥ n−1 ⎦<br />
A n−1 0 1×m A n−1 A 0<br />
Lemma 2.2.2 Let (V, π, r) be a finite n-fold vector-space over F p . When m is greater than or<br />
equal to <strong>the</strong> dimension <strong>of</strong> <strong>the</strong> vector-space π(V) <strong>the</strong>n <strong>the</strong>re exists a n-fold linear transformation h from<br />
M p (n, m) to (V, π, r) that is onto, i.e. surjective.<br />
Pro<strong>of</strong>: Since <strong>the</strong> dimension <strong>of</strong> π(V) is less than or equal to m <strong>the</strong>re exists a surjective linear<br />
transformation from F m p to π(V). From this we conclude that <strong>the</strong>re is a mapping h 0 from matrices<br />
⎡ ⎤<br />
A 0<br />
0 1×m<br />
<strong>of</strong> <strong>the</strong> form<br />
onto π(V). Write M for this sub-vector-space <strong>of</strong> M<br />
⎢<br />
⎣ . ⎥<br />
p (n, m). Let r ′ denote<br />
⎦<br />
0 1×m<br />
<strong>the</strong> operation <strong>of</strong> M p (n, m) that corresponds to r in <strong>the</strong> lemma.<br />
Let h 0 map M onto π(r n (V)).<br />
Let h 1 map r ′ (M) onto r(π(r n−1 (V))).<br />
.<br />
Let h n−1 map r ′n−1 (M)) onto r n−1 (π(r(V))).<br />
Now every n × m matrix is <strong>the</strong> sum <strong>of</strong> n matrices that are <strong>non</strong>-zero in at most one row so <strong>the</strong>re is<br />
an isomorphism f from M p (n, m) to <strong>the</strong> vector-space M×r ′ (M)×· · ·×r ′n−1 (M)). Using this<br />
isomorphism we easily turn this product into an n-fold vector-space in such a way that f becomes<br />
an n-fold linear transformation. Now h 0 + h 1 + · · · + h n−1 is an n-fold linear transformation from<br />
M × r ′ (M) × · · · × r ′n−1 (M)) to π(r n (V)) + r(π(r n−1 (V))) + · · · + r n−1 (π(r(V))). This<br />
mapping is onto by <strong>the</strong> axiom for n-fold vector-spaces which reads: π(r n (x)) + r(π(r n−1 (x))) +<br />
· · · + r n−1 (π(r(x))) = x. Therefore (h 0 + h 1 + · · · + h n−1 ) ◦ f is <strong>the</strong> desired surjective n-fold<br />
linear transformation.<br />
qed<br />
We add <strong>the</strong> following slightly less general <strong>version</strong> <strong>of</strong> <strong>the</strong> above.<br />
Lemma 2.2.3 Let V = (V ′ , π, r) be a finite n-fold vector-space over F p such that elements <strong>of</strong> V are<br />
determined by <strong>the</strong>ir projections, i.e.,<br />
V |= π(r n (x)) = π(r n (y))∧r(π(r n−1 (x)) = r(π(r n−1 (y))∧· · ·∧r n−1 (π(r(x)) = r n−1 (π(r(y)) →<br />
x = y.<br />
46
Let m be equal to <strong>the</strong> dimension <strong>of</strong> <strong>the</strong> vector-space π(V). Then V is isomorphic to M p (n, m).<br />
Pro<strong>of</strong>: Imediate from <strong>the</strong> lemma 2.2.2 and <strong>the</strong> statement that elements <strong>of</strong> V are determied by<br />
<strong>the</strong>ir projections.<br />
qed<br />
2.3 p-automata with abstract alphabets<br />
We now define p-automata using possibly abstract n-fold vector-spaces as alphabets. The p-<br />
automata <strong>of</strong> section 2.1.3 now become a special case with concrete vector-spaces M p (n, 1) as<br />
alphabets. In light <strong>of</strong> lemmas 2.1.5 and 2.2.1 we have a sense in which this is isomorphic.<br />
By lemma 2.2.2 we can view <strong>the</strong> elements <strong>of</strong> an n-fold vector-space over F p as a set <strong>of</strong> n × m<br />
matrices where n · m is <strong>the</strong> dimension. We shall define n-tapes on which automata operate by<br />
concatenating such matrices.<br />
Definition Let A and B be n × m matrices over F p . Then A ⌢ B denotes <strong>the</strong> n × (2 · m) matrix<br />
obtained by concatenating A and B. Likewise if A 0 ⌢ · · · ⌢A k−1 is a n × (k · m) matrix and B is<br />
as before <strong>the</strong>n A 0 , ⌢ · · · ⌢A k−1 ⌢ B is <strong>the</strong> n × ((k + 1) · m) matrix obtained by concatenation.<br />
We now define <strong>the</strong> set <strong>of</strong> n-tapes as a set <strong>of</strong> matrices closed under concatenation.<br />
Definition Let n ∈ N. Then for each m ∈ N a n × m matrix with entries from F p is called an<br />
n-symbol or an n-tape depending on <strong>the</strong> context. In a context where M p (n, m) is <strong>the</strong> set <strong>of</strong> symbols<br />
<strong>the</strong> set <strong>of</strong> n-tapes is <strong>the</strong> set ⋃ k∈N M p (n, k · m) for which we write M p (n, m) ∗ .<br />
2.3.1 n-tape p-automata<br />
We give a two-sorted definition <strong>of</strong> p-automata with abstract alphabets.<br />
Definition An n-tape p-automaton is a structure (V, K, δ, ι, T ) where<br />
V = (V, +, −, 0, . . .) is an n-fold vector-space over F p , V is called <strong>the</strong> alphabet,<br />
K is a set whose elements are called states,<br />
δ : K × V → K is called <strong>the</strong> transition function,<br />
T ⊆ K is called <strong>the</strong> set <strong>of</strong> terminal states,<br />
ι ∈ K is called <strong>the</strong> initial state,<br />
moreover <strong>the</strong> following axiom holds<br />
q ∈ T iff δ(q, 0) ∈ T .<br />
To see how <strong>the</strong>se relate to <strong>the</strong> classical p-automata as in e.g. V.Bruyère et al. [BHMV94] or<br />
W.Thomas [Tho96], note that <strong>the</strong>se would be n-tape p-automata with <strong>the</strong> concrete vector-space<br />
M p (n, 1) as alphabet.<br />
We extend <strong>the</strong> transition function to map from tapes to states, ra<strong>the</strong>r than symbols to states.<br />
We give a slightly more abstract <strong>version</strong> than <strong>the</strong> standard in automata <strong>the</strong>ory. Symbols are in<br />
now n × m matrices for some m depending on <strong>the</strong> size <strong>of</strong> <strong>the</strong> carrier-set. So for each k ∈ N an<br />
n × (k · m) matrix is a tape.<br />
47
Definition Let (V, K, δ, ι, T ) be an n-tape p-automaton, where V is an n-fold vector-space over<br />
F p . Let h be an n-fold linear transformation from M p (n, m) onto V. Then δ ∗ h : K×M p (n, m) ∗ →<br />
K is defined by recursion as follows:<br />
δ ∗ h(q, 0 n×0 ) = q, here 0 n×0 is <strong>the</strong> unique n × 0 matrix, i.e. <strong>the</strong> tape <strong>of</strong> length 0,<br />
δ ∗ h(q, B ⌢ A) = δ(δ ∗ h(q, B), h(A)), here B ∈ M p (n, m) ∗ and A ∈ M p (n, m).<br />
2.3.2 Infinite tapes with finite support<br />
Here we define infinite n-tapes with finite support and what it means for an automaton to accept<br />
such a tape. We also prove a lemma which eventually provides us with a sense in which automata<br />
with slightly different alphabets can be said to accept <strong>the</strong> same finitely supported n-tapes.<br />
Definition 0 n×N denotes <strong>the</strong> n×N matrix whose entries are all zero, and O n×N <strong>the</strong> unique n-fold<br />
vector-space whose carrier set consists <strong>of</strong> 0 n×N .<br />
Definition Let n, m ∈ N be positive. Then M p (n, m) ∗ ⌢ 0 n×N denotes <strong>the</strong> set <strong>of</strong> infinite n-<br />
tapes with m-support. When A is an element <strong>of</strong> M p (n, m) ∗ , <strong>the</strong>n A ⌢ 0 n×N denotes an element<br />
M p (n, m) ∗ ⌢ O n×N .<br />
It is left to <strong>the</strong> reader to verify that <strong>the</strong> following defines an n-fold vector-space.<br />
Definition Let n, m ∈ N be positive. Then M p (n, m) ∗ ⌢ O n×N denotes <strong>the</strong> n-fold vector-space<br />
((M p (n, m) ∗ ⌢ 0 n×N , +, −, 0, . . .), π, r), where <strong>the</strong> vector-space operations are defined<br />
component-wise and where π and r operate on rows.<br />
Lemma 2.3.1 If m, m ′ ∈ N are positive <strong>the</strong>n M p (n, m) ∗ ⌢ O n×N and M p (n, m ′ ) ∗ ⌢ O n×N are<br />
isomorphic as n-fold vector-spaces.<br />
Pro<strong>of</strong>: The dimension <strong>of</strong> M p (n, m) m′ is n · m · m ′ . Likewise <strong>the</strong> dimension <strong>of</strong><br />
M p (n, m ′ ) m is n · m ′ · m. By lemma 2.2.3 we may <strong>the</strong>refore conclude that <strong>the</strong> mentioned n-<br />
fold vector-spaces are isomorphic. In <strong>the</strong> same manner we conclude that <strong>the</strong>re are isomorphisms<br />
between M p (n, m) k·m′ and M p (n, m ′ ) k·m for each k ∈ N. Being isomorphic <strong>the</strong>y are for each k<br />
<strong>the</strong> same set <strong>of</strong> matrices. For this pro<strong>of</strong> only, write X k for this set. The sought after isomorphism<br />
is obtained by taking <strong>the</strong> colimit <strong>of</strong> <strong>the</strong>se isomorphisms under <strong>the</strong> inclusions A ↦→ A ⌢ 0 n×(m·m ′ ),<br />
where A ∈ X k and A ⌢ 0 n×(m·m ′ ) ∈ X k+1<br />
qed<br />
By <strong>the</strong> last lemma <strong>the</strong> following is well-defined.<br />
Definition For p, n ∈ N <strong>the</strong> finitely supported n-tapes (over F p ) is <strong>the</strong> vector-space<br />
M p (n, m) ∗ ⌢ O n×N for any positive m ∈ N.<br />
The following is well defined since we have defined our automata in such a way that <strong>the</strong> terminal<br />
states T are invariant under adding zero-vectors at <strong>the</strong> most significant end <strong>of</strong> tapes.<br />
Definition Let W = (V, K, δ, ι, T ) be an n-tape p-automaton and h an n-fold linear transformation<br />
from M p (n, m) onto <strong>the</strong> alphabet V. We say that <strong>the</strong> pair (W, h) accepts A ⌢ 0 n×N if<br />
A ∈ M p (n, m) and δ ∗ h(ι, A) ∈ T .<br />
48
Definition Let W = (V, K, δ, ι, T ) be an n-tape p-automaton and h an n-fold linear transformation<br />
from M p (n, m) onto <strong>the</strong> alphabet V. Let L ⊆ M p (n, m) ∗ ⌢ 0 n×N . We say that <strong>the</strong> pair<br />
(W, h) recognises L if L is <strong>the</strong> set <strong>of</strong> A ⌢ 0 n×N accepted by (W, h).<br />
It should be clear that for positive m, n ∈ N <strong>the</strong>re is a one-to-one correspondence f : N n →<br />
M p (n, m) ∗ ⌢ 0 n×N . Using this we also say that (W, h) recognises L ⊆ N n if (W, h) recognises<br />
f(L) ⊆ M p (n, m) ∗ ⌢ 0 n×N .<br />
2.4 Transitive automata and reachability in general<br />
In this section transitive automata are introduced. We look at one- and two-sorted variants. Twosorted<br />
automata appear easier to work with. For given n, p ∈ N <strong>the</strong> class <strong>of</strong> one-sorted transitive<br />
n-tape p-automata is formally axiomatised by a finite set <strong>of</strong> first-order sentences. In one-sorted<br />
transitive n-tape p-automata <strong>the</strong> reachability relation is definable using a first-order formula.<br />
First we define a notion <strong>of</strong> equivalence with <strong>the</strong> property that every automaton is equivalent to<br />
a transitive automaton. Equivalent automata accept <strong>the</strong> same finitely supported tapes.<br />
2.4.1 A not quite classical notion <strong>of</strong> equivalence<br />
Classically two automata are equivalent if <strong>the</strong>y have <strong>the</strong> same alphabet, and <strong>the</strong>y accept <strong>the</strong> same<br />
tapes. We introduce a slightly looser notion <strong>of</strong> equivalence where <strong>the</strong> alphabets aren’t required<br />
to be exactly <strong>the</strong> same. The alphabets <strong>of</strong> classical automata are <strong>of</strong> <strong>the</strong> form M p (n, 1). We allow<br />
for comparison where one automaton has an alphabet <strong>of</strong> <strong>the</strong> form M p (n, m) and <strong>the</strong> o<strong>the</strong>r an<br />
alphabet <strong>of</strong> <strong>the</strong> form M p (n, m ′ ) where m, m ′ ∈ N are possibly different. By lemma 2.3.1 it is still<br />
meaningful to talk about recognising <strong>the</strong> same finitely supported tapes.<br />
We shall only be concerned with equivalence <strong>of</strong> automata with concrete alphabets, i.e. <strong>the</strong><br />
alphabets <strong>of</strong> <strong>the</strong> form M p (n, m). These are a sub-class <strong>of</strong> <strong>the</strong> class <strong>of</strong> n-tape p-automata. To make<br />
<strong>the</strong> concreteness explicit consider pairs (W, h) where h is an n-fold linear transformation from a<br />
concrete n-fold vector-space to <strong>the</strong> possibly abstract alphabet <strong>of</strong> W.<br />
Definition Let W = (V, K, δ, ι, T ) and W ′ = (V ′ , K ′ , δ ′ , ι ′ , T ′ ) be two n-tape p-automata. Let<br />
m, m ′ ∈ N and h, h ′ be two n-fold linear transformations:<br />
h from M p (n, m) onto V and<br />
h ′ from M p (n, m ′ ) onto V ′<br />
Then (W, h) and (W ′ , h ′ ) are equivalent provided that for tapes A <strong>of</strong> M p (n, 1) ∗ we have δh(ι, ∗ A) ∈<br />
T if and only if δ ′ ∗<br />
h ′(ι ′ , A) ∈ T ′ whenever δh ∗ and δ ′ ∗<br />
h ′ are both defined.<br />
Note that in <strong>the</strong> definition δ ∗ h and δ ′ ∗<br />
h ′ are both defined on M p (n, k) whenever k is a common<br />
multiple <strong>of</strong> m and m ′ .<br />
Proposition 2.4.1 If (W, h) and (W ′ , h ′ ) are equivalent <strong>the</strong>n <strong>the</strong>y accept <strong>the</strong> same finitely supported<br />
n-tapes.<br />
Pro<strong>of</strong>: Left to <strong>the</strong> reader.<br />
qed<br />
49
2.4.2 Transitive p-automata, <strong>the</strong> two-sorted case<br />
We single out a class <strong>of</strong> concrete n-tape p-automata in which a state is reachable from ano<strong>the</strong>r iff<br />
it is reachable in exactly one step. This makes reachability first-order definable quite trivially. Less<br />
trivially <strong>the</strong> class turns out to suffice for deciding Presburger arithmetic and <strong>the</strong> <strong>the</strong>ories <strong>of</strong> <strong>the</strong> o<strong>the</strong>r<br />
automatic structures.<br />
For <strong>the</strong> rest <strong>of</strong> this section we fixate n, p ∈ N where p is a prime power. Let W = (V, K, δ, ι, T )<br />
be a finite n-tape p-automaton. Moreover let m ∈ N and let h be an n-fold linear transformation<br />
from M p (n, m) onto V.<br />
Definition Let W and h be as above. Let k ∈ N. Then <strong>the</strong> k-outreach <strong>of</strong> (W, h) is <strong>the</strong> <strong>the</strong><br />
pair (W ′ , h ′ ) where W ′ = (M p (n, m · k), K, δ ′ , ι, T ). Here δ ′ is <strong>the</strong> restriction <strong>of</strong> δ ∗ h to K ×<br />
M p (n, m · k) and h ′ : M p (n, m · k) → M p (n, m · k) is <strong>the</strong> identity mapping and thus an n-fold<br />
linear transformation.<br />
Note that <strong>the</strong> k-outreach (W ′ , h ′ ) <strong>of</strong> (W, h) is a finite automaton as long as W is.<br />
Proposition 2.4.2 Let W and h be as above. Let k ∈ N. Then <strong>the</strong> k-outreach (W ′ , h ′ ) <strong>of</strong> (W, h) is<br />
equivalent to (W, h).<br />
Pro<strong>of</strong>: Observe that δ ′ and h ′ are defined by means <strong>of</strong> δh ∗ in such a way that δh(q, ∗ A) and δ ′ ∗<br />
h ′(q, A)<br />
are both defined and elements <strong>of</strong> T whenever δ ′ ∗<br />
h ′(q, A) is defined.<br />
qed<br />
We now define a notion <strong>of</strong> reachability that depends on an n-fold linear transformation from a<br />
concrete n-fold vector-space to <strong>the</strong> alphabet <strong>of</strong> an automaton.<br />
Definition Let W and h be as above. Let q, q ′ ∈ K be two states. Then q ′ is h-reachable from q if<br />
<strong>the</strong>re is a k ∈ N and a matrix A <strong>of</strong> M p (n, k · m) such that δ ∗ h(q, A) = q ′ .<br />
We now define <strong>the</strong> notion that makes reachability first-order definable.<br />
Definition Let W and h be as above. Then (W, h) is said to be transitive if<br />
for all states q ∈ K and tapes A, B ∈ M p (n, m)<br />
<strong>the</strong>re exists a tape C ∈ M p (n, m) such that<br />
δ ∗ h(q, A ⌢ B) = δ ∗ h(q, C).<br />
Now <strong>the</strong> main result on two-sorted transitive automata.<br />
Proposition 2.4.3 Let W and h be as above. Then <strong>the</strong>re exists a k ∈ N such that <strong>the</strong> k-outreach<br />
(W ′ , h ′ ) <strong>of</strong> (W, h) is transitive. This implies that if a state is h ′ -reachable from ano<strong>the</strong>r, <strong>the</strong>n it is<br />
h ′ -reachable in exactly one step.<br />
Pro<strong>of</strong>: To find <strong>the</strong> sought k we shall use lemma 2.1.4. Consider <strong>the</strong>refore X = P(K) K , <strong>the</strong> set <strong>of</strong><br />
mappings from <strong>the</strong> set <strong>of</strong> states to <strong>the</strong> power-set <strong>of</strong> <strong>the</strong> set <strong>of</strong> states. Define α ∈ X by α(q) = {q}.<br />
We now define a function f : X → X that, when beginning with α and iterating, we can use to<br />
keep track <strong>of</strong> <strong>the</strong> states reachable from a given state q. This is to say that [f i (α)](q) is <strong>the</strong> set <strong>of</strong><br />
50
states reachable in i steps from <strong>the</strong> state q. Formally for ξ ∈ X let [f(ξ)](q) = {δ(q ′ , h(A)) : q ′ ∈<br />
ξ(q) ∧ A ∈ M p (n, m)}.<br />
By lemma 2.1.4 we obtain a k such that f k (α) = f 2·k (α). Using this k, consider <strong>the</strong> k-<br />
outreach (W ′ , h ′ ) <strong>of</strong> (W, h). Let q ∈ K be any state <strong>of</strong> W ′ and let A, B ∈ M p (n, k · m) be any<br />
symbols in <strong>the</strong> alphabet <strong>of</strong> W ′ . Now δh(q, ∗ A ⌢ B) ∈ [f 2·k (α)](q). Since by lemma 2.1.4 we have<br />
f k (α) = f 2·k (α), it is <strong>the</strong> case that δh(q, ∗ A ⌢ B) ∈ [f k (α)](q). Accordingly by <strong>the</strong> definition <strong>of</strong><br />
f and α <strong>the</strong>re is a symbol C ∈ M p (n, k · m), <strong>the</strong> alphabet, such that δh(q, ∗ A ⌢ B) = δh(q, ∗ C).<br />
Since δ ′ , <strong>the</strong> transition function <strong>of</strong> W ′ , is defined as <strong>the</strong> restriction <strong>of</strong> δh ∗ to K × M p (n, k · m) <strong>the</strong><br />
proposition follows.<br />
qed<br />
Corollary 2.4.4 Every concrete n-tape p-automaton is equivalent to a transitive automaton.<br />
Pro<strong>of</strong>: Using <strong>the</strong> k <strong>of</strong> proposition 2.4.3, <strong>the</strong> k-outreach <strong>of</strong> a classical automaton is a transitive<br />
automaton, moreover it is equivalent to <strong>the</strong> classical one by proposition 2.4.2.<br />
qed<br />
Corollary 2.4.5 Finite transitive n-tape p-automata, are versatile enough to replace classical automata<br />
in Büchis decision procedure.<br />
Pro<strong>of</strong>: This follows from <strong>the</strong> previous corollary and <strong>the</strong> fact that classical p-automata are sufficient<br />
for deciding Presburger arithmetic and <strong>the</strong> <strong>the</strong>ories <strong>of</strong> <strong>the</strong> o<strong>the</strong>r automatic structures. Here a classical<br />
automaton is an n-tape p-automaton where <strong>the</strong> alphabet is <strong>the</strong> concrete vector-space M p (n, 1).<br />
qed<br />
2.4.3 Transitive p-automata, <strong>the</strong> one-sorted case<br />
By corollary 3.3.5 a class <strong>of</strong> two-sorted automata sufficient for deciding <strong>the</strong>ories <strong>of</strong> automatic structure<br />
was singled out. Namely <strong>the</strong> transitive and hence concrete n-tape p-automata. In <strong>the</strong>se automata<br />
a state is reachable from ano<strong>the</strong>r iff it is reachable exactly one step.<br />
Recall that we applied <strong>the</strong> Ehrenfeucht-Fraïssé method to one-sorted automata. To make sure<br />
that first-order definability <strong>of</strong> reachability is not a property <strong>of</strong> two-sorted automata or <strong>of</strong> concreteness,<br />
we provide a finite set <strong>of</strong> axioms for a one-sorted <strong>version</strong> <strong>of</strong> transitive automata. In <strong>the</strong>se,<br />
transitivity and reachability is first-order definable, independently <strong>of</strong> a particular concrete alphabet.<br />
The elements <strong>of</strong> <strong>the</strong> carrier-set serve both as states and as symbols <strong>of</strong> <strong>the</strong> alphabet. The initial state<br />
and <strong>the</strong> 0 symbol are identified.<br />
Definition Let n, p ∈ N where p is a prime power. A (one-sorted) n-tape p-automaton is a structure<br />
W = (V, δ, T ) where<br />
V = (V, +, −, 0, . . .) is an n-fold vector-space over <strong>the</strong> field F p where <strong>the</strong> elements <strong>of</strong> V serve both<br />
as symbols and states,<br />
δ : V × V → V is <strong>the</strong> transition function,<br />
T ⊆ V , is <strong>the</strong> set <strong>of</strong> terminal states.<br />
Moreover<br />
T is closed under adding zero-vectors at <strong>the</strong> most significant end, i.e.<br />
51
W |= ∀q[T (q) ↔ T (δ(q, 0))].<br />
Definition An n-tape p-automaton W = (V, δ, T ) is said to be transitive if every state that is<br />
reachable from a given state q, is reachable from q in one step, i.e.<br />
W |= ∀qxy∃z[δ(δ(q, x), y) = δ(q, z)].<br />
Note that, as opposed to <strong>the</strong> two sorted case, one-sorted transitive automata need not be concrete.<br />
Proposition 2.4.6 Over structures <strong>of</strong> suitable similarity-type transitive n-tape p-automata are axiomatisable<br />
by a finite set <strong>of</strong> first-order axioms.<br />
Pro<strong>of</strong>: Stating <strong>the</strong> first axiom, “V is an n-fold vector-space over <strong>the</strong> field F p ” with a finite number<br />
<strong>of</strong> first-order sentences can be done as in section 2.2. The rest <strong>of</strong> <strong>the</strong> axioms are obviously finite in<br />
number and first-order.<br />
qed<br />
We will now regard various classes <strong>of</strong> automata we have defined in this paper, and provide notation<br />
for <strong>the</strong>m. For <strong>the</strong> rest <strong>of</strong> this section fixate n, p ∈ N such that p is a prime power.<br />
Definition The following denote classes <strong>of</strong> finite one-sorted n-tape p-automata.<br />
K 1,cla classical automata, i.e those with alphabet M p (n, 1)<br />
K 1,con concrete automata, i.e those <strong>of</strong> <strong>the</strong> form (W, h)<br />
K 1,ctr concrete and transitive automata.<br />
The following denote two-sorted <strong>version</strong>s <strong>of</strong> <strong>the</strong> same.<br />
K 2,cla classical automata, i.e those with alphabet M p (n, 1)<br />
K 2,con concrete automata, i.e those <strong>of</strong> <strong>the</strong> form (W, h)<br />
K 2,ctr concrete and transitive automata.<br />
We will now define equivalence for members <strong>of</strong> K 1,con in such a way that equivalent automata accept<br />
<strong>the</strong> same n-tapes. We could do this by adapting <strong>the</strong> definitions <strong>of</strong> equivalence and recognition<br />
we have on K 2,con , but we shall instead use a trick and embed K 1,con into K 2,con and have <strong>the</strong><br />
needed notions reflected by <strong>the</strong> embedding.<br />
Definition The mapping ∆ : K 1,con → K 2,con is defined as follows. Let W = (V, δ, T ) an<br />
automaton and h an n-fold linear transformation making (W, h) concrete. Let V have carrier-set<br />
V and zero-vector 0. Then ∆((V, δ, T ), h) = ((V, V, δ, 0, T ), h)<br />
Definition (W, h), (W ′ , h ′ ) ∈ K 1,con are equivalent iff ∆(W, h), ∆(W ′ , h ′ ) ∈ K 2,con are equivalent.<br />
Definition Let (W, h) ∈ K 1,con Let V be <strong>the</strong> alphabet <strong>of</strong> W and M p (n, m) <strong>the</strong> domain <strong>of</strong> h. Let<br />
L ⊆ M p (n, m) ∗ ⌢ 0 n×N . Then (W, h) recognises L iff ∆(W, h) recognises L.<br />
Proposition 2.4.7 If (W, h), (W ′ , h ′ ) ∈ K 1,con are equivalent <strong>the</strong>n <strong>the</strong>y accept <strong>the</strong> same finitely<br />
supported n-tapes.<br />
52
Pro<strong>of</strong>: Left to <strong>the</strong> reader.<br />
qed<br />
Definition Let K X and K Y be two classes <strong>of</strong> automata and let f : K X → K Y . Then f is said<br />
to be recognition-invariant if for all W ∈ K X and all A ∈ M p (n, 1) ∗ we have that W accepts<br />
A ⌢ 0 n×N iff f(W) accepts A ⌢ 0 n×N .<br />
Again effective means computable and not necessarily very fast, assuming some reasonable way <strong>of</strong><br />
encoding automata.<br />
Proposition 2.4.8 There are effective recognition-invariant mappings for each <strong>of</strong> <strong>the</strong> types:<br />
1. K 1,cla → K 2,cla<br />
2. K 2,cla → K 2,con<br />
3. K 2,con → K 2,ctr<br />
4. K 2,ctr → K 1,ctr<br />
5. K 1,ctr → K 2,ctr<br />
6. K 2,ctr → K 2,con<br />
7. K 2,con → K 2,cla<br />
8. K 2,cla → K 1,cla<br />
Pro<strong>of</strong>:<br />
1. This is a statement about classical automata and left to <strong>the</strong> reader.<br />
2. K 2,cla ⊆ K 2,con so <strong>the</strong> identity mapping will do.<br />
3. For this we map automata to <strong>the</strong>ir k-outreach where k is as in proposition 2.4.3. Such a<br />
mapping is recognition-invariant by proposition 2.4.2.<br />
4. Given a two-sorted transitive ((V, K, δ, ι, T ), h) we leave V and h as <strong>the</strong>y are and define a<br />
one-sorted counterpart ((V, δ ′ , T ′ ), h). We are only interested in states that are reachable in<br />
one step and we shall use <strong>the</strong> members <strong>of</strong> V as states. First we choose some well-ordering ≤<br />
on V making 0 minimal. Second define δ ′ and T ′ as follows.<br />
For each pair x, y <strong>of</strong> V let δ ′ (x, y) be <strong>the</strong> ≤-minimal z s.t. δ(δ(ι, x), y) = δ(ι, z).<br />
Moreover let x ∈ T ′ iff δ(ι, x) ∈ T .<br />
5. This is <strong>the</strong> opposite direction. Here we use <strong>the</strong> recently defined mapping ∆ which maps<br />
structures<br />
((V, δ, T ), h) to ((V, V, δ, 0, T ), h). <strong>On</strong>e easily verifies that transitivity is preserved and that<br />
∆ is recognition-invariant.<br />
6. K 2,ctr ⊆ K 2,con so <strong>the</strong> identity mapping will do.<br />
53
7. It is known from classical automata <strong>the</strong>ory that <strong>non</strong>deterministic and partially defined automata<br />
can effectively be transformed into equivalent deterministic and totally defined automata.<br />
The members <strong>of</strong> K 2,con can be seen as partially defined members <strong>of</strong> K 2,cla , and <strong>the</strong><br />
mentioned determination-operation as <strong>the</strong> sought mapping.<br />
8. Again this is a statement about classical automata and left to <strong>the</strong> reader.<br />
Now two corollaries. Note that by <strong>the</strong> proposition we have effective recognition-invariant transformations<br />
to and from classical automata. Since <strong>the</strong>re are effective constructions on classical automata<br />
to do negation, disjunction, variable substitutions and existential quantification we get <strong>the</strong> same for<br />
concrete n-tape p-automata, be <strong>the</strong>y transitive or not. Therefore:<br />
Corollary 2.4.9 Transitive n-tape p-automata are versatile enough to replace classical automata in<br />
Büchis decision procedure. More specifically transitive automata that are both concrete and finite suffice.<br />
By <strong>the</strong> definition <strong>of</strong> transitive automaton we get <strong>the</strong> following.<br />
Corollary 2.4.10 In concrete finite transitive n-tape p-automata (W, h), <strong>the</strong> relation <strong>of</strong> being h-<br />
reachable is definable by <strong>the</strong> first-order formula ∃z[δ(q, z) = q ′ ].<br />
Note that <strong>the</strong>re is no reference to <strong>the</strong> homomorphism h in <strong>the</strong> first-order formula. This means that<br />
<strong>the</strong> formula defines reachability regardless <strong>of</strong> which surjection h one chooses.<br />
qed<br />
54
2.5 Reachability refined<br />
Here we introduce three properties <strong>of</strong> automata, namely being Projectively Transitive, having Projections<br />
and having Substitutions. We refer to automata with all three <strong>of</strong> <strong>the</strong>se properties as PTPS automata.<br />
We look at one- and two-sorted variants. For given n, p ∈ N <strong>the</strong> class <strong>of</strong> one-sorted n-tape p-entry<br />
PTPS automata is definable by a finite set <strong>of</strong> first-order sentences. In each one-sorted PTPS automaton<br />
we are able to express, in first-order language, that states are reachable from one ano<strong>the</strong>r<br />
using tapes that represent various projections <strong>of</strong> N n . We are for instance be able to express, in firstorder<br />
language, that <strong>the</strong> state q ′ is reachable from q using an n-tape whose second row represents<br />
<strong>the</strong> number 0.<br />
We introduce PTPS automata as multi-automata, i.e. automata with several sets <strong>of</strong> terminal<br />
states. We don’t actually use <strong>the</strong> fact that PTPS automata are multi-automata before <strong>the</strong> next<br />
section. So on a first reading one may assume that PTPS automata have one set <strong>of</strong> terminal states,<br />
in which case PTPS automata form a subclass <strong>of</strong> <strong>the</strong> transitive automata.<br />
2.5.1 Two-sorted multi-automata<br />
We introduce multi-automata to make it possible to compare automata with <strong>the</strong> same transition<br />
function quite directly. For example we will see how an automaton recognising <strong>the</strong> relation defined<br />
by <strong>the</strong> formula φ ∨ ψ, compares to <strong>the</strong> automata recognising <strong>the</strong> relations defined by φ and ψ.<br />
Definition An n-tape p-multi-automaton is a tuple (V, K, δ, ι, T 0 , . . . , T t−1 ) where each<br />
(V, K, δ, ι, T 0 ), . . . , (V, K, δ, ι, T t−1 ) is a two-sorted n-tape p-automaton.<br />
Each W i = (V, K, δ, ι, T i ) is said to be a component <strong>of</strong> W.<br />
We can now carry defined notions to multi-automata via <strong>the</strong>ir components.<br />
Definition Let W = (V, . . .) and W ′ = (V ′ , . . .) be n-tape p-multi-automata. Let h be an n-fold<br />
linear transformation from M p (n, m) onto V. Let h ′ be an n-fold linear transformation from<br />
M p (n, m) onto V ′ . Then<br />
(W, h) is concrete if each component (W i , h) is concrete.<br />
(W ′ , h ′ ) is <strong>the</strong> k-outreach <strong>of</strong> (W, h) if each component (W ′ i, h ′ ) is <strong>the</strong> k-outreach <strong>of</strong> (W i , h).<br />
(W, h) is equivalent to (W ′ , h ′ ) if W and W ′ are tuples <strong>of</strong> <strong>the</strong> same length and each corresponding<br />
pair <strong>of</strong> components (W i , h) and (W ′ i, h) is equivalent.<br />
The following multi-automaton <strong>version</strong> <strong>of</strong> Proposition 2.4.2 is now a fairly immediate consequence.<br />
Proposition 2.5.1 Let (W, h) and (W ′ , h ′ ) be concrete n-tape p-multi-automata. Let k ∈ N. Then<br />
<strong>the</strong> k-outreach (W ′ , h ′ ) <strong>of</strong> (W, h) is equivalent to (W, h).<br />
Pro<strong>of</strong>: Immediate from <strong>the</strong> last definition and proposition 2.4.2.<br />
qed<br />
The following lemma allows us to merge multi-automata in a way that resembles concatenation.<br />
The present author was unable to adapt <strong>the</strong> pro<strong>of</strong> <strong>of</strong> <strong>the</strong> lemma to <strong>the</strong> one-sorted counterpart <strong>of</strong><br />
multi-automata. This is <strong>the</strong> reason we bo<strong>the</strong>r with two-sorted automata at this point.<br />
55
Lemma 2.5.2 If (W, h) and (W ′ , h ′ ) are finite concrete n-tape p-multi-automata, <strong>the</strong>n <strong>the</strong>re exists a<br />
finite concrete n-tape p-multi-automaton (W ′′ , h ′′ ) such that<br />
(W 0 , h) is equivalent to (W ′′<br />
0 , h ′′ )<br />
.<br />
(W l−1 , h) is equivalent to (W ′′<br />
l−1, h ′′ )<br />
(W ′ 0, h ′ ) is equivalent to (W ′′<br />
l , h ′′ )<br />
.<br />
(W ′ l ′ −1, h ′ ) is equivalent to (W ′′<br />
l+l ′ −1, h ′′ )<br />
Moreover h ′′ can be chosen so as to make it an isomorphism.<br />
Pro<strong>of</strong>: To introduce required notation, let (W, h) = (V, K, δ, ι, T 0 , . . . , T l−1 , h) and<br />
(W ′ , h ′ ) = (V ′ , K ′ , δ ′ , ι, T 0, ′ . . . , T l ′<br />
−1, h ′ ).<br />
′<br />
By definition <strong>of</strong> concrete automaton <strong>the</strong>re are m, m ′ ∈ N s.t. h and h ′ are surjective linear<br />
transformations from M p (n, m) onto V and M p (n, m ′ ) onto V ′ respectively. Let m ′′ be<br />
<strong>the</strong> least common multiple <strong>of</strong> m and m ′ . We now define <strong>the</strong> concrete automaton (W ′′ , h ′′ ) =<br />
(V ′′ , K ′′ , δ ′′ , ι ′′ , T 0 ′′ , . . . , T l+l ′′<br />
−1, h ′′ ) which proves <strong>the</strong> lemma.<br />
′<br />
V ′′ = M p (n, m ′′ )<br />
K ′′ = K × K ′<br />
δ ′′ ((q, q ′ ), A) = (δ ∗ h(q, A), δ ′∗<br />
h ′(q′ , A))<br />
ι ′′ = (ι, ι ′ )<br />
(q, q ′ ) ∈ T ′′<br />
0 iff q ∈ T 0<br />
.<br />
(q, q ′ ) ∈ T ′′<br />
l−1 iff q ∈ T l−1<br />
(q, q ′ ) ∈ T ′′<br />
l iff q ∈ T ′ 0<br />
.<br />
(q, q ′ ) ∈ T ′′<br />
l+l ′ −1 iff q ∈ T l ′ −1<br />
h ′′ (A) = A<br />
. . . <strong>the</strong> n-fold vector-space <strong>of</strong> n × m ′′ matrices, which is finite<br />
. . . cartesian product <strong>of</strong> sets which preserves finiteness<br />
. . . <strong>the</strong> identity mapping on M p (n, m ′′ ), clearly an isomorphism.<br />
The crucial thing to note here is that δ ′′ is well defined, which it is by choice <strong>of</strong> m ′′ . The equivalences<br />
<strong>the</strong>n are immediate by <strong>the</strong> definition <strong>of</strong> <strong>the</strong> T i ′′ ’s.<br />
qed<br />
2.5.2 Definable linear transformations<br />
Here we introduce two kinds <strong>of</strong> linear transformations that allow us to define certain semigroups<br />
<strong>of</strong> symbols for n-tapes in <strong>the</strong> language <strong>of</strong> n-fold vector-spaces. We will eventually be able to give<br />
first-order definitions <strong>of</strong> reachability by tapes built from <strong>the</strong>se semigroups.<br />
The first kind comprises n n special linear transformations. When <strong>the</strong> alphabet happens to be<br />
concrete, i.e. a set <strong>of</strong> matrices, a given linear transformation <strong>of</strong> this class corresponds to a particular<br />
way <strong>of</strong> swapping and/or overwriting rows with o<strong>the</strong>r rows.<br />
56
Definition For each n-fold vector-space V = (V, +.−, 0, . . . , π, r) and mapping σ : n → n we<br />
define ˆσ : V → V as follows. ˆσ(x) = r −0 (π(r σ(0) (x))) + . . . + r −(n−1) (π(r σ(n−1) (x))).<br />
The second kind comprises n linear transformations. When <strong>the</strong> alphabet happens to be concrete,<br />
i.e. a set <strong>of</strong> matrices, <strong>the</strong> i’th linear transformation replaces each entry on <strong>the</strong> i’th row with a<br />
0.<br />
Definition For for each n-fold vector-space V = (V, +, −, 0, . . . , π, r) and i < n we define<br />
ˆπ i : V → V as follows<br />
ˆπ 0 (x) = 0 + r 1 (π(r −1 (x))) + · · · + r n−1 (π(r −(n−1) (x)))<br />
ˆπ 1 (x) = r 0 (π(r −0 (x))) + 0 + · · · + r n−1 (π(r −(n−1) (x)))<br />
.<br />
ˆπ n−1 (x) = r 0 (π(r −0 (x))) + · · · + r n−2 (π(r −(n−2) (x))) + 0<br />
The ˆσ’s and ˆπ i ’s are clearly linear transformations since <strong>the</strong>y are built using sums and <strong>the</strong> linear<br />
transformations π and r. They need not be n-fold linear transformations.<br />
Given an n-fold vector-space <strong>the</strong> ˆσ’s and ˆπ i ’s each generate a semigroup under composition.<br />
We introduce notation for <strong>the</strong> carrier set <strong>of</strong> <strong>the</strong>se two semigroups.<br />
Definition Let V = (V, . . .) be an n-fold vector space over F p . Then we define SL(V) ⊆ V V as<br />
follows<br />
SL(V) = {ˆσ : σ ∈ n n }<br />
The letters SL stand for Substitution and Linear.<br />
Definition Let V = (V, . . .) be an n-fold vector space over F p . Then we define P L(V) ⊆ V V as<br />
follows<br />
P L(V) = <strong>the</strong> set <strong>of</strong> mappings generated by { ˆπ i : i < n} under composition.<br />
The letters P L stand for Projection and Llinear.<br />
Lemma 2.5.3 For finite n-fold vector-spaces V = (V, . . .) over F p , <strong>the</strong> sets SL(V) and P L(V) are<br />
finite.<br />
Pro<strong>of</strong>: Since SL(V) ⊆ V V and P L(V) ⊆ V V and V V is finite.<br />
qed<br />
2.5.3 Projective transitivity<br />
Here we define a property that enables us to express notions such as: <strong>the</strong> state q ′ is reachable from q<br />
using an n-tape whose second row represents <strong>the</strong> number 0. We will be using linear transformations<br />
in P L(M p (n, m)) for this. We shortly prove a variant <strong>of</strong> Proposition 2.4.3. A definition is handy<br />
first.<br />
Definition Let W = (V, K, δ, . . .) be an n-tape p-multi-automaton. Let h be an n-fold linear<br />
transformation from M p (n, m) onto V. Then (W, h) is said to be projectively transitive if<br />
57
for each t ∈ P L(M p (n, m)),<br />
for all states q ∈ K and tapes A, B ∈ M p (n, m),<br />
<strong>the</strong>re exists a C ∈ M p (n, m) such that<br />
δ ∗ h(δ ∗ h(q, t(A)), t(B)) = δ ∗ h(q, t(C)).<br />
By repeated use <strong>of</strong> this definition, projectively transitive automata do for example have <strong>the</strong> property<br />
that if a state is h-reachable by a tape in M p (n, m) ∗ whose second row represents <strong>the</strong> number 0,<br />
<strong>the</strong>n it is h-reachable by a symbol in M p (n, m) whose second row represents 0.<br />
Lemma 2.5.4 Let W = (V, K, δ, . . .) be a finite n-tape p-multi-automaton. Let h be an n-fold<br />
linear transformation from M p (n, m) onto V. Then <strong>the</strong>re exists a k ∈ N such that <strong>the</strong> k-outreach<br />
(W ′ , h ′ ) <strong>of</strong> (W, h) is projectively transitive.<br />
Pro<strong>of</strong>: To find <strong>the</strong> sought k we shall use lemma 2.1.4. Consider <strong>the</strong>refore<br />
X = P(K) K×P L(Mp(n,m)) , <strong>the</strong> set <strong>of</strong> mappings from <strong>the</strong> set <strong>of</strong> pairs <strong>of</strong> states and projections to<br />
<strong>the</strong> power-set <strong>of</strong> <strong>the</strong> set <strong>of</strong> states. The set X is a finite set since K is finite and by lemma 2.5.3,<br />
P L(M p (n, m)) is finite. Define <strong>the</strong> mapping α ∈ X by α(q, t) = {q}. We now define a function<br />
f : X → X that, when beginning with α and iterating, we can use to keep track <strong>of</strong> <strong>the</strong> states<br />
reachable form a given state q using symbols <strong>of</strong> <strong>the</strong> form t(A). This is to say that [f i (α)](q, t) is<br />
<strong>the</strong> set <strong>of</strong> states reachable in i steps from <strong>the</strong> state q using symbols <strong>of</strong> <strong>the</strong> form t(A). Formally for<br />
ξ ∈ X let [f(ξ)](q, t) = {δ(q ′ , h(t(A))) : q ′ ∈ ξ(q, t) ∧ A ∈ M p (n, m)}.<br />
By lemma 2.1.4 we obtain a k such that f k (α) = f 2·k (α). Using this k consider <strong>the</strong> k-outreach<br />
(W ′ , h ′ ) <strong>of</strong> (W, h). Let q ∈ K be any state <strong>of</strong> W ′ . Let A, B ∈ M p (n, k · m) be any symbols in<br />
<strong>the</strong> alphabet <strong>of</strong> W ′ . Let t ∈ P L(M p (n, k · m) be a transformation on <strong>the</strong> alphabet <strong>of</strong> W ′ . Now<br />
δh(q, ∗ t(A) ⌢ t(B)) ∈ [f 2·k (α)](q, t). Since by lemma 2.1.4 we have f k (α) = f 2·k (α), it is <strong>the</strong> case<br />
that δh(q, ∗ t(A) ⌢ t(B)) ∈ [f k (α)](q, t). Accordingly, by <strong>the</strong> definition <strong>of</strong> f and α, <strong>the</strong>re exists a<br />
symbol C ∈ M p (n, k · m), <strong>the</strong> alphabet, such that δh(q, ∗ t(A) ⌢ t(B)) = δh(q, ∗ t(C)). Since δ ′ by<br />
definition <strong>of</strong> W ′ , is <strong>the</strong> restriction <strong>of</strong> δh ∗ to K × M p (n, k · m), <strong>the</strong> lemma follows. qed<br />
Keep in mind that <strong>the</strong> k-outreach <strong>of</strong> a finite automaton is a finite equivalent automaton. Hence<br />
with each finite automaton, we may associate an equivalent finite automaton that is projectively<br />
transitive. By examining <strong>the</strong> pro<strong>of</strong> <strong>of</strong> <strong>the</strong> previous proposition we obtain <strong>the</strong> following.<br />
Corollary 2.5.5 For k ∈ N <strong>the</strong> k-outreach <strong>of</strong> a projectively transitive automaton is again projectively<br />
transitive.<br />
2.5.4 Projection automata<br />
For each linear transformation in P L(M p (n, m)) we now define a relation on pairs <strong>of</strong> states <strong>of</strong><br />
a given automaton which for example we can use as follows. From <strong>the</strong> fact that q and q ′ are in<br />
relation, we may infer that <strong>the</strong>re exist two symbols,that differ only in <strong>the</strong> <strong>the</strong> second row, which,<br />
from <strong>the</strong> initial state ι, we can use to reach q and q ′ respectively. This property turns out to be useful<br />
when comparing an automaton recognising an interpretation <strong>of</strong> <strong>the</strong> formula φ to an automaton<br />
recognising an interpretation <strong>of</strong> <strong>the</strong> formula ∃v i φ.<br />
58
Definition Let W = (V, K, δ, ι, . . .) be an n-tape p-multi-automaton. Let h be an n-fold linear<br />
transformation from M p (n, m) onto V. Let t ∈ P L(M p (n, m)). Then t ∼⊆ K × K is defined as<br />
follows.<br />
The relation q t ∼ q ′ holds if<br />
<strong>the</strong>re exist A, B ∈ M p (n, m) such that<br />
t(x) = t(y) and<br />
δ ∗ h(ι, A) = q and<br />
δ ∗ h(ι, B) = q ′ .<br />
Note that if q t ∼ q ′ <strong>the</strong>n <strong>the</strong> states q and q ′ can both be reached from <strong>the</strong> initial state in one step.<br />
Definition Let W = (V, K, δ, ι, . . .) be an n-tape p-multi-automaton. Let h be an n-fold linear<br />
transformation from M p (n, m) onto V. Then (W, h) is said to be a projection automaton or to<br />
have projections if<br />
for each t ∈ P L(M p (n, m))<br />
{(q, q ′ ) : q t ∼ q ′ } = {(δ ∗ h(q, A), δ ∗ h(q ′ , B)) : q t ∼ q ′ ∧ A, B ∈ M p (n, m) ∧ t(A) = t(B)}<br />
So in a projection automaton if q t ∼ q ′ <strong>the</strong>n <strong>the</strong> states q and q ′ can both be reached from <strong>the</strong> initial<br />
state in one step, moreover both can be reached in exactly two steps, and so forth. We now use<br />
lemma 2.1.4 to show that every automaton is equivalent to a projection automaton.<br />
Lemma 2.5.6 Let W = (V, K, δ, ι, . . .) be a finite n-tape p-multi-automaton. Let h be an n-fold<br />
linear transformation from M p (n, m) onto V. Then <strong>the</strong>re exists a k ∈ N such that <strong>the</strong> k-outreach<br />
(W ′ , h ′ ) <strong>of</strong> (W, h) is a projection automaton.<br />
Pro<strong>of</strong>: To use lemma 2.1.4 consider X = P(K × K) P L(Mp(n,m)) , <strong>the</strong> set <strong>of</strong> mappings from<br />
P L(M p (n, m)) to <strong>the</strong> set <strong>of</strong> binary relations on states. The set X is finite since by lemma 2.5.3,<br />
P L(M p (n, m)) is finite. Define <strong>the</strong> mapping α ∈ X by α(t) = {(ι, ι)}. We now define a<br />
function f : X → X that, when beginning with α and iterating, we can use to keep track <strong>of</strong> pairs<br />
<strong>of</strong> states that are reachable from <strong>the</strong> initial state by a pair <strong>of</strong> tapes that differ in, for example, <strong>the</strong><br />
second row only. This is to say that for t ∈ P L(M p (n, m)) we have<br />
(q, q ′ ) ∈ [f i (α)](t)<br />
iff<br />
<strong>the</strong>re exist A 0 , . . . , A i−1 , B 0 , . . . , B i−1 ∈ M p (n, m)<br />
such that<br />
t(A 0 ) ⌢ · · · ⌢t(A i−1 ) = t(B 0 ) ⌢ · · · ⌢t(B i−1 ) and<br />
δ ∗ h(ι, A 0 ⌢ · · · ⌢A i−1 ) = q and<br />
δ ∗ h(ι, B 0 ⌢ · · · ⌢B i−1 ) = q ′ .<br />
59
A way <strong>of</strong> defining f is for ξ ∈ X to let [f(ξ)](t) = {(δ(q, h(A)), δ(q ′ , h(B))) : (q, q ′ ) ∈<br />
ξ(t) ∧ A, B ∈ M p (n, m) ∧ t(A) = t(B)}.<br />
By lemma 2.1.4 we obtain a k such that f k (α) = f 2·k (α). Using this k consider <strong>the</strong> k-<br />
outreach (W ′ , h ′ ) <strong>of</strong> (W, h). With each t ∈ P L(M p (n, m)) associate t ′ ∈ P L(M p (n, k · m))<br />
by <strong>the</strong> equation t ′ (A 0 ⌢ · · · ⌢A k−1 ) = t(A 0 ) ⌢ · · · ⌢t(A k−1 ). Note that this association sets up<br />
a bijection between P L(M p (n, m)) and P L(M p (n, k · m)), since both are sets <strong>of</strong> operations<br />
on n rows. We now have what we need to show that (W ′ , h ′ ) fulfils <strong>the</strong> requirement <strong>of</strong> being a<br />
projection automaton. So letting δ ′ be as in <strong>the</strong> definition <strong>of</strong> k-outreach we have:<br />
{(δ ′ (q, A), δ ′ (q ′ , B)) : q ∼ t′<br />
q ′ ∧ A, B ∈ M p (n, k · m) ∧ t ′ (A) = t ′ (B)}<br />
= {(δ ∗ h(ι, A ⌢ A ′ ), δ ∗ h(ι, B ⌢ B ′ )) : A, A ′ , B, B ′ ∈ M p (n, k · m) ∧ t ′ (A) = t ′ (B) ∧ t ′ (A ′ ) =<br />
t ′ (B ′ )} by definition <strong>of</strong> δ ′ and ∼<br />
t′<br />
= {(δ ∗ h(ι, A 0 ⌢ · · · ⌢A 2k−1 ), δ ∗ h(ι, B 0 ⌢ · · · ⌢B 2k−1 )) : A 0 ⌢ · · · ⌢A 2k−1 , B 0 ⌢ · · · ⌢B 2k−1 ∈<br />
M p (n, m)<br />
∧t(A 0 ) = t(B 0 ) ∧ · · · ∧ t(A 2k−1 ) = t(B 2k−1 )} by definition <strong>of</strong> t ′<br />
= [f 2·k (α)](t) by design <strong>of</strong> f<br />
= [f k (α)](t) by choosing <strong>the</strong> k <strong>of</strong> lemma 2.1.4<br />
= {(q, q ′ ) : q ∼ t′<br />
q ′ } again by definition <strong>of</strong> ∼<br />
t′<br />
The left and right side <strong>of</strong> this sequence <strong>of</strong> equations is what we require <strong>of</strong> a projection automaton.<br />
qed<br />
As before <strong>the</strong> k-outreach <strong>of</strong> a finite automaton is a finite equivalent automaton. Hence with each<br />
finite automaton, we may associate an equivalent finite automaton that is a projection automaton.<br />
By examining <strong>the</strong> pro<strong>of</strong> <strong>of</strong> <strong>the</strong> previous proposition we ga<strong>the</strong>r <strong>the</strong> following.<br />
Corollary 2.5.7 For k ∈ N <strong>the</strong> k-outreach <strong>of</strong> a projection automaton is again a projection automaton.<br />
2.5.5 Substitution automata<br />
For each linear transformation in SL(M p (n, m)) we now define a relation on pairs <strong>of</strong> states <strong>of</strong><br />
a given automaton which for example we can use as follows. From <strong>the</strong> fact that q and q ′ are<br />
in relation, we may infer that <strong>the</strong>re exist two symbols, one whose rows are a permutation <strong>of</strong> <strong>the</strong><br />
o<strong>the</strong>r symbols rows, which from <strong>the</strong> initial state we can use to reach q and q ′ respectively. This<br />
property turns out to be useful when comparing <strong>the</strong> automata that recognise <strong>the</strong> interpretations <strong>of</strong><br />
for example <strong>the</strong> atomic formulas R(v 0 , v 1 ) and R(v 1 , v 0 ).<br />
Definition Let (V, K, δ, ι, . . .) be a two-sorted n-tape p-multi-automaton. Let h be an n-fold<br />
linear transformation from M p (n, m) onto V. Let ˆσ ∈ SL(M p (n, m)). Then ⊲ˆσ ⊆ K × K is<br />
defined as follows.<br />
q ⊲ˆσ q ′ if<br />
<strong>the</strong>re exists a symbol A ∈ M p (n, m) such that<br />
δ ∗ h(ι, A) = q and<br />
60
δ ∗ h(ι, ˆσ(A)) = q ′<br />
Also ⋄ˆσ ⊆ K × K is defined by<br />
q ′ ⋄ˆσ q ′′ if<br />
<strong>the</strong>re exists a q ∈ K such that<br />
q ⊲ˆσ q ′ and<br />
q ⊲ˆσ q ′′<br />
Note that if q ⊲ˆσ q ′ <strong>the</strong>n <strong>the</strong> states q and q ′ are h-reachable from <strong>the</strong> initial state in one step.<br />
Definition Let W = (V, K, δ, ι, . . .) be a two-sorted n-tape p-multi-automaton. Let h be an n-<br />
fold linear transformation from M p (n, m) onto V. Then <strong>the</strong> concrete automaton (W, h) is said<br />
to be a substitution automaton or to have substitutions if<br />
for each ˆσ ∈ SL(M p (n, m))<br />
{(q, q ′ ) : q ⊲ˆσ q ′ } = {(δ ∗ h(q, A), δ ∗ h(q ′ , ˆσ(A)) : q ⊲ˆσ q ′ ∧ A ∈ M p (n, m)}<br />
So in an automaton which has substitutions, if q ⊲ˆσ q ′ <strong>the</strong>n <strong>the</strong> states q and q ′ are h-reachable from<br />
<strong>the</strong> initial state in one step, moreover <strong>the</strong>y are h-reachable from <strong>the</strong> initial state in exactly two steps,<br />
and so forth.<br />
Lemma 2.5.8 Let W = (V, . . .) be a finite two-sorted n-tape p-multi-automaton. Let h be an n-fold<br />
linear transformation from M p (n, m) onto V. Then <strong>the</strong>re exists a k ∈ N such that <strong>the</strong> k-outreach<br />
(W ′ , h ′ ) <strong>of</strong> (W, h) has substitutions.<br />
Pro<strong>of</strong>: We use lemma 2.1.4, so consider X = P(K × K) SL(Mp(n,m)) , <strong>the</strong> set <strong>of</strong> mappings<br />
from SL(M p (n, m)) to <strong>the</strong> set <strong>of</strong> binary relations on states. The set X is finite since by lemma<br />
2.5.3, <strong>the</strong> set SL(M p (n, m)) is finite. Define <strong>the</strong> mapping α ∈ X by α(t) = {(ι, ι)}. We<br />
now define a function f : X → X that, when beginning with α and iterating, we can use to<br />
keep track <strong>of</strong> pairs <strong>of</strong> states that are reachable from <strong>the</strong> initial state by a pair <strong>of</strong> tapes <strong>of</strong> <strong>the</strong> form<br />
(A 0 , . . . , A i−1 , ˆσ(A 0 ), . . . , ˆσ(A i−1 )).<br />
This is to say that for ˆσ ∈ SL(M p (n, m)) we have<br />
(q, q ′ ) ∈ [f i (α)](ˆσ)<br />
iff<br />
<strong>the</strong>re exist A 0 , . . . , A i−1 ∈ M p (n, m) such that<br />
δ ∗ h(ι, A 0 ⌢ · · · ⌢A i−1 ) = q<br />
δ ∗ h(ι, ˆσ(A 0 ) ⌢ · · · ⌢ˆσ(A i−1 )) = q ′<br />
A way <strong>of</strong> defining such an f is for ξ ∈ X to let<br />
[f(ξ)](ˆσ) = {(δ(q, h(A)), δ(q ′ , h(ˆσ(A)))) : (q, q ′ ) ∈ ξ(ˆσ) ∧ A ∈ M p (n, m)}.<br />
61
By lemma 2.1.4 we obtain a k ∈ N such that f k (α) = f 2·k (α). Using this k consider <strong>the</strong> k-<br />
outreach (W ′ , h ′ ) <strong>of</strong> (W, h). For σ ∈ n n let ˆσ denote <strong>the</strong> corresponding operation on M p (n, m)<br />
and ˆσ <strong>the</strong> same on M p (n, k · m).<br />
We now have what we need to show that <strong>the</strong> k-outreach (W ′ , h ′ ) fulfills <strong>the</strong> requirement <strong>of</strong><br />
being a substitution automaton. So letting δ ′ be as in <strong>the</strong> definition <strong>of</strong> k-outreach we have.<br />
{(δ ′ (q, A), δ ′ (q ′ , ˆσ(A))) : q ⊲ˆσ q′ ∧ A ∈ M p (n, k · m)}<br />
= {(δ ∗ h(ι, A ⌢ A ′ ), δ ∗ h(ι, ˆσ(A) ⌢ˆσ(A ′ ))) : A, A ′ ∈ M p (n, k · m)} by definition <strong>of</strong> δ ′ and ⊲ˆσ<br />
= {(δ ∗ h(ι, A 0 ⌢ · · · ⌢A 2·k−1 ), δ ∗ h(ι, ˆσ(A 0 ) ⌢ · · · ⌢ˆσ(A 2·k−1 ))) : A 0 , . . . , A 2·k−1 ∈ M p (n, m)}<br />
by definition <strong>of</strong> ˆσ and ˆσ<br />
= [f 2·k (α)](ˆσ) by design <strong>of</strong> f<br />
= [f k (α)](ˆσ) by choosing <strong>the</strong> k <strong>of</strong> lemma 2.1.4<br />
{(q, q ′ ) : q ⊲ˆσ q′ }<br />
by definition <strong>of</strong> ⊲ˆσ<br />
The left and right side <strong>of</strong> this sequence <strong>of</strong> equations is what we require <strong>of</strong> a substitution automaton.<br />
qed<br />
By examining <strong>the</strong> pro<strong>of</strong> <strong>of</strong> <strong>the</strong> previous proposition we obtain <strong>the</strong> following.<br />
Corollary 2.5.9 For k ∈ N <strong>the</strong> k-outreach <strong>of</strong> a substitution automaton is again a substitution automaton.<br />
2.5.6 Properties <strong>of</strong> <strong>the</strong> two-sorted transition function<br />
Here we prove two properties <strong>of</strong> two-sorted automata with projections, which shortly will be used<br />
as defining properties for <strong>the</strong> one-sorted <strong>version</strong> <strong>of</strong> having projections.<br />
Lemma 2.5.10 Let W = (V, K, δ, ι, . . .) be an n-tape p-multi-automaton. Let h be an n-fold linear<br />
transformation from M p (n, m) onto V. Let (W, h) be a projection automaton. Then<br />
1. for each t ∈ P L(M p (n, m))<br />
for each pair <strong>of</strong> states q, q ′ ∈ K and symbols A, B ∈ M p (n, m)<br />
if q t ∼ q ′ and t(A) = t(B) <strong>the</strong>n<br />
δ ∗ h(q, A) t ∼ δ ∗ h(q ′ , B) .<br />
2. for each t ∈ P L(M p (n, m))<br />
Pro<strong>of</strong>:<br />
for each pair <strong>of</strong> states q, q ′ ∈ K and symbol A ∈ M p (n, m)<br />
if δ ∗ h(q, A) t ∼ q ′<br />
<strong>the</strong>n <strong>the</strong>re exist q ′′ ∈ K and B ∈ M p (n, m) such that<br />
q t ∼ q ′′ and t(A) = t(B) and δ ∗ h(q ′′ , B) = q ′<br />
62
1. Immediate from <strong>the</strong> definition <strong>of</strong> projection automaton.<br />
2. Note that, by definition, <strong>the</strong> relation t ∼ is an equivalence relation on δ ∗ h(ι, M p (n, m)), <strong>the</strong> set<br />
<strong>of</strong> states h-reachable from ι in one step. Therefore δ ∗ h(ι, M p (n, m)) can be partitioned into<br />
equivalence classes. We use [q] to denote <strong>the</strong> equivalence class <strong>of</strong> q ∈ δ ∗ h(ι, M p (n, m)).<br />
By <strong>the</strong> definition <strong>of</strong> projection automaton we get <strong>the</strong> following equality: [δ ∗ h(q, A)] =<br />
{δ ∗ h(q ′′ , B) : q t ∼ q ′′ ∧ t(A) = t(B)}. To prove <strong>the</strong> lemma, assume now that δ ∗ h(q, A) t ∼ q ′ .<br />
Then q ′ ∈ [δ ∗ h(q, A)] and by <strong>the</strong> equality <strong>the</strong>re exist q ′′ and B such that q t ∼ q ′′ and<br />
t(A) = t(B) and δ ∗ h(q ′′ , B) = q ′ .<br />
Here we prove two properties <strong>of</strong> two-sorted automata with substitutions which shortly will be<br />
used as defining properties for <strong>the</strong> one-sorted <strong>version</strong> <strong>of</strong> having substitutions.<br />
Lemma 2.5.11 Let W = (V, K, δ, ι, . . .) be a two-sorted n-tape p-multi-automaton. Let h be an n-<br />
fold linear transformation from M p (n, m) onto V. Let <strong>the</strong> concrete automaton (W, h) have substitions.<br />
Let ˆσ ∈ SL(M p (n, m)). Then<br />
1. for each pair <strong>of</strong> states q, q ′ ∈ K and symbol A ∈ M p (n, m)<br />
if q ⊲ˆσ q ′<br />
<strong>the</strong>n δ ∗ h(q, A) ⊲ˆσ δ ∗ h(q ′ , ˆσ(A))<br />
2. for each pair <strong>of</strong> states q, q ′ ∈ K and symbol A ∈ M p (n, m)<br />
if δ ∗ h(q, A) ⊲ˆσ q ′<br />
<strong>the</strong>n <strong>the</strong>re exists a q ′′ ∈ K such that<br />
q ⊲ˆσ q ′′ and δ ∗ h(q ′′ , ˆσ(A)) = q ′ .<br />
Pro<strong>of</strong>:<br />
1. By definition <strong>of</strong> (W, h) having substitutions we conclude that<br />
{(δ ∗ h(q, A), δ ∗ h(q ′ , ˆσ(A)) : q⊲ˆσ q ′ ∧A ∈ M p (n, m)} ⊆ {(q, q ′ ) : q⊲ˆσ q ′ }. Using this inclusion<br />
toge<strong>the</strong>r with <strong>the</strong> assumptions A ∈ M p (n, m) and q ⊲ˆσ q ′ we conclude that δ ∗ h(q, A) ⊲ˆσ<br />
δ ∗ h(q ′ , ˆσ(A)).<br />
2. By definition <strong>of</strong> (W, h) having substitutions we also conclude that<br />
{(q, q ′ ) : q ⊲ˆσ q ′ } ⊆ {(δ ∗ h(q, A), δ ∗ h(q ′ , ˆσ(A)) : q ⊲ˆσ q ′ ∧ A ∈ M p (n, m)}. By <strong>the</strong> assumptions<br />
A ∈ M p (n, m) and δ ∗ h(q, A) ⊲ˆσ q ′ <strong>of</strong> <strong>the</strong> lemma, we conclude that (δ ∗ h(q, A), q ′ ) ∈<br />
{(δ ∗ h(q, A), δ ∗ h(q ′ , ˆσ(A)) : q ⊲ˆσ q ′ ∧ A ∈ M p (n, m)}. If we rename variables we conclude<br />
that (δ ∗ h(q, A), q ′ ) ∈ {(δ ∗ h(q, A), δ ∗ h(q ′′ , ˆσ(A)) : q ⊲ˆσ q ′′ ∧ A ∈ M p (n, m)}. It follows that<br />
<strong>the</strong>re exists a q ′′ ∈ K such that q ⊲ˆσ q ′′ and δ ∗ h(q ′′ , ˆσ(A)) = q ′ .<br />
qed<br />
qed<br />
2.6 Moving to <strong>the</strong> abstract and to one sort<br />
We now define a class <strong>of</strong> one-sorted automata that are projectively transitive, have projections and<br />
substitutions, namely PTPS automata. As opposed to two-sorted automata, one-sorted automata<br />
need not be concrete to be in possetion <strong>of</strong> ei<strong>the</strong>r <strong>of</strong> <strong>the</strong>se tree properties.<br />
63
2.6.1 <strong>On</strong>e-sorted multi-automata<br />
Definition A (one-sorted) n-tape p-multi-automaton is a tuple W = (V, δ, {T i } i∈I ) where<br />
V = (V, +, −, 0, . . .) is an n-fold vector-space over F p<br />
δ : V × V → V<br />
I is a finite set and<br />
for each i ∈ I it is <strong>the</strong> case that T i ⊆ V .<br />
moreover <strong>the</strong> terminal states are invariant under adding zero-vectors at <strong>the</strong> most significant end, i.e.<br />
for each i ∈ I<br />
W |= ∀q[T i (q) ↔ T i (δ(q, 0))].<br />
We use <strong>the</strong> variables q, q ′ , q ′′ and x to range over <strong>the</strong> carrier-set <strong>of</strong> one-sorted automata. We use<br />
q, q ′ , q ′′ when <strong>the</strong> elements are used as states and x, y, . . . when <strong>the</strong>y are used as symbols.<br />
We adapt some notions from two-sorted automata to one-sorted n-tape p-automata.<br />
Definition If W = (V, δ, {T i } i∈I ) is an n-tape p-multi-automaton and i ∈ I <strong>the</strong>n <strong>the</strong> automaton<br />
W i = (V, δ, T i ) is said to be a component <strong>of</strong> W.<br />
We use (W, h) to denote concrete automata also in <strong>the</strong> one sorted case. The definitions <strong>of</strong> δ ∗ h,<br />
acceptence and recoginition are as in <strong>the</strong> two-sorted case.<br />
We define <strong>the</strong> relation q ⊲ˆσ q ′ for pairs <strong>of</strong> states q, q ′ and mappings ˆσ ∈ SL(M p (n, m)). This<br />
time without reference to a linear transformation h.<br />
Definition Let W = (V, δ, . . .) be a one-sorted n-tape p-multi-automaton.<br />
Let V = (V, . . . , 0, . . .). Let ˆσ ∈ SL(M p (n, m)). Then ⊲ˆσ ⊆ V × V is defined as follows.<br />
The relation q ⊲ˆσ q ′ holds if<br />
<strong>the</strong>re exists an x ∈ V such that<br />
δ(0, x) = q and<br />
δ(0, ˆσ(x)) = q ′ .<br />
Also ⋄ˆσ ⊆ V × V is defined as follows.<br />
q ′ ⋄ˆσ q ′′ if<br />
<strong>the</strong>re exists a q ∈ V such that<br />
q ⊲ˆσ q ′ and<br />
q ⊲ˆσ q ′′ .<br />
We again define <strong>the</strong> relation q t ∼ q ′ for pairs <strong>of</strong> states q, q ′ and projections t. This time without<br />
reference to a linear transformation h.<br />
Definition Let W = (V, δ, . . .) be an n-tape p-multi-automaton. Let V = (V, . . . , 0, . . .). Let<br />
t ∈ P L(V). Then t ∼⊆ V × V is defined as follows.<br />
The relation q t ∼ q ′ holds if<br />
64
<strong>the</strong>re exist x, y ∈ V such that<br />
t(x) = t(y) and<br />
δ(0, x) = q and<br />
δ(0, y) = q ′ .<br />
Using this binary relation we are able to define <strong>the</strong> one-sorted counterpart <strong>of</strong> projection automaton.<br />
In <strong>the</strong> following we treat terms like t(x), where t ∈ P L(V), as shorthand for a longer defining<br />
term such as for instance: ˆπ 1 (x) = r 0 (π(r −0 (x))) + 0 + · · · + r n−1 (π(r −(n−1) (x))). Moreover<br />
we treat expressions like q t ∼ q ′ as shorthand for a longer defining formula such as for instance:<br />
∃z, z ′ [t(z) = t(z ′ ) ∧ δ(0, z) = q ∧ δ(0, z ′ ) = q ′ ].<br />
Definition Let W = (V, δ, . . .) be a one-sorted n-tape p-multi-automaton. Then W is said to be<br />
a PTPS automaton if<br />
1. it is projectively transitive, i.e. for each t ∈ P L(V),<br />
W |= ∀qxy∃z[δ(δ(q, t(x)), t(y)) = δ(q, t(z))].<br />
2. it has <strong>the</strong> property <strong>of</strong> lemma 2.5.10.1, i.e. for each t ∈ P L(V),<br />
W |= ∀qq ′ xy[q t ∼ q ′ ∧ t(x) = t(y) → δ(q, x) t ∼ δ(q ′ , y)].<br />
3. it has <strong>the</strong> property <strong>of</strong> lemma 2.5.10.2, i.e. for each t ∈ P L(V),<br />
W |= ∀qq ′ x[δ(q, x) t ∼ q ′ → ∃q ′′ , y[q t ∼ q ′′ ∧ t(x) = t(y) ∧ δ(q ′′ , y) = q ′ ]].<br />
4. it has <strong>the</strong> property <strong>of</strong> lemma 2.5.11.1, i.e. for each σ ∈ n n ,<br />
W |= ∀xqq ′ [q ⊲ˆσ q ′ → δ(q, x) ⊲ˆσ δ(q ′ , ˆσ(x))],<br />
5. it has <strong>the</strong> property <strong>of</strong> lemma 2.5.11.2, i.e. for each σ ∈ n n ,<br />
W |= ∀xqq ′ [δ(q, x) ⊲ˆσ q ′ → ∃q ′′ [q ⊲ˆσ q ′′ ∧ δ(q ′′ , ˆσ(x)) = q ′ ]].<br />
As with one-sorted transitive automata, one-sorted PTPS automata are finitely axiomatisable.<br />
Proposition 2.6.1 Let n, p ∈ N and I ⊂ N where p is a prime power and I is finite. PTPS<br />
automata are axiomatisable by a finite set <strong>of</strong> first-order axioms in <strong>the</strong> language <strong>of</strong> (V, δ, {T i } i∈I ).<br />
Pro<strong>of</strong>: The fact that PTPS automata are n-tape p-multi-automata can be finitely axiomatised<br />
much as <strong>the</strong> case <strong>of</strong> one-sorted transitive automata. To state <strong>the</strong> remaining properties <strong>of</strong> PTPS<br />
automata recall that SL(V) and P L(V) are finite for finite V. Also recall that P L(V) is <strong>the</strong><br />
closure, under composition, <strong>of</strong> operations definable in <strong>the</strong> language for PTPS automata. qed<br />
We will soon focus on concrete automata (W, h) where h is an isomorphism. It is <strong>the</strong>rfore worth<br />
noting that <strong>the</strong> property <strong>of</strong> being isomorphic to some M p (n, m) is expressible with a first-order<br />
axiom, provided W is finite.<br />
Corollary 2.6.2 Let n, p ∈ N and I ⊂ N where p is a prime number and I is finite. Over finite<br />
structures, <strong>the</strong> class <strong>of</strong> PTPS automata whose alphabet is isomorphic to M p (n, m) for some m ∈ N, is<br />
axiomatisable by a finite set <strong>of</strong> first-order axioms in <strong>the</strong> language <strong>of</strong> (V, δ, {T i } i∈I ).<br />
65
Pro<strong>of</strong>: To prove this we we add to <strong>the</strong> axioms <strong>of</strong> PTPS automata that <strong>the</strong> elements <strong>of</strong> <strong>the</strong> alphabet<br />
are determined by <strong>the</strong>ir components as in lemma 2.2.3.<br />
qed<br />
We now prove a counterpart <strong>of</strong> proposition 2.4.8 for PTPS automata.<br />
Proposition 2.6.3 There exists an effective mapping between finite two-sorted concrete n-tape p-multiautomata<br />
(W, h) and finite one-sorted concrete PTPS automata (W ′ , h ′ ) that is recognition-invariant<br />
in <strong>the</strong> components. Moreover h ′ can be chosen so as to be an isomorphism.<br />
Pro<strong>of</strong>: We describe a procedure to transform a given two-sorted multi-automaton (W, h) into one<br />
with <strong>the</strong> desired properties. First transform (W, h) into its k-outreach (W ′ , h ′ ) where k ∈ N is <strong>the</strong><br />
result <strong>of</strong> doing <strong>the</strong> k-outreach three times. First as in lemma 2.5.4 to make it projectively transitive,<br />
<strong>the</strong>n as in lemma 2.5.6 to make it a projection automaton and finally as in lemma 2.5.8 to make<br />
it have substitutions. The resulting automaton (W ′ , h ′ ) has all three <strong>of</strong> <strong>the</strong> above properties by <strong>the</strong><br />
corollarys to <strong>the</strong> three lemmas. Moreover it is finite when (W, h) is finite. This transformation is<br />
recognition-invariant in <strong>the</strong> components by proposition 2.5.1. Moreover h ′ is an isomorphism by<br />
<strong>the</strong> definition <strong>of</strong> k-outreach. We <strong>the</strong>n transform (W ′ , h ′ ) into a one-sorted automaton (W ′′ , h ′′ ), as<br />
in proposition 2.4.8, <strong>the</strong> step for K 2,ctr → K 1,ctr . Here h ′′ = h ′ and <strong>the</strong>refore is an isomorphism.<br />
qed<br />
As with proposition 2.4.8 we have two corollaries. The first regarding <strong>the</strong> versatility <strong>of</strong> PTPS<br />
automata.<br />
Corollary 2.6.4 Finite one-sorted PTPS automata are versatile enough to replace classical automata in<br />
Büchis decision procedure. More specifically finite and concrete one-sorted projection automata (W, h)<br />
where h is an isomorphism suffice.<br />
The second regarding reachability.<br />
Corollary 2.6.5 Let W = (V, . . .) be a finite concrete projection automaton. Let h from M p (n, m)<br />
onto V be an isomorphism. Define for each t ∈ P L(V) a corresponding t ′ ∈ P L(M p (n, m))<br />
by t ′ = h −1 (t(h(A))). Then in (W, h) <strong>the</strong> relation <strong>of</strong> being h-reachable by a word <strong>of</strong> <strong>the</strong> form<br />
t ′ (A 0 ) ⌢ · · · ⌢t ′ (A l−1 ) is definable by <strong>the</strong> first-order formula ∃z[δ(q, t(z)) = q ′ ].<br />
Note that <strong>the</strong>re is no reference to <strong>the</strong> isomorphism h in <strong>the</strong> first-order formula. This means that<br />
<strong>the</strong> formula defines <strong>the</strong> relevant reachability relation regardless <strong>of</strong> which particular isomorphism, h,<br />
one chooses.<br />
2.6.2 Properties <strong>of</strong> <strong>the</strong> one-sorted transition function<br />
By proposition 2.6.3 one-sorted PTPS automata (W, h) where h is an isomorphism are as versatile<br />
as <strong>the</strong> o<strong>the</strong>r classes <strong>of</strong> concrete automata we have looked at so far. Also by corollary 2.6.2 having<br />
an alphabet that is isomorphic to some M p (n, m) is first-order definable over finite structures. We<br />
may <strong>the</strong>refore restrict attention to (W, h) where h is an isomorphism from now on.<br />
Lemma 2.6.6 Let W = (V, δ, . . .) be a one-sorted n-tape p-multi-automaton. Let h be an n-fold<br />
isomorphism from M p (n, m) to V. Let <strong>the</strong> concrete automaton (W, h) have substitutions. Let ˆσ ∈<br />
SL(M p (n, m)). Let A 0 , . . . , A k−1 ∈ M p (n, m). Then both <strong>the</strong> following hold for positive k ∈ N.<br />
66
1. δ ∗ h(0, A 0 ⌢ · · · ⌢A k−1 ) ⊲ˆσ δ ∗ h(0, ˆσ(A 0 ) ⌢ · · · ⌢ˆσ(A k−1 )).<br />
2. If δ ∗ h(0, A 0 ⌢ · · · ⌢A k−1 ) ⊲ˆσ q <strong>the</strong>n δ ∗ h(0, ˆσ(A 0 ) ⌢ · · · ⌢ˆσ(A k−1 )) ⋄ˆσ q.<br />
Pro<strong>of</strong>:<br />
1. Trivial by repeated use <strong>of</strong> 1. in <strong>the</strong> definition <strong>of</strong> having substitution.<br />
2. This follows from <strong>the</strong> definition <strong>of</strong> ⋄ˆσ , i.e. definition 3.3.1, which by a suitable instantiation<br />
<strong>of</strong> <strong>the</strong> variables says that<br />
(a) if δ ∗ h(0, A 0 ⌢ · · · ⌢A k−1 ) ⊲ˆσ δ ∗ h(0, ˆσ(A 0 ) ⌢ · · · ⌢ˆσ(A k−1 )) and<br />
(b) δ ∗ h(0, A 0 ⌢ · · · ⌢A k−1 ) ⊲ˆσ q <strong>the</strong>n<br />
(c) δ ∗ h(0, ˆσ(A 0 ) ⌢ · · · ⌢ˆσ(A k−1 )) ⋄ˆσ q<br />
Here (a) follows from part 1. <strong>of</strong> this lemma and (b) is <strong>the</strong> assumption <strong>of</strong> part 2. <strong>of</strong> this<br />
lemma.<br />
The following two lemmas are useful when in <strong>the</strong> next section we look at automata for recognising<br />
interpretations <strong>of</strong> formulae beginning with an existential quantifier.<br />
Lemma 2.6.7 Let (V, δ, . . .) be a PTPS automaton. Let h be an isomorphism from M p (n, m) to V.<br />
Let i < n and A 0 , . . . , A k−1 ∈ M p (n, m). Then for each state q <strong>the</strong> following are equivalent.<br />
1. There exist B 0 , . . . , B l−1 ∈ M p (n, m) s.t.<br />
ˆπ i (A 0 ) ⌢ · · · ⌢ˆπ i (A k−1 ) ⌢ 0 n×m ⌢ · · · ⌢0 n×m =<br />
ˆπ i (A 0 ) ⌢ · · · ⌢ˆπ i (A k−1 ) ⌢ˆπ i (B 0 ) ⌢ · · · ⌢ˆπ i (B l−1 ) and s.t.<br />
δ ∗ h(0, A 0 ⌢ · · · ⌢A k−1 ⌢ B 0 ⌢ · · · ⌢B l−1 ) = q.<br />
2. There exists C ∈ M p (n, m) s.t.<br />
ˆπ i (A 0 ) ⌢ · · · ⌢ˆπ i (A k−1 ) ⌢ 0 n×m =<br />
ˆπ i (A 0 ) ⌢ · · · ⌢ˆπ i (A k−1 ) ⌢ˆπ i (C) and s.t. δ ∗ h(0, A 0 ⌢ · · · ⌢A k−1 ⌢ C) = q.<br />
Pro<strong>of</strong>: 1. ⇒ 2. : Let q = δ ∗ h(0, A 0 ⌢ · · · ⌢A k−1 ⌢ B 0 ⌢ · · · ⌢B l−1 ). Recall that ˆπ i replaces every<br />
entry on <strong>the</strong> i’th row with a 0, <strong>the</strong>refore from 1. we may conclude that B 0 ⌢ · · · ⌢B l−1 is an n-tape<br />
whose entries are 0 except possibly on <strong>the</strong> i’th row. Let t ∈ P L(V) be <strong>the</strong> linear transformation that<br />
replaces every entry not on <strong>the</strong> i’th row with a 0. Then t(B 0 ) ⌢ · · · ⌢t(B l−1 ) = B 0 ⌢ · · · ⌢B l−1 .<br />
Now q is reachable from δ ∗ h(0, A 0 ⌢ · · · ⌢A k−1 ) by a word <strong>of</strong> <strong>the</strong> form<br />
t(B 0 ) ⌢ · · · ⌢t(B l−1 ). Since (W, h) is projectively transitive <strong>the</strong>re exists a C s.t. t(C) = C and<br />
such that δ ∗ h(0, A 0 ⌢ · · · ⌢A k−1 ⌢ C) ∈ T φ .<br />
2. ⇒ 1. : Immediate when letting l = 1 and B 0 = C. qed<br />
Lemma 2.6.8 Let (V, δ, . . .) be an PTPS automaton. Let h be an isomorphism from M p (n, m) to<br />
V. Associate t ′ ∈ P L(M p (n, m)) with each t ∈ P L(V) by <strong>the</strong> equation<br />
t ′ (A) = h −1 (t(h(A))). Let A 0 , . . . , A k−1 ∈ M p (n, m). Then δ ∗ h(0, A 0 ⌢ . . . ⌢ A k−1 ) t ∼ q iff<br />
<strong>the</strong>re exist B 0 , . . . , B k−1 ∈ M p (n, m) such that<br />
t ′ (A 0 ) ⌢ · · · ⌢t ′ (A k−1 ) = t ′ (B 0 ) ⌢ · · · ⌢t ′ (B k−1 ) and δ ∗ h(0, B 0 ⌢ . . . ⌢ B k−1 ) = q.<br />
67<br />
qed
Pro<strong>of</strong>: (If): For tapes <strong>of</strong> length 1 we prove this as follows. By <strong>the</strong> definition <strong>of</strong> ∼ t <strong>the</strong>re is an x s.t.<br />
δ(0, x) = q. Since h is an isomorphism this case <strong>of</strong> <strong>the</strong> lemma follows by letting B 0 = h −1 (x).<br />
For longer tapes we combine this with repeated use <strong>of</strong> <strong>the</strong> property <strong>of</strong> lemma 2.5.10.2.<br />
(<strong>On</strong>ly if): This follows by repeated use <strong>of</strong> <strong>the</strong> property <strong>of</strong> lemma 2.5.10.1.<br />
qed<br />
2.7 The automata for a formula and its sub-formulae<br />
Here we investigate how PTPS automata used in Büchis procedure for deciding a first-order sentence<br />
about <strong>the</strong> structure (N, +, | p ) relate. We are in particular interested in <strong>the</strong> relationship between<br />
<strong>the</strong> automaton associated with a formula and <strong>the</strong> automata associated with its sub-formulae.<br />
We provide a sense in which this relationship is first-order.<br />
The fact that PTPS automata are multi-automata is now put to use. We switch to using a set<br />
<strong>of</strong> first-order formulae as index-set for <strong>the</strong> terminal states, as this is convenient when we set up a<br />
correspondence between automata and relations defined by formulae. The following proposition<br />
states that concrete one-sorted projection multi-automata are as versatile as finite sets <strong>of</strong> classical<br />
automata.<br />
Definition Let W = (V, δ, T ) be an n-tape p-automaton. Let h be an isomorphism from<br />
M p (n, m) to V. Then L(W, h) ⊆ N n denotes is <strong>the</strong> relation recongised by (W, h).<br />
Proposition 2.7.1 There is an effective mapping from <strong>the</strong> set <strong>of</strong> finite sets X <strong>of</strong> classical finite n-<br />
tape p-automata (one- or two-sorted) to finite concrete PTPS automata, (W, h), such that for each<br />
(W ′ , h ′ ) ∈ X <strong>the</strong>re exists a component (W φ , h) <strong>of</strong> (W, h) such that L(W ′ , h ′ ) = L(W φ , h).<br />
Pro<strong>of</strong>: Let X be a finite set <strong>of</strong> classical finite n-tape p-automata. We now describe a procedure<br />
to build a one-sorted n-tape p-multi-automaton out <strong>of</strong> X. By proposition 2.4.8 <strong>the</strong>re exists a<br />
recognition-invariant mapping, f from <strong>the</strong> classical automata (one- or two-sorted) to <strong>the</strong> finite<br />
two-sorted concrete automata K 2,con . By lemma 2.5.2 we are able to concatenate <strong>the</strong> elements <strong>of</strong><br />
f(X) into a finite two-sorted n-tape p-multi automaton, which by proposition 2.6.3 we can turn<br />
into a one-sorted PTPS automaton with <strong>the</strong> desired property.<br />
qed<br />
Part <strong>of</strong> <strong>the</strong> procedure to decide a given n-variable formula, φ, in <strong>the</strong> language <strong>of</strong> (N, +, | p ), is<br />
to associate one concrete n-tape p-automaton (W ′ ψ, h ′ ) with each sub-formula ψ <strong>of</strong> φ. By <strong>the</strong> just<br />
proven proposition 2.7.1 <strong>the</strong>re is for all φ in <strong>the</strong> language <strong>of</strong> (N, +, | p ), a concrete PTPS automaton<br />
(W, h) such that for each sub-formula ψ <strong>of</strong> φ <strong>the</strong>re is a component (W ψ , h) that recognises <strong>the</strong><br />
interpretation <strong>of</strong> ψ in (N, +, | p ). The components <strong>of</strong> (W, h) that recognise interpretations for<br />
formulae and immediate sub-formulae are related as follows.<br />
Proposition 2.7.2 Let W = (V, δ, {T φ } φ∈Γ ) be an PTPS automaton. Let h be an isomorphism from<br />
M p (n, m) to V. Let R be <strong>the</strong> set <strong>of</strong> states reachable from <strong>the</strong> initial state, which since W is projectively<br />
transitive is <strong>the</strong> set <strong>of</strong> states q defined by <strong>the</strong> formula ∃x[δ(0, x) = q]. Consider <strong>the</strong> operations ˆπ i<br />
∼, ⊲ˆσ<br />
and ⋄ˆσ as defined in <strong>the</strong> language <strong>of</strong> PTPS automata.<br />
1. Negation: Let {φ, ¬φ} ⊆ Γ. Then<br />
W |= ∀q ∈ R[T ¬φ (q) ↔ ¬T φ (q)]<br />
iff<br />
(W ¬φ , h) recognises <strong>the</strong> complement <strong>of</strong> <strong>the</strong> relation recognised by (W φ , h).<br />
68
2. Disjunction: Let {φ, ψ, φ ∨ ψ} ⊆ Γ. Then<br />
W |= ∀q ∈ R[T φ∨ψ (q) ↔ (T φ (q) ∨ T ψ (q))]<br />
iff<br />
(W φ∧ψ , h) recognises <strong>the</strong> union <strong>of</strong> <strong>the</strong> relations recognised by (W φ , h) and (W ψ , h).<br />
3. Substituting variables for variables:<br />
Let ˆσ ∈ SL(M p (n, m)). Let {R(v 0 , . . . , v n−1 ), R(v σ(0) , . . . , v σ(n−1) )} ⊆ Γ <strong>the</strong>n<br />
W |= ∀q ∈ R[T R(vσ(0) ,...,v σ(n−1) )(q) ↔ ∃q ′ [q ⊲ˆσ q ′ ∧ T R(v0 ,...,v n−1 )(q ′ )]]<br />
∧∀q ′ q ′′ ∈ R[q ′ ⋄ˆσ q ′′ → (T R(v0 ,...,v n−1 )(q ′ ) ↔ T R(v0 ,...,v n−1 )(q ′′ ))]<br />
iff<br />
L(W R(vσ(0) ,...,v σ(n−1) ), h) = {(x 0 , . . . , x n−1 ) : (x σ(0) , . . . , x σ(n−1) ) ∈ L(W R(v0 ,...,v n−1 ), h)}<br />
4. Existential quantification:<br />
Let i ≤ n. Let {φ, ∃v i φ} ⊆ Γ. Then<br />
W |= ∀q ∈ R[(T ∃vi φ(q) ↔ ∃q ′ , x[q ˆπ i<br />
∼ q ′ ∧ 0 = ˆπ i (x) ∧ T φ (δ(q ′ , x))]]<br />
iff<br />
(W ∃vi φ, h) recognises <strong>the</strong> set <strong>of</strong> (x 0 , . . . , x n−1 ) ∈ N n s.t. <strong>the</strong>re exists a<br />
(y 0 , . . . , y n−1 ) ∈ N n that is equal to (x 0 , . . . , x n−1 ) except possibly in <strong>the</strong> i’th component and<br />
s.t. (W φ , h) recognises (y 0 , . . . , y n−1 ).<br />
Pro<strong>of</strong>: We treat n-ary relations on N as subsets <strong>of</strong> M p (n, m) ∗ ⌢ O n×N . Recall that <strong>the</strong> elements<br />
<strong>of</strong><br />
M p (n, m) ∗ ⌢ O n×N are n-tapes where <strong>the</strong> entries on each row signifies <strong>the</strong> p-nary expansion <strong>of</strong> a<br />
natural number. In <strong>the</strong> following we let A 0 , . . . , A k−1 ∈ M p (n, m).<br />
1. Negation: Make <strong>the</strong> assumptions <strong>of</strong> <strong>the</strong> lemma. Then (V, δ, T ¬φ , h) recognises <strong>the</strong> tuple represented<br />
by A 0 ⌢ · · · ⌢A k−1 iff δ ∗ h(0, A 0 ⌢ · · · ⌢A k−1 ) ∈ T ¬φ , which by <strong>the</strong> assumptions<br />
<strong>of</strong> <strong>the</strong> lemma is iff it is not <strong>the</strong> case that δ ∗ h(A 0 ⌢ · · · ⌢A k−1 ) ∈ T φ , which is iff <strong>the</strong> tuple<br />
represented by A 0 ⌢ · · · ⌢A k−1 is not recognised by (V, δ, T φ , h).<br />
2. Disjunction: This is fairly similar to <strong>the</strong> negation case and left to <strong>the</strong> reader.<br />
3. Substituting variables for variables: Let Rv and Rσv be short for <strong>the</strong> two atomic formulae<br />
occurring in <strong>the</strong> proposition.<br />
(if) We prove <strong>the</strong> equality by showing that <strong>the</strong> sets are included in one ano<strong>the</strong>r under <strong>the</strong><br />
assumption that W |= ∀q ∈ R[T Rσv (q) ↔ ∃q ′ [q ⊲ˆσ q ′ ∧ T Rv (q ′ )]]. and <strong>the</strong> assumption that<br />
W |= ∀q ′ q ′′ ∈ R[q ′ ⋄ˆσ q ′′ → (T R(v0 ,...,v n−1 )(q ′ ) ↔ T R(v0 ,...,v n−1 )(q ′′ ))] We refer <strong>the</strong> first <strong>of</strong><br />
<strong>the</strong>se this as <strong>the</strong> assumed equivalence and <strong>the</strong> second as <strong>the</strong> assumed coherence.<br />
(⊆) Let A 0 ⌢ · · · ⌢A k−1 represent a tuple (x 0 , . . . , x n−1 ) ∈ L(W Rσv , h). By definition<br />
<strong>of</strong> acceptance and recognition this translates to δ ∗ h(0, A 0 ⌢ · · · ⌢A k−1 ) ∈ T Rσv . By <strong>the</strong><br />
assumed equivalence <strong>the</strong>re exists a q ′ such that δ ∗ h(0, A 0 ⌢ · · · ⌢A k−1 ) ⊲ˆσ q ′ and such that<br />
q ′ ∈ T Rv . By part two <strong>of</strong> lemma 2.6.6 it is <strong>the</strong> case that δ ∗ h(0, ˆσ(A 0 ) ⌢ · · · ⌢ˆσ(A k−1 )) ⋄ˆσ q ′<br />
and by <strong>the</strong> assumed coherence δ ∗ h(0, ˆσ(A 0 ) ⌢ · · · ⌢ˆσ(A k−1 )) ∈ T Rv .<br />
Hence (x σ(0) , . . . , x σ(n−1) ) ∈ L(W Rv , h).<br />
69
(⊇) Let ˆσ(A 0 ) ⌢ · · · ⌢ˆσ(A k−1 ) represent a tuple (x σ(0) , . . . , x σ(n−1) ) ∈ L(W Rv , h). By<br />
definition <strong>of</strong> acceptance and recognition this translates to δ ∗ h(0, ˆσ(A 0 ) ⌢ · · · ⌢ˆσ(A k−1 )) ∈<br />
T Rv . By part one <strong>of</strong> lemma 2.6.6 we get<br />
δ ∗ h(0, A 0 ⌢ · · · ⌢A k−1 ) ⊲ˆσ δ ∗ h(0, ˆσ(A 0 ) ⌢ · · · ⌢ˆσ(A k−1 )). By <strong>the</strong> assumed equivalence we<br />
conclude that δ ∗ h(0, A 0 ⌢ · · · ⌢A k−1 ) ∈ T Rσv and hence that (x 0 , . . . , x n−1 ) ∈ L(W Rσv , h).<br />
(only if) For this direction we assume that<br />
L(W R(σv) , h) = {(x 0 , . . . , x n−1 ) : (x σ(0) , . . . , x σ(n−1) ) ∈ L(W Rv , h)}. Then we show<br />
coherence, i.e. that W |= ∀q ′ q ′′ ∈ R[q ′ ⋄ˆσ q ′′ → (T R(v0 ,...,v n−1 )(q ′ ) ↔ T R(v0 ,...,v n−1 )(q ′′ ))]<br />
and finally <strong>the</strong> rest. Our assumption we translate into a statement about infinite tapes as<br />
follows: A ⌢ 0 n×N ∈ L(W Rσv , h) iff ˆσ(A) ⌢ 0 n×N ∈ L(W Rv , h).<br />
To show coherence we proceed as follows. Let q be an arbitrary reachable state, and q ′ , q ′′ such<br />
that q ′ ⋄ˆσ q ′′ . By def <strong>of</strong> ⋄ <strong>the</strong>re must be symbols A and B such that δ(0, hA) = δ(0, hB) = q<br />
and such that δ(0, ˆσhA) = q ′ and δ(0, ˆσhB) = q ′′ . Now<br />
q ′ ∈ T Rv<br />
iff δ(0, ˆσhA) ∈ T Rv by choosing δ(0, ˆσhA) = q ′<br />
iff δ ∗ h(0, ˆσA) ∈ T Rv<br />
iff ˆσA ⌢ 0 n×N ∈ L(W Rv , h)<br />
iff A ⌢ 0 n×N ∈ L(W Rσv , h)<br />
by <strong>the</strong> translated assumption<br />
iff δ ∗ h(0, A) ∈ T Rσv<br />
iff δ ∗ h(0, B) ∈ T Rσv<br />
since by choice <strong>of</strong> A and B we have δ(0, hA) = δ(0, hB) = q<br />
iff B ⌢ 0 n×N ∈ L(W Rσv , h)<br />
iff ˆσB ⌢ 0 n×N ∈ L(W Rv , h)<br />
iff δ ∗ h(0, ˆσB) ∈ T Rv<br />
iff δ(0, ˆσhB) ∈ T Rv<br />
iff q ′′ ∈ T Rv by choosing δ(0, ˆσhB) = q ′′<br />
Now we show that each <strong>of</strong> <strong>the</strong> directions in <strong>the</strong> equivalence in W |= ∀q ∈ R[T Rσv (q) ↔<br />
∃q ′ [q ⊲ˆσ q ′ ∧ T Rv (q ′ )]] holds.<br />
(→) Let q be a reachable state such that q ∈ T Rσv . Let A ∈ M p (n, m) be a symbol by<br />
which q is reached. This means that δ(0, h(A)) = q. From q ∈ T Rσv we infer that <strong>the</strong><br />
number represented by A is in L(W Rσv , h). We now define <strong>the</strong> q ′ that fulfils <strong>the</strong> right side<br />
<strong>of</strong> ↔ by q ′ = δ(0, h(ˆσ(A))). By definition <strong>of</strong> ⊲ˆσ we have q ⊲ˆσ q ′ . By <strong>the</strong> assumed equality<br />
<strong>the</strong> tuple <strong>of</strong> natural numbers represented by ˆσ(A) is a member <strong>of</strong> L(W Rv , h). Therefore<br />
δ(0, h(ˆσ(A))) ∈ T Rv . Since q ′ = δ(0, h(ˆσ(A))) we get q ′ ∈ T Rv .<br />
70
(←) Again we let q be reachable and A ∈ M p (n, m) such that δ(0, h(A)) = q. For this<br />
direction we assume that <strong>the</strong>re exists a q ′ such that q ⊲ˆσ q ′ and q ′ ∈ T Rv . By <strong>the</strong> definition<br />
<strong>of</strong> ⋄ˆσ we get δ(0, h(ˆσ(A))) ⋄ˆσ q ′ and by <strong>the</strong> recently shown coherence it follows<br />
that δ(0, h(ˆσ(A))) ∈ T Rv . Therefore <strong>the</strong> tuple <strong>of</strong> numbers which ˆσ(A) represents is in<br />
L(W Rv , h). By <strong>the</strong> assumed equality this means that <strong>the</strong> tuple <strong>of</strong> numbers which A represents<br />
is in L(W Rσv , h). Hence δ(0, h(A)) ∈ T Rσv wich by definition <strong>of</strong> A means that<br />
q ∈ T Rσv .<br />
4. Existential quantification:<br />
(If): Assume W |= ∀q ∈ R[T ∃vi φ(q) ↔ ∃q ′ , x[q ˆπ i<br />
∼ q ′ ∧ 0 = ˆπ i (x) ∧ T φ (δ(q ′ , x))]].<br />
(T ∃vi φ not too big): Assume that <strong>the</strong>re are A 0 , . . . , A k−1 ∈ M p (n, m) such that<br />
δh(0, ∗ A ⌢ 0 · · · A k−1 ) ∈ T ∃vi φ. We now wish to display l ≥ k and B 0 , . . . , B l−1 ∈ M p (n, m)<br />
such that<br />
A ⌢ 0 · · · ⌢A ⌢ k−1 0 ⌢ n×m · · · ⌢0 n×m and B ⌢ 0 · · · ⌢B l−1 are equal except possibly on <strong>the</strong><br />
i’th row and such that δh(0, ∗ B ⌢ 0 · · · ⌢B l−1 ) ∈ T φ . To do this let l = k + 1. By <strong>the</strong> first <strong>of</strong><br />
<strong>the</strong> current two assumptions <strong>the</strong>re exists a q ′ such that δh(0, ∗ A ⌢ 0 · · · ⌢A k−1 ) ˆπ i<br />
∼ q ′ . Using<br />
this equivalence we do by lemma 2.6.8 get B 0 , . . . , B k−1 ∈ M p (n, m) with <strong>the</strong> property<br />
that that A ⌢ 0 · · · ⌢A k−1 and B ⌢ 0 · · · ⌢B k−1 are equal except possibly on <strong>the</strong> i’th row.<br />
Moreover δh(0, ∗ B ⌢ 0 · · · ⌢B k−1 ) = q ′ . By <strong>the</strong> firs assumption <strong>the</strong>re also is an x such that<br />
δ(q ′ , x) ∈ T φ . Since h is an isomorphism we can define B l−1 = h −1 (x) and we get <strong>the</strong><br />
B 0 , . . . , B l−1 we sought to display.<br />
(T ∃vi φ big enough): Assume that δh(0, ∗ B ⌢ 0 · · · ⌢B k−1 ) ∈ T φ . By <strong>the</strong> definition <strong>of</strong> onesorted<br />
n-tape p-multi-automaton this implies that δh(0, ∗ B ⌢ 0 · · · ⌢B ⌢ k−1 0 n×m ) ∈ T φ . If<br />
we now let both q and q ′ equal δh(0, ∗ B ⌢ 0 · · · ⌢B k−1 ) and let x = 0 we can from <strong>the</strong> first<br />
assumption infer (from <strong>the</strong> right to <strong>the</strong> left side) that δh(0, ∗ B ⌢ 0 · · · ⌢B k−1 ) ∈ T ∃vi φ.<br />
(<strong>On</strong>ly if): Assume now that we have a PTPS automaton with two sets <strong>of</strong> terminal states that<br />
incidentally are indexed like T ∃vi φ and T φ . Assume also that (V, δ, T ∃vi φ, h) recognises <strong>the</strong><br />
set set <strong>of</strong> tapes A ⌢ 0 · · · ⌢A ⌢ k−1 0 n×N such that <strong>the</strong>re exists a<br />
B ⌢ 0 · · · ⌢B ⌢ l−1 0 n×N different from A ⌢ 0 · · · ⌢A ⌢ k−1 0 n×N in at most <strong>the</strong> i’th row<br />
and such that δh(0, ∗ B ⌢ 0 · · · ⌢B l−1 ) ∈ T φ . Note that if l < k + 1 we can extend<br />
B ⌢ 0 · · · ⌢B l−1 by adding 0 n×m ’s at <strong>the</strong> most significant end obtaining a word whose membership<br />
status in T φ is equivalent. Also if l > k + 1 we use <strong>the</strong> fact that our automaton is<br />
projectively transitive as in lemma 2.6.7 and shorten B ⌢ 0 · · · ⌢B l−1 so as to obtain a word<br />
whose membership status in T φ is equivalent and that differs from A ⌢ 0 · · · ⌢A ⌢ k−1 0 n×N<br />
in at most <strong>the</strong> i’th row. Without loss <strong>of</strong> generality we <strong>the</strong>refore assume that l = k + 1.<br />
Recall that ˆπ i ∈ P L(M p (n, m)) is <strong>the</strong> projection that replaces every entry on <strong>the</strong> i’th row<br />
with a 0. Using ˆπ i we translate <strong>the</strong> present assumptions to <strong>the</strong> following equivalence.<br />
For all states <strong>of</strong> <strong>the</strong> form δ ∗ h(0, A 0 ⌢ · · · ⌢A k−1 ) it is <strong>the</strong> case that<br />
δ ∗ h(0, A 0 ⌢ · · · ⌢A k−1 ) ∈ T ∃vi φ iff <strong>the</strong>re exists a state <strong>of</strong> <strong>the</strong> form δ ∗ h(0, B 0 ⌢ · · · ⌢B k ) such<br />
that <strong>the</strong> conjunction <strong>of</strong><br />
(a) ˆπ i (A 0 ) ⌢ · · · ⌢ˆπ i (A k−1 ) = ˆπ i (B 0 ) ⌢ · · · ⌢ˆπ i (B k−1 )<br />
(b) 0 n×m = ˆπ i (B k )<br />
71
(c) δ ∗ h(0, B 0 ⌢ · · · ⌢B k ) ∈ T φ<br />
holds.<br />
Since for each k <strong>the</strong> matrices {A i } i
1. If Γ has an automatic model <strong>the</strong>n to each φ ∈ Γ we can assign a finite concrete twosorted<br />
n-tape p-(multi-)automaton (W φ , h φ ) that recognises <strong>the</strong> definition <strong>of</strong> φ in <strong>the</strong> automatic<br />
model. Using proposition 2.7.1 we can merge <strong>the</strong>se into one finite two-sorted multiautomaton<br />
(W, h) where each component is equivalent to a (W φ , h φ ). The structure W is<br />
finite and clearly a model for Γ ′′ ∪ Γ ′ ∪ {∀q ∈ R[T φ (q)]}.<br />
2. If Γ ′′ ∪ Γ ′ ∪ {∀q ∈ R[T φ (q)]} has a finite model W <strong>the</strong>n W is a multi-automaton. To make<br />
W concrete we use a homomomorphism provided by lemma 2.2.2.<br />
qed<br />
2.8 Concluding remarks<br />
With proposition 2.4.6, corollary 2.4.9 and corollary 2.4.10 we have shown that in finite transitive<br />
automata reachability is first-order, we have shown that <strong>the</strong> class <strong>of</strong> transitive automata is finitely<br />
axiomatisable. Moreover we have shown that <strong>the</strong> finite and concrete representatives <strong>of</strong> <strong>the</strong> class are<br />
versatile enough to replace automata with fixed alphabets in J.R. Büchis procedure for deciding <strong>the</strong><br />
<strong>the</strong>ory <strong>of</strong> automatic structures. With proposition 2.6.1, corollary 2.6.4 and corollary 2.6.5 we have<br />
proven <strong>the</strong> same for PTPS automata.<br />
We have investigated <strong>the</strong> relationship between a PTPS automaton that recognises <strong>the</strong> interpretation<br />
<strong>of</strong> a formula with <strong>the</strong> PTPS automata that recognise <strong>the</strong> interpretation <strong>of</strong> its sub-formulae.<br />
With proposition 3.3.2 we have shown that this relationship is in a sense first-order. The proposition<br />
relies on first-order definability <strong>of</strong> various variants <strong>of</strong> reachability in PTPS automata.<br />
PTPS-automata can be used to construct a semi-desicion procedure that terminates on input<br />
on consistent sentences only, as follows.<br />
1. Let Γ be <strong>the</strong> set <strong>of</strong> sub-formulae <strong>of</strong> <strong>the</strong> input sentence.<br />
2. Let Γ ′′ ∪ Γ ′ be <strong>the</strong> definition <strong>of</strong> <strong>the</strong> PTPS-automaton in corollary 2.7.3.<br />
3. Search for a finite model for Γ ′′ ∪ Γ ′ ∪ {∀q ∈ R[T φ (q)]} and terminate if one is found.<br />
In step 3 we use <strong>the</strong> fact that <strong>the</strong> class <strong>of</strong> PTPS-automata is basic elementary to make <strong>the</strong> procedure<br />
terminate when <strong>the</strong> input sentence has a model definable in an automatic structure. This procedure<br />
terminates on input on at least those infinity axioms that are true in Presburger arithmetic.<br />
<strong>On</strong>e idea <strong>of</strong> transforming a first-order sentence into a first-order description <strong>of</strong> automata and<br />
<strong>the</strong>n to search for a finite model <strong>of</strong> <strong>the</strong> transformed sentence has been proposed by N. Peltier<br />
[Pel09]. To define reachability N. Peltiers transformation introduces <strong>the</strong> element-relation and lets<br />
<strong>the</strong> carrier-set consist <strong>of</strong> sets <strong>of</strong> states. In contrast, <strong>the</strong> transformation outlined in <strong>the</strong> present paper<br />
has a carrier-set whose members are symbols, some <strong>of</strong> which are also used as states. To define<br />
reachability we allow for some flexibility in <strong>the</strong> set <strong>of</strong> symbols. N. Peltier says <strong>of</strong> his transformation<br />
that, “The main interest <strong>of</strong> <strong>the</strong> present work is to prove that <strong>the</strong> translation is feasible from a<br />
<strong>the</strong>oretical point <strong>of</strong> view”. The same can be said <strong>of</strong> <strong>the</strong> transformation proposed in <strong>the</strong> present<br />
paper. To make search for satisfiable interpretations in automatic models remotely feasible from a<br />
practical point <strong>of</strong> view <strong>the</strong> present author proposes to use <strong>the</strong> atom-strucures <strong>of</strong> finite representable<br />
polyadic algebras. How to use PTPS-automata to compute such atom-structures is <strong>the</strong> subject <strong>of</strong><br />
<strong>the</strong> second paper.<br />
73
Chapter 3<br />
Automata for <strong>the</strong> computation <strong>of</strong> finite<br />
representable polyadic algebras<br />
75
3.1 Introduction<br />
This is <strong>the</strong> second <strong>of</strong> two papers on a kind <strong>of</strong> automaton that is suitable for consistency pro<strong>of</strong>s and<br />
for <strong>the</strong> computation <strong>of</strong> atom-structures <strong>of</strong> finite representable polyadic algebras, including some<br />
which have purely infinite spectrum. Finite representable algebras with infinite spectrum are <strong>of</strong><br />
interest in regards to Hilberts Entscheidungsproblem as <strong>the</strong> algebras can be used to construct semidecision<br />
procedures that recognise not only finitely satisfiable sentences as consistent but also some<br />
infinity axioms, see A. Rognes [Rog09]. Infinity axioms are consistent first-order sentences that have<br />
infinite models only. Devising reasonably natural semi-decision procedures that terminate on input<br />
<strong>of</strong> even a single infinity axiom is a challenge. Note, for instance, that an eventually periodic infinite<br />
branch <strong>of</strong> a semantic tree implies finite satisfiability. Similar effects occur with finitely presented<br />
Herbrand models.<br />
Polyadic algebras were introduced by P. R. Halmos who was inspired by A.Tarskis closely related<br />
cylindric algebras. Finite and representable polyadic algebras are ma<strong>the</strong>matical objects that can be<br />
used much like structures and models when recognising given first-order sentences as consistent.<br />
As with structures we may interpret relational sentences in <strong>the</strong>se algebras and only consistent sentences<br />
have satisfying interpretations in representable algebras. Curiously some <strong>of</strong> <strong>the</strong> finite and<br />
representable algebras have satisfying interpretations for infinity axioms. In computations involving<br />
infinity axioms explicitly presented models are generally unsuitable as arguments to computable<br />
functions by virtue <strong>of</strong> being infinite. Finite representable algebras with satisfying interpretations for<br />
infinity axioms however, work well by virtue <strong>of</strong> being finite. The atom-structures <strong>of</strong> finite algebras<br />
are even more suitable as <strong>the</strong>y are considerably smaller that <strong>the</strong> algebra it self. An atom-structure<br />
plays <strong>the</strong> same role for a finite polyadic algebra as does a basis for a vector space or a topology.<br />
The present paper, i.e. part two <strong>of</strong> two papers, can be read independently from part one if one<br />
is willing to accept a few propositions without pro<strong>of</strong>.<br />
Part one makes no use <strong>of</strong> algebraic logic. In it we introduce a basic elementary class <strong>of</strong> multiautomata<br />
and see how <strong>the</strong>se can be used for showing consistency <strong>of</strong> first-order sentences, including<br />
some infinity axioms. Part two requires some knowledge <strong>of</strong> algebraic logic which we apply to <strong>the</strong><br />
multi-automata <strong>of</strong> part one. In part two we show that in <strong>the</strong> finite <strong>of</strong> <strong>the</strong> multi-automata we can<br />
define atom-structures whose complex algebra are representable polyadic algebras. The class <strong>of</strong><br />
atom-structures in question is seen to be definable by a given finite set <strong>of</strong> first-order sentences in<br />
<strong>the</strong> language <strong>of</strong> <strong>the</strong> multi-automata. The fact that <strong>the</strong>se classes are basic elementary implies that<br />
we are able to recursively enumerate <strong>the</strong> atom-structures <strong>of</strong> finite simple representable polyadic<br />
algebras, without computing <strong>the</strong> <strong>the</strong> whole polyadic algebra in question. Simple algebras are <strong>of</strong><br />
interest as <strong>the</strong>se are <strong>the</strong> basic building blocks <strong>of</strong> <strong>the</strong> class at hand. In <strong>the</strong> context <strong>of</strong> algebraic<br />
logic, finite representable algebras are <strong>of</strong> interest in <strong>the</strong>m selves, see H. Andréka and R.D. Maddux<br />
[AM94]. In <strong>the</strong> context <strong>of</strong> Hilberts Entscheidungsproblem <strong>the</strong> atom-structures <strong>of</strong> finite simple<br />
algebras whose spectrum is purely infinite, form a key component in <strong>the</strong> implementation <strong>of</strong> semidecision<br />
procedures that go beyond search for finite models, see A. Rognes[Rog09].<br />
In <strong>the</strong> present paper we consider, <strong>the</strong> atom-structures <strong>of</strong>, n-dimensional one- and many-sorted<br />
variants <strong>of</strong> polyadic algebras. For each kind <strong>of</strong> algebra we have a notion <strong>of</strong> homomorphism, embedding<br />
and sub-algebra. In accordance with <strong>the</strong> pattern <strong>of</strong> H. Andréka and R.D. Maddux [AM94] we<br />
define <strong>the</strong> following. A set algebra is an algebra whose carrier set is a subset <strong>of</strong> P(U n ), <strong>the</strong> power-set<br />
<strong>of</strong> U n . An algebra is representable if it is embeddable into a product <strong>of</strong> set algebras. A representable<br />
76
algebra is simple if it is embeddable into a set algebra. The spectrum <strong>of</strong> a simple representable algebra<br />
is <strong>the</strong> class <strong>of</strong> cardinalities κ such that <strong>the</strong>re exists an embedding <strong>of</strong> <strong>the</strong> simple algebra to a set algebra<br />
with carrier set P(κ). To this we add that a simple representable algebra is said to have purely<br />
infinite spectrum if <strong>the</strong>re is no embedding <strong>of</strong> <strong>the</strong> algebra to any set algebra with carrier set P(U n )<br />
for any finite U. The latter class <strong>of</strong> algebras has satisfying interpretations for infinity axioms.<br />
Note that although <strong>the</strong> algebras considered here are boolean algebras with operators, shortened<br />
BAO’s, <strong>the</strong> notion <strong>of</strong> representability used for BAO’s is to weak for consistency pro<strong>of</strong>s for first-order<br />
language in general. Indeed it follows from a well known result <strong>of</strong> B.Jónsson and A.Tarski that finite<br />
BAO’s are representable over finite sets and so <strong>the</strong>ir spectra necessarily contain finite cardinals, see<br />
B.Jónsson and A.Tarski [JT51] or a survey such as R. Goldblatt [Gol00], Theorem 3.1.<br />
3.1.1 Outline <strong>of</strong> paper<br />
The rest <strong>of</strong> section 1 introduces notation and recalls some definitions on directed many-sorted<br />
polyadic algebras <strong>of</strong> dimension n, or dMsPs n for short, as defined in A. Rognes [Rog09].<br />
In section 2 we define <strong>the</strong> polyadic atom-structure as known from <strong>the</strong> literature. We do however<br />
introduce <strong>the</strong> h-complex algebra <strong>of</strong> a polyadic atom-structure, where h is a homomorphism <strong>of</strong><br />
polyadic atom-structures. The h-complex algebra generalises <strong>the</strong> complex algebra known from <strong>the</strong><br />
literature. A criterion on atom-structures is introduced and we prove that <strong>the</strong> h-complex algebra <strong>of</strong><br />
an atom-structure meeting <strong>the</strong> criterion is representable for a ca<strong>non</strong>ical h.<br />
In section 3 we recall <strong>the</strong> definition <strong>of</strong> PTPS-automata as known from part one, i.e., A.Rognes<br />
[Rog11]. We introduce properly partitioned automata which serve as a kind <strong>of</strong> normal form for<br />
PTPS-automata. We identify a polyadic atom-structure with each properly partitioned automaton<br />
and show that for any finite automaton this atom-structure necessarily meets <strong>the</strong> criterion <strong>of</strong> section<br />
2, and <strong>the</strong>refore has a representable h-complex algebra.<br />
In section 4 we show that every dMsPs n that is generated by a finite set <strong>of</strong> first-order definable<br />
relations over <strong>the</strong> structure (N, +, | p ) is embeddable in <strong>the</strong> h-complex algebra <strong>of</strong> a properly<br />
partitioned finite automaton. Here N are <strong>the</strong> natural numbers, ’+’ is addition and | p is a binary<br />
relation such that x| p y if y is <strong>the</strong> greatest power <strong>of</strong> p such that y divides x. We also show that <strong>the</strong><br />
above embeddability criterion does not hold for n-dimensional polyadic algebras, Ps n , nor does it<br />
hold for many-sorted algebras, MsPs n , unless <strong>the</strong>y are directed. The situation is unchanged when<br />
we consider <strong>the</strong> corresponding classes with diagonals, namely dMsPEs n , MsPEs n and PEs n .<br />
In section 5 we provide finite sets <strong>of</strong> first-order axioms for four variants <strong>of</strong> properly partitioned<br />
automata. These four axiom sets ensure that <strong>the</strong> h-complex algebra <strong>of</strong> (<strong>the</strong> atom-structure <strong>of</strong>)<br />
a finite automaton is a finite dMsPs n , dMsPEs n , Ps n or PEs n respectively. We prove some<br />
result that are <strong>of</strong> interest in regards to recursively enumerating representatives for <strong>the</strong> finite members<br />
<strong>of</strong> <strong>the</strong> four classes.<br />
3.1.2 Sets, relations and mappings<br />
Uppercase letters such as A, B, C, R, X, Y and Z are used for sets. X × Y , X ∪ Y , X ∩ Y , X\Y<br />
and X ⊆ Y are as usual. A set R is a binary relation if it is a subset <strong>of</strong> X × Y . The domain <strong>of</strong> R is<br />
<strong>the</strong> first projection <strong>of</strong> R, making it a, possibly proper, subset <strong>of</strong> X. R is said to be total if its domain<br />
is X. If A ⊆ X <strong>the</strong>n <strong>the</strong> image <strong>of</strong> A under R is {y : x ∈ A, (x, y) ∈ R}. We write R(A) for <strong>the</strong><br />
image <strong>of</strong> A under R. If B ⊆ Y <strong>the</strong>n <strong>the</strong> inverse image <strong>of</strong> B under R is {x : y ∈ B, (x, y) ∈ R}.<br />
77
We write R −1 (B) for <strong>the</strong> inverse image <strong>of</strong> B under R. The range <strong>of</strong> R is <strong>the</strong> image <strong>of</strong> X under R.<br />
If (x, y) ∈ R and (x, y ′ ) ∈ R implies that y = y ′ <strong>the</strong>n R is said to be a partial function from X<br />
to Y . If a partial function happens to be total we call it a mapping from X to Y , or an operation.<br />
The set <strong>of</strong> mappings from X to Y is written (X → Y ). The power-set <strong>of</strong> a set U is written P(U).<br />
If U is a set <strong>the</strong>n B(U) = (P(U), ∪, −) is <strong>the</strong> full boolean set algebra over U. The closure <strong>of</strong> a<br />
family F ⊆ P(U) under finite unions is written Cl ∪ (F). The notions signature, homomorphism,<br />
embedding, (direct) product, expansion and reduction are used as is common in model <strong>the</strong>ory.<br />
3.1.3 Finite-dimensional (quasi) polyadic algebras<br />
We consider finite-dimensional polyadic algebras in <strong>the</strong> present papers and we fixate a dimension<br />
n ∈ N throughout. We also assume that 3 ≤ n, since for 0 ≤ n < 3 representable polyadic<br />
algebras <strong>of</strong> dimension n are representable over finite sets. We make no distinction between quasi<br />
polyadic algebras and polyadic algebras since <strong>the</strong> two notions coincide for finite-dimensions. There<br />
is a variation <strong>of</strong> quasi polyadic algebras due to C. Pinter [Pin73] called quantifier algebras. These<br />
in turn are called substitution-cylindric algebras by I. Németi [Ném91] and H. Andréka and I. Sain<br />
and I. Németi [ASN01].<br />
For cylindric algebras <strong>the</strong> result on representability over finite sets is due to L. Henkin, see<br />
[MHT85] corollary 3.2.66. For (quasi) polyadic algebras it was proven independently by L.<br />
Henkin, and by H. Andréka and I. Németi, see [MHT85] <strong>the</strong>orem 5.4.23. For simplified pro<strong>of</strong>s<br />
<strong>of</strong> both <strong>the</strong>se results and some history see M. Marx and S. Mikulás [MM99].<br />
We shortly define a many-sorted variant <strong>of</strong> quasi polyadic algebras . We begin with <strong>the</strong> signature.<br />
Definition The tuple A = (B n , . . . , B 0 , r, p, s, c 0 , . . . , c n−1 ) is said to have <strong>the</strong> signature <strong>of</strong> an<br />
n-dimensional directed many-sorted polyadic algebra, or to be <strong>of</strong> dMsP n -signature for short, when<br />
B n , · · · , B 0 are algebras which have <strong>the</strong> signature <strong>of</strong> a boolean algebra, i.e. for 0 ≤ i ≤ n we have<br />
that B i = (B i , ∨ B i<br />
, ¬ B i<br />
, ⊥ B i<br />
),<br />
r, p, s : B n → B n are operations that are called rotation, permutation and substitution and relate<br />
to variable substitutions,<br />
for 0 ≤ i < n it is <strong>the</strong> case that c i : B i+1 → B i is an operation called cylindrification and it relates<br />
to existential quantification.<br />
Note that this signature differs slightly from that found in <strong>the</strong> literature in that we use only r, p and<br />
s for <strong>the</strong> operations that relate to variable substitutions. This difference is not important as long as<br />
<strong>the</strong> intended interpretation <strong>of</strong> <strong>the</strong> symbols suffices to generate all variable substitutions. Example<br />
Let (K,
defined by <strong>the</strong> formulae v 0 = v 1 , v 1 = v 2 and v 0 = v 2 . <strong>On</strong>e <strong>the</strong>n ends up with 6 tetrahedra,<br />
wedged between <strong>the</strong>se <strong>the</strong>re are 6 triangles and finally <strong>the</strong>re is a diagonal where <strong>the</strong> three<br />
defining planes meet.<br />
B 2 = (B A 2 , ∪, −, ∅) is <strong>the</strong> set algebra where B 2 ⊆ P(K 3 ) is <strong>the</strong> set <strong>of</strong> ternary relations definable<br />
using sentences <strong>of</strong> <strong>the</strong> form ∃v 2 φ where φ defines a relation in B 3 . This algebra has 3 atoms,<br />
which can be visualised by slicing a cube along <strong>the</strong> plane defined by v 0 = v 1 .<br />
B 1 = (B A 1 , ∪, −, ∅) is <strong>the</strong> set algebra where B 1 ⊆ P(K 3 ) is <strong>the</strong> set <strong>of</strong> ternary relations definable<br />
using sentences <strong>of</strong> <strong>the</strong> form ∃v 1 φ where φ defines a relation in B 2 . This algebra has 2<br />
elements.<br />
B 0 = (B A 0 , ∪, −, ∅) is <strong>the</strong> set algebra where B 0 ⊆ P(K 3 ) is <strong>the</strong> set <strong>of</strong> ternary relations definable<br />
using sentences <strong>of</strong> <strong>the</strong> form ∃v 0 φ where φ defines a relation in B 1 . This algebra has 2<br />
elements.<br />
When X ⊆ K 3 we let<br />
c A 0 (X) = {(x 0 , x 1 , x 2 )|∃y(y ∈ K ∧ (y, x 1 , x 2 ) ∈ X)},<br />
c A 1 (X) = {(x 0 , x 1 , x 2 )|∃y(y ∈ K ∧ (x 0 , y, x 2 ) ∈ X)},<br />
c A 2 (X) = {(x 0 , x 1 , x 2 )|∃y(y ∈ K ∧ (x 0 , x 1 , y) ∈ X)}.<br />
These operations are called cylindrifications for <strong>the</strong> reason that geometrically <strong>the</strong>y would turn a ball<br />
into a cylinder. In this example <strong>the</strong>re are no balls, but<br />
tetrahedra are turned into prisms,<br />
<strong>the</strong> diagonal is turned into planes,<br />
prisms are turned into prisms or <strong>the</strong> cube depending on <strong>the</strong> axis we cylindrify along,<br />
planes are turned into planes or <strong>the</strong> cube depending on <strong>the</strong> axis we cylindrify along,<br />
triangles are turned into prisms or planes depending on <strong>the</strong> axis we cylindrify along,<br />
<strong>the</strong> cube and <strong>the</strong> empty set remain as <strong>the</strong>y are.<br />
When X ⊆ K 3 we let<br />
r A (X) = {(x 0 , x 1 , x 2 )|(x 2 , x 0 , x 1 ) ∈ X} geometrically X is rotated around <strong>the</strong> axis x 0 = x 1 =<br />
x 2 an angle a third <strong>of</strong> <strong>the</strong> full circle,<br />
p A (X) = {(x 0 , x 1 , x 2 )|(x 1 , x 0 , x 2 ) ∈ X} geometrically X is mirrored in <strong>the</strong> xy plane,<br />
s A (X) = {(x 0 , x 1 , x 2 )|(x 1 , x 1 , x 2 ) ∈ X} geometrically <strong>the</strong> part <strong>of</strong> X that meets <strong>the</strong> xy-diagonal<br />
and that is orthogonal to xy plane is cylindrified.<br />
As usual in algebraic logic we now generalise <strong>the</strong> last example to n dimensions and to as many n-ary<br />
relations as possible on a single set U.<br />
79
Definition Let U be a set. The full n-dimensional directed many-sorted polyadic set algebra over U or<br />
<strong>the</strong> full dMsPs n over U for short, is <strong>the</strong><br />
U = (B(U n ), . . . , B(U n ), r U , p U , s U , c U 0 , . . . , c U n−1) <strong>of</strong> dMsP n -signature where<br />
.<br />
B(U n ) = (P(U n ), ∪, −, ∅) is <strong>the</strong> boolean algebra <strong>of</strong> subsets <strong>of</strong> <strong>the</strong> set U n .<br />
r U (X) = {(x 0 , . . . , x n−1 ) ∈ U n |(x n−1 , x 0 , x 1 . . . , x n−2 ) ∈ X}, i.e. subtract one from each<br />
index modulo n,<br />
p U (X) = {(x 0 , . . . , x n−1 ) ∈ U n |(x 1 , x 0 , x 2 . . . , x n−1 ) ∈ X}, i.e. swap <strong>the</strong> first and <strong>the</strong> second<br />
component,<br />
s U (X) = {(x 0 , . . . , x n−1 ) ∈ U n |(x 1 , x 1 , x 2 . . . , x n−1 ) ∈ X}, i.e. overwrite <strong>the</strong> first component<br />
with <strong>the</strong> second.<br />
c U 0 (X) = {(x 0 , . . . , x n−1 ) ∈ U n |∃y(y ∈ U ∧ (y, x 1 , . . . , x n−1 ) ∈ X}<br />
.<br />
c U i (X) = {(x 0 , . . . , x n−1 ) ∈ U n |∃y(y ∈ U ∧ (x 0 , . . . , x i−1 , y, x i+1 , . . . , x n−1 ) ∈ X}<br />
c U n−1(X) = {(x 0 , . . . , x n−1 ) ∈ U n |∃y(y ∈ U ∧ (x 0 , . . . , x n−2 , y) ∈ X}<br />
Example The full dMsPs 3 over K. This algebra has four uncountable sorts, all equal to P(K 3 ).<br />
The operations <strong>of</strong> this algebra are defined as in example 3.1.3. We are particularly interested in<br />
finitely generated sub-algebras <strong>of</strong> full dMsPs n ’s over infinite sets, as <strong>the</strong>se are suitable as objects <strong>of</strong><br />
computation as long as <strong>the</strong>y are abstract, see A. Rognes [Rog09].<br />
Definition Let A = (B n , . . . , B 0 , . . .) and A ′ = (B ′ n, . . . , B ′ 0, . . .) be two algebras <strong>of</strong> dMsP n -<br />
signature. A dMsP n -homomorphism from A to A ′ is a tuple f = (f n , . . . , f 0 ) <strong>of</strong> mappings such<br />
that<br />
f n : B n → B ′ n is a boolean homomorphism,<br />
.<br />
f 0 : B 0 → B ′ 0 is a boolean homomorphism,<br />
f n preserves <strong>the</strong> operations r, p and s,<br />
for i < n it is <strong>the</strong> case that f i (c i (x)) = c ′ i(f i+1 (x)).<br />
The homomorphism f is said to be a dMsP n -embedding if each <strong>of</strong> f 0 , . . . , f n is a boolean embedding.<br />
Using embeddings we now define <strong>the</strong> objects which our main result is about.<br />
80
Definition An algebra A <strong>of</strong> dMsP n -signature is said to be an n-dimensional many-sorted polyadic<br />
set algebra, or a dMsPs n for short, if <strong>the</strong>re exists a full dMsPs n , say U, and a dMsP n -embedding<br />
from A to U.<br />
Note that dMsPs n ’s are not required to be subsets <strong>of</strong> full algebras, only isomorphic to such. Example<br />
The algebra <strong>of</strong> dMsP 3 -signature generated by <strong>the</strong> ternary relations definable by quantifier<br />
free formulae in <strong>the</strong> language <strong>of</strong> (K,
Pro<strong>of</strong>: We turn a fragment <strong>of</strong> first-order language into an algebra L crc<br />
n <strong>of</strong> dMsP n -signature as<br />
follows. The sort with index n consist <strong>of</strong> boolean combinations <strong>of</strong> atomic formulae built from<br />
a countably infinite supply <strong>of</strong> n-ary relation symbols. The sort with index i consist <strong>of</strong> boolean<br />
combinations <strong>of</strong> formulae <strong>of</strong> <strong>the</strong> form ∃x i φ, where φ is <strong>of</strong> <strong>the</strong> sort with index i + 1. See see<br />
A. Rognes [Rog09] for fur<strong>the</strong>r details. Interpretations <strong>of</strong> L crc<br />
n -formulae are nothing more than<br />
dMsP n -homomorphisms from L crc<br />
n to o<strong>the</strong>r algebras <strong>of</strong> dMsP n -signature. An interpretation f<br />
is satisfying for a sentence φ if f 0 (φ) = ⊤.<br />
(⇒): Assume that A = (B n , . . .) has purely infinite spectrum. We construct an infinity axiom<br />
φ <strong>of</strong> L crc<br />
n , that has a satisfying interpretation f in A, by describing A up to isomorphism. For<br />
this construction we need one n-ary relation symbol, R, for each element <strong>of</strong> B n . The satisfying<br />
interpretation <strong>of</strong> φ is uniquely determined by mapping <strong>the</strong> atomic formula R a (x 0 , . . . , x n−1 ) to<br />
<strong>the</strong> element a ∈ B n . Since A is simple <strong>the</strong>re is for each element a ∈ B i a formula ψ such that<br />
f i (ψ) = a. This allows us to describe, in L crc<br />
n , what <strong>the</strong> result <strong>of</strong> any operation <strong>of</strong> A is, on any<br />
arguments. Since A is finite, <strong>the</strong> conjunction <strong>of</strong> <strong>the</strong>se descriptions is again a sentence in L crc<br />
n . The<br />
conjunction is clearly satisfied by A and if it had a model over a finite set U <strong>the</strong>n A would be<br />
embeddable into <strong>the</strong> full dMsPs n over U. But A was assumed to have purely infinite spectrum<br />
so <strong>the</strong> conjunction is an infinity axiom.<br />
(⇐): Assume now that φ is an infinity axiom <strong>of</strong> L crc<br />
n and let f be a satisfying interpretation, i.e.,<br />
a homomorphism from L crc<br />
n to A with <strong>the</strong> property that f 0 (φ) = ⊤. Assume, for <strong>the</strong> purpose <strong>of</strong><br />
arriving at a contradiction, that <strong>the</strong>re is a finite cardinal number in spec(A). This means that <strong>the</strong>re<br />
is a finite set U and a homomorphism f ′ from A into <strong>the</strong> full dMsPs n over U. The composition<br />
<strong>of</strong> f with f ′ now is an interpretation <strong>of</strong> φ in <strong>the</strong> full dMsPs n over U. The composition maps φ<br />
to ⊤ since homomorphisms preserve ⊤. But <strong>the</strong>n φ has a model whose carrier set is <strong>the</strong> finite set<br />
U. This contradicts <strong>the</strong> assumption that φ is an infinity axiom.<br />
qed<br />
3.2 The polyadic atom-structure and <strong>the</strong> h-complex algebra<br />
We define <strong>the</strong> polyadic analog <strong>of</strong> atom-structure, following <strong>the</strong> terminology <strong>of</strong> R. Hirsch and I. M.<br />
Hodkinson, [Hod97] [HH09], who work with relation and cylindric algebras ra<strong>the</strong>r than polyadic<br />
algebras. The polyadic <strong>version</strong> <strong>of</strong> atom-structures are used in <strong>the</strong> definition <strong>of</strong> complex algebra in<br />
<strong>the</strong> book <strong>of</strong> L.Henkin, J.D.Monk and A.Tarski [MHT85], but called relational structure. Finite<br />
polyadic atom-structures correspond to (finite many-sorted) polyadic algebras but atom-structures<br />
are preferable as objects <strong>of</strong> computation as an atom-structure with k elements has a corresponding<br />
algebra with 2 k elements. Since n is fixed throughout, we simply write atom-structure to mean<br />
n-dimensional polyadic atom-structure in this section.<br />
3.2.1 Atom-structures and n-homomorphisms<br />
Here <strong>the</strong> atom-structure is formally defined. Moreover we define a special kind <strong>of</strong> homomorphism<br />
designed to correspond to dMsP n -embeddings. The homomorphisms are called n-homomorphisms<br />
and are believed to be new.<br />
Definition Let n ∈ N. An n-dimensional polyadic atom-structure is a tuple<br />
(H, ⊳ r , ⊳ p , ⊳ s , E 0 , . . . , E n−1 ), where<br />
82
H is a set <strong>of</strong> elements thought <strong>of</strong> as atoms <strong>of</strong> a boolean algebra.<br />
⊳ r , ⊳ p , ⊳ s are binary relations, which correspond to substitution <strong>of</strong> variables for variables.<br />
E 0 , . . . , E n−1 are binary relations, which correspond to existential quantifiers.<br />
By <strong>the</strong> following we associate an atom-structure with each atomic dMsPs n . The associated atomstructure<br />
plays <strong>the</strong> same role as does a basis associated with a vector space. Example Let A =<br />
(B n , . . . , B 0 , r, p, s, c 0 , . . . , c n−1 ) be a dMsPs n such that B n , . . . , B 0 are atomic. Then <strong>the</strong> atomstructure<br />
<strong>of</strong> A is <strong>the</strong> n-dimensional polyadic atom-structure (H, ⊳ r , ⊳ p , ⊳ s , E 0 , . . . , E n−1 ), defined<br />
by <strong>the</strong> following.<br />
H = At(B n )∪. . .∪At(B 0 ), i.e. H is <strong>the</strong> union <strong>of</strong> <strong>the</strong> atoms <strong>of</strong> <strong>the</strong> boolean algebras B n , . . . , B 0 ,<br />
for σ ∈ {r, p, s} it is <strong>the</strong> case that x ⊳ σ y iff σ(x) is defined and y ≤ σ(x), here ≤ is <strong>the</strong> usual<br />
ordering <strong>of</strong> <strong>the</strong> boolean algebra B n<br />
for i ≤ n it is <strong>the</strong> case that xE i y iff c i (x) is defined and y ≤ c i (x).<br />
The following is an explicit definition <strong>of</strong> <strong>the</strong> atom-structure <strong>of</strong> a full dMsPs n .<br />
Definition Let U be a set. The n-dimensional set polyadic atom-structure over U is <strong>the</strong> tuple<br />
(U n , ⊳ U r , ⊳ U p , ⊳ U s , E0 U , . . . , En−1), U where<br />
U n is <strong>the</strong> set <strong>of</strong> n-tuples <strong>of</strong> elements <strong>of</strong> U.<br />
⊳ U r ⊆ U n × U n is defined by ⊳ U r = {(u, u ′ )|(u 0 , u 1 , u 2 . . . , u n−1 ) = (u ′ n−1, u ′ 0, u ′ 1, . . . , u ′ n−2)},<br />
i.e. subtract one from each index modulo n.<br />
⊳ U p ⊆ U n × U n is defined by ⊳ U p = {(u, u ′ )|(u 0 , u 1 , u 2 . . . , u n−1 ) = (u ′ 1, u ′ 0, u ′ 2, . . . , u ′ n−1)}, i.e.<br />
swap <strong>the</strong> first and <strong>the</strong> second component.<br />
⊳ U s ⊆ U n × U n is defined by ⊳ U s = {(u, u ′ )|(u 0 , u 1 , u 2 . . . , u n−1 ) = (u ′ 1, u ′ 1, u ′ 2, . . . , u ′ n−1)}, i.e.<br />
overwrite <strong>the</strong> first component with <strong>the</strong> second.<br />
E U 0<br />
= {(u, u ′ )|(u 1 , . . . , u n−1 ) = (u ′ 1, . . . , u ′ n−1)}<br />
.<br />
E U i<br />
= {(u, u ′ )|(u 0 , . . . , u i−1 ) = (u ′ 0, . . . , u ′ i−1) ∧ (u i+1 , . . . , u n−1 ) = (u ′ i+1, . . . , u ′ n−1)}<br />
.<br />
E U n−1 = {(u, u ′ )|(u 0 , . . . , u n−2 ) = (u ′ 0, . . . , u ′ n−2)}<br />
Definition Let {r, p, s} be a set <strong>of</strong> formal symbols. Let H and H ′ be atom-structures. An n-<br />
homomorphism from H to H ′ is an n + 1 tuple <strong>of</strong> mappings h = (h n , . . . , h 0 ) such that<br />
for σ ∈ {r, p, s}, if q ⊳ σ q ′ <strong>the</strong>n h n (q) ⊳ ′ σ h n (q ′ ).<br />
for i < n, if qE i q ′ <strong>the</strong>n h i+1 (q)E ′ ih i (q ′ ).<br />
83
3.2.2 The complex algebra tailored for many sorts<br />
We introduce a, believed to be new, generalisation <strong>of</strong> <strong>the</strong> complex algebra <strong>of</strong> an atom-structure<br />
which is suitable for many-sorted polyadic algebras. The definition depends on an n-homomorphism,<br />
h, on an atom-structure H and is denoted H h + . If each component <strong>of</strong> h = (h n, . . . , h 0 )<br />
is <strong>the</strong> identity mapping on H <strong>the</strong>n H h + is nothing more than <strong>the</strong> complex algebra H+ know form<br />
<strong>the</strong> literature, see eg. R.Goldblatt [Gol00].<br />
In <strong>the</strong> following definition we use <strong>the</strong> fact that a mapping h on a set H induces a partition <strong>of</strong><br />
H. A partition is a family <strong>of</strong> pairwise disjoin and <strong>non</strong>empty subsets <strong>of</strong> H whose union is all <strong>of</strong> H.<br />
The elements <strong>of</strong> <strong>the</strong> partition are called parts and <strong>the</strong> part that a given q ∈ H belongs to is written<br />
[q] h .<br />
Definition Let {r, p, s} be a set <strong>of</strong> formal symbols. Let H = (H, ⊳ r , ⊳ p , ⊳ s , E 0 , . . . , E n−1 ) and<br />
H ′ = (H ′ , ⊳ ′ r, ⊳ ′ p, ⊳ ′ s, E 0, ′ . . . , E n−1) ′ be polyadic atom-structures. Let h be an n-homomorphism<br />
from H to H ′ . Then <strong>the</strong> h-complex algebra <strong>of</strong> H , denoted H h + , is <strong>the</strong> algebra<br />
(B n , . . . , B 0 , r + , p + , s + , c + 0 , . . . , c + n−1) <strong>of</strong> dMsP n -signature defined as follows.<br />
1. For i < n let B i = (B i , ∪, −, ∅), where B i = Cl ∪ ({h −i<br />
i h i q|q ∈ H}). Here B i is <strong>the</strong><br />
closure under unions <strong>of</strong> <strong>the</strong> partition induced by h i . This makes B i <strong>the</strong> boolean set algebra<br />
generated by <strong>the</strong> partition induced by h i .<br />
2. For σ ∈ {r, p, s} each σ + : B n → B n is defined first on parts, <strong>the</strong>n on unions <strong>of</strong> parts as<br />
follows<br />
(a) for q ∈ H we let σ + ([q] hn ) = ⊳ σ ([q] hn ), i.e. <strong>the</strong> image <strong>of</strong> [q] hn under ⊳ σ .<br />
(b) for Q ⊆ H let σ + ( ⋃ {[q] hn |q ∈ Q}) = ⋃ {σ + ([q] hn )|q ∈ Q}, here <strong>the</strong> last occurrence<br />
<strong>of</strong> σ + is applied to parts on which σ + is already defined.<br />
3. for i < n each c + i : B i+1 → B i is defined first on parts, <strong>the</strong>n on unions <strong>of</strong> parts as follows<br />
(a) for q ∈ H we let c + i ([q] hi+1 ) = E i ([q] hi+1 ), i.e. <strong>the</strong> image <strong>of</strong> [q] hi+1 under E i .<br />
(b) for Q ⊆ H let c + i ( ⋃ {[q] hi+1 |q ∈ Q}) = ⋃ {c + i ([q] hi+1 ))|q ∈ Q}.<br />
Definition Let H = (H, . . .) be an n-dimensional polyadic atom-structure. Let id : H → H be<br />
<strong>the</strong> identity mapping. Let h = (id, . . . , id) be <strong>the</strong> identity n-homomorphism on H. Then <strong>the</strong><br />
complex algebra <strong>of</strong> H, denoted H + , is <strong>the</strong> algebra H + h .<br />
Lemma 3.2.1 Let U be a set. Let U be <strong>the</strong> n-dimensional set polyadic atom-structure over U. Then<br />
U + is <strong>the</strong> full dMsPs n over U.<br />
Pro<strong>of</strong>: Left to <strong>the</strong> reader.<br />
qed<br />
We now give a criterion for a h-complex algebra to be a dMsPs n , and <strong>the</strong>refore representable,<br />
ra<strong>the</strong>r than any odd algebra <strong>of</strong> dMsP n -signature. The criterion depends on a possibly infinite<br />
external set polyadic atom-structure. The existence <strong>of</strong> such an infinite external set polyadic atomstructure<br />
may sometimes be inferred from intrinsic criteria. We consider one such intrinsic criterion<br />
later in <strong>the</strong> present paper where we apply <strong>the</strong> following proposition to automata.<br />
84
Proposition 3.2.2 Let {r, p, s} be a set <strong>of</strong> formal symbols. Let U be a set.<br />
Let U = (U n , ⊳ U r , . . . , E U 0 , . . .) be <strong>the</strong> n-dimensional set polyadic atom-structure over U. Let H =<br />
(H, ⊳ r , . . . , E 0 , . . .) and H ′ = (H ′ , ⊳ ′ r, . . . , E ′ 0, . . .) be polyadic atom-structures such that <strong>the</strong>re exists<br />
an n-homomorphism g from U to H, and an n-homomorphism h from H to H ′ with <strong>the</strong> following two<br />
properties.<br />
1. For σ ∈ {r, p, s} and q ∈ H we have gn<br />
−1 (⊳ σ ([q] hn )) = ⊳ σ (gn −1 ([q] hn )).<br />
2. For i < n, q ∈ H we have gi<br />
−1 (E i ([q] hi+1 )) = E i (gi+1([q] −1<br />
hi+1 )).<br />
Then <strong>the</strong> h-complex algebra H + h is representable (by an embedding into U + ).<br />
Pro<strong>of</strong>: We intend to show that H h + ∈ dMsPs n by displaying an embedding f from H h + to U + .<br />
This implies representability. So define f = (f n , . . . , f 0 ) from H h<br />
+ to U + as follows. For each<br />
i ≤ n we define f i : Cl ∪ {[q] hi |q ∈ H} → Cl ∪ {gi<br />
−1 ([q] hi )|q ∈ H} first on parts <strong>the</strong>n on unions<br />
<strong>of</strong> parts.<br />
1. For q ∈ H let f i [q] hi = g −1<br />
i [q] hi .<br />
2. For Q ⊆ H let f i ( ⋃ {[q] hi |q ∈ Q}) = ⋃ {gi<br />
−1 [q] hi |q ∈ Q}.<br />
The fact that each f i is a boolean embedding is easy to see, so we skip <strong>the</strong> formal pro<strong>of</strong>. For<br />
σ ∈ {r, p, s} we now prove that f n (σ(X)) = σ(f n (X)), first for parts <strong>the</strong>n for unions <strong>of</strong> parts.<br />
The first occurrence <strong>of</strong> σ is defined on H + h and <strong>the</strong> second on U + .<br />
1. Let q ∈ H. Now<br />
u ∈ f n σ[q] hn<br />
iff u ∈ g −1<br />
n σ[q] hn by definition <strong>of</strong> f<br />
iff u ∈ g −1<br />
n (⊳ σ [q] hn ) by definition <strong>of</strong> σ on parts <strong>of</strong> H + h<br />
iff u ∈ ⊳ U σ (g −1<br />
n [q] hn ) by <strong>the</strong> first <strong>of</strong> <strong>the</strong> assumed properties <strong>of</strong> g and h<br />
iff u ∈ σg −1<br />
n [q] hn σ is defined in <strong>the</strong> full dMsPs n over U, see lemma 3.2.1<br />
iff u ∈ σf n [q] hn .<br />
From this we conclude that f n (σ(X)) = σ(f n (X)) when X is a part.<br />
2. Let Q ⊆ H. Now<br />
f n (σ( ⋃ q∈Q[q] hn ))<br />
= f n ( ⋃ q∈Q σ([q] hn )) by definition <strong>of</strong> σ on unions <strong>of</strong> parts <strong>of</strong> H + h<br />
= ⋃ q∈Q f n (σ([q] hn )) f n was just seen to be boolean embedding<br />
= ⋃ q∈Q σ(f n ([q] hn )) f n and σ were just seen to commute on parts<br />
= σ( ⋃ q∈Q f n ([q] hn )) by definition <strong>of</strong> σ on unions <strong>of</strong> parts <strong>of</strong> U + , recall lemma 3.2.1<br />
= σ(f n ( ⋃ q∈Q[q] hn )). f n was just seen to be a boolean embedding<br />
From this we conclude that f n (σ(X)) = σ(f n (X)) when X is a union <strong>of</strong> parts.<br />
For each i ∈ N we can prove that f i+1 (c i (X)) = c i (f i (X)) for parts and unions <strong>of</strong> parts, X, in<br />
exactly <strong>the</strong> same way.<br />
qed<br />
85
3.3 The h-complex algebra <strong>of</strong> a multi-automaton<br />
We define PTPS-automata as introduced in A. Rognes [Rog11] and define operations on <strong>the</strong>m<br />
so as to make <strong>the</strong>m a polyadic atom-structure. This allows us to define <strong>the</strong> h-complex algebra<br />
<strong>of</strong> a PTPS-automaton. We use proposition 3.2.2 to show that <strong>the</strong> h-complex algebra <strong>of</strong> a finite<br />
PTPS-automaton necessarily is representable.<br />
3.3.1 Concrete PTPS-automata defined<br />
We recall definitions and some basic facts on PTPS-automata here, see A.Rognes [Rog11] for fur<strong>the</strong>r<br />
details. PTPS-automata use matrices over finite fields both as alphabet and as states. We write<br />
F p for <strong>the</strong> unique finite field whose order is <strong>the</strong> prime power p. Moreover M p (n, m) is <strong>the</strong> set <strong>of</strong> n<br />
times m matrices with entries form F p . We let 0 n×m ∈ M p (n, m) be <strong>the</strong> matrix whose entries are<br />
all 0 ∈ F p . Likewise 0 n×N is <strong>the</strong> matrix with n infinite rows whose entries are all 0 ∈ F p .<br />
By definition <strong>the</strong> set M p (n, m) ∗ = ⋃ k∈N M p (n, k · m). The concatenation <strong>of</strong> <strong>the</strong> two matrices<br />
A, A ′ ∈ M p (n, m) ∗ is written A ⌢ A ′ and is an element <strong>of</strong> M p (n, m) ∗ . We write M p (n, m) ∗ ⌢ {0 n×N }<br />
for <strong>the</strong> set <strong>of</strong> n times N matrices with finite support and call <strong>the</strong>m n-tapes.<br />
The set M p (n, m) is a vector-space over F p when <strong>the</strong> operations are defined component-wise<br />
and <strong>the</strong> tuple M p (n, m) stands for this vector-space. We let P L(M p (n, m)) stand for <strong>the</strong> set <strong>of</strong><br />
linear-transformations on M p (n, m) that replace every entry on a given set <strong>of</strong> rows with 0’s. The<br />
following is an example <strong>of</strong> an element <strong>of</strong> P L(M p (n, m)) which is <strong>of</strong> special interest to us.<br />
Definition The mapping π : M p (n, m) → M p (n, m) is defined by<br />
⎡<br />
π(<br />
⎢<br />
⎣<br />
⎤<br />
A 0<br />
A 1<br />
. ⎥<br />
⎦<br />
A n−1<br />
⎡<br />
) =<br />
⎢<br />
⎣<br />
⎤<br />
A 0<br />
0 1×m<br />
. ⎥<br />
⎦<br />
0 1×m<br />
We let SL(M p (n, m)) stand for <strong>the</strong> set <strong>of</strong> linear-transformations on M p (n, m) that swaps or<br />
overwrites rows with o<strong>the</strong>r rows according to some mapping σ on row-indices. The following is an<br />
example <strong>of</strong> an element <strong>of</strong> SL(M p (n, m)) which also is <strong>of</strong> special interest to us.<br />
Definition The mapping r : M p (n, m) → M p (n, m) is defined by<br />
⎡<br />
r(<br />
⎢<br />
⎣<br />
⎤<br />
A 0<br />
A 1<br />
. ⎥<br />
⎦<br />
A n−1<br />
⎡<br />
) =<br />
⎢<br />
⎣<br />
A 1<br />
.<br />
A n−1<br />
A 0<br />
⎤<br />
⎥<br />
⎦<br />
Using <strong>the</strong>se two definition we now define a class <strong>of</strong> special vector-spaces which serve as alphabet<br />
(and states) <strong>of</strong> PTPS-automata.<br />
Definition A concrete n-fold vector-space is a tuple<br />
V p,n,m = (M p (n, m), π, r) where M p (n, m), π, r are defined as above.<br />
86
We consider p, n, m as fixed throughout so we mostly write V ra<strong>the</strong>r than V p,n,m .<br />
We write O n×N for <strong>the</strong> concrete one-element n-fold vector-space over F p whose sole element is<br />
<strong>the</strong> infinite n-tape whose entries are all 0. What follows are <strong>the</strong> objects that PTPS-automata devour.<br />
It is <strong>the</strong> set <strong>of</strong> infinite n-tapes with operations that allow us to compare and overwrite tracks on <strong>the</strong><br />
tape, i.e. rows <strong>of</strong> matrices with finite support.<br />
Definition V ∗ p,n,m ⌢ O n×N is <strong>the</strong> concrete n − fold vector-space whose carrier set is<br />
M p (n, m) ∗ ⌢ {0 n×N }, where <strong>the</strong> vector-space operations are defined component-wise and where<br />
π(A 0 ⌢ · · · ⌢A k−1 ⌢ 0 n×N ) = π(A 0 ) ⌢ · · · ⌢π(A k−1 ) ⌢ 0 n×N<br />
and<br />
r(A 0 ⌢ · · · ⌢A k−1 ⌢ 0 n×N ) = r(A 0 ) ⌢ · · · ⌢r(A k−1 ) ⌢ 0 n×N<br />
Each track holds <strong>the</strong> p-nary expansion <strong>of</strong> a natural number. Thus <strong>the</strong>re is a natural one to one<br />
correspondence between V ∗ p,n,m ⌢ O n×N and n-tuples <strong>of</strong> natural numbers.<br />
We shall define PTPS-automata in two steps. First we define its signature <strong>the</strong>n after having<br />
defined some notions on <strong>the</strong> signature we introduce <strong>the</strong> axioms in a second step.<br />
Definition Let J be a finite set. A concrete automaton <strong>of</strong> PTPS-signature is a<br />
W = (V, δ, {T j } j∈J ) s.t.<br />
V = (V, +, . . .) is a concrete n-fold vector-space thought <strong>of</strong> as both alphabet and state set,<br />
δ : V × V → V is <strong>the</strong> transition function,<br />
for each j ∈ J <strong>the</strong> set T j ⊆ V is a set <strong>of</strong> terminal states.<br />
We mention that <strong>the</strong> zero-vector <strong>of</strong> V is used as <strong>the</strong> initial state. We will use variables such as<br />
q, q ′ , q ′′ . . . for elements <strong>of</strong> <strong>the</strong> vector-space when we think <strong>of</strong> <strong>the</strong>m as states. When we think<br />
<strong>of</strong> <strong>the</strong>m as symbols we use A, A ′ , B, C, A 0 , . . . instead. Automata recognise sets <strong>of</strong> n-tapes, and<br />
<strong>the</strong>refore sets <strong>of</strong> n-tuples <strong>of</strong> numbers, by means <strong>of</strong> <strong>the</strong> following.<br />
Definition Let W = (V, δ, {T j } j∈J ) be <strong>of</strong> PTPS-signature. Let V = (V, . . .). Then:<br />
1. δ ∗ : V × V ∗ → V is defined by δ ∗ (q, A ⌢ A ′ ) = δ(δ ∗ (q, A), A ′ ),<br />
2. W j = (V, δ, T j ) and is called <strong>the</strong> j-th projection <strong>of</strong> W.<br />
3. W j recognises A ∈ M p (n, m) ∗ if δ ∗ (0, A) ∈ T j .<br />
We now define two relations on automata that relate to <strong>the</strong> existential quantifier.<br />
Definition Let ˆπ i ∈ P L(V) be <strong>the</strong> linear transformation that replaces every entry on <strong>the</strong> i-th row<br />
with a 0. Let t ∈ P L(V) be arbitrary. Then,<br />
1.<br />
t<br />
∼⊆ V × V is defined by q t ∼ q ′ if <strong>the</strong>re exist A, A ′ ∈ V such that t(A) = t(A ′ ) and<br />
δ(0, A) = q and δ(0, A ′ ) = q ′ ,<br />
2. E i ⊆ V × V is defined by qE i q ′ if δ(q, 0) ˆπ i<br />
∼ q ′ and q can be reached from <strong>the</strong> initial state<br />
in one step, i.e., <strong>the</strong>re exists an A ∈ V such that q = δ(0, A).<br />
87
Similarly we define two relations on automata that relate to substitution <strong>of</strong> variables for variables.<br />
Definition Let ˆσ ∈ SL(V), i.e., ˆσ is a linear transformation that swaps and overwrites columns<br />
according to some mapping on track indices. Then,<br />
1. ⊳ˆσ ⊆ V × V is defined by q ⊳ˆσ q ′ is <strong>the</strong>re exists an A ∈ V such that δ(0, ˆσ(A)) = q and<br />
δ(0, A) = q ′ ,<br />
2. ⋄ˆσ ⊆ V × V is defined by q ⋄ˆσ q ′ if <strong>the</strong>re exists a q ′′ ∈ V such that q ⊳ˆσ q ′′ and q ′ ⊳ˆσ q ′′ .<br />
As was seen in A.Rognes [Rog11] <strong>the</strong> elements <strong>of</strong> P L(V) and SL(V) are definable using <strong>the</strong><br />
operations <strong>of</strong> automata <strong>of</strong> PTPS-signature, more specifically <strong>the</strong> operations π, r and +. It follows<br />
that for each t and σ <strong>the</strong> relations t ∼, E i , ⋄ˆσ , ⊳ˆσ are definable as well. We use this observation to<br />
shorten <strong>the</strong> definition <strong>of</strong> PTPS-automata now.<br />
Definition A concrete PTPS-automaton is a W = (V, δ, {T j } j∈J ) <strong>of</strong> concrete PTPS-signature such<br />
that, For each j ∈ J, each t ∈ P L(V) and ˆσ ∈ SL(V) <strong>the</strong> following axioms hold:<br />
1. W |= T j (q) ↔ T j (δ(q, 0)), to ensure well definedness <strong>of</strong> relations recognised by automata,<br />
2. W |= ∀qAB∃C(δ(δ(q, t(A)), t(B)) = δ(q, t(C))), i.e., a state reachable by a tape <strong>of</strong> <strong>the</strong><br />
form t(A) ⌢ t(B) is also reachable by a tape <strong>of</strong> <strong>the</strong> form t(C),<br />
3. W |= ∀qq ′ AB(q t ∼ q ′ ∧ t(A) = t(B) → δ(q, A) t ∼ δ(q ′ , B)), e.g, if t = ˆπ i and A ⌢ A ′ and<br />
B ⌢ B ′ are tapes that are <strong>the</strong> same except on <strong>the</strong> i-th row <strong>the</strong>n δ ∗ (0, A ⌢ A ′ ) ˆπ i<br />
∼ δ ∗ (0, B ⌢ B ′ ),<br />
4. W |= ∀qq ′ A(δ(q, A) t ∼ q ′ → ∃q ′′ B(q t ∼ q ′′ ∧ t(A) = t(B) ∧ δ(q ′′ , B) = q ′ )), e.g., if<br />
t = ˆπ i and δ ∗ (0, A ⌢ A ′ ) ˆπ i<br />
∼ q ′ <strong>the</strong>n <strong>the</strong>re exists a tape B ⌢ B ′ that is <strong>the</strong> same as A ⌢ A ′<br />
except possibly on <strong>the</strong> i-th row such that δ ∗ (0, B ⌢ B ′ ) = q ′ see A.Rognes [Rog11],<br />
5. W |= ∀qq ′ A(q ′ ⊳ˆσ q → δ(q ′ , ˆσ(A)) ⊳ˆσ δ(q, A)), this is similar to 3,<br />
6. W |= ∀qq ′ A(q ′ ⊳ˆσ δ(q, A) → ∃q ′′ (q ′′ ⊳ˆσ q ∧ δ(q ′′ , ˆσ(A)) = q ′ )) this is similar to 4.<br />
The first axiom schema here is to ensure that we can extend <strong>the</strong> definition <strong>of</strong> recognition from finite<br />
tapes to infinite tapes with finite support, and thus tuples <strong>of</strong> natural numbers, as follows.<br />
Definition For A ⌢ 0 n×N ∈ V ∗ p,n,m ⌢ O n×N we say that W j = (V, δ, T j ) recognises A ⌢ 0 n×N if<br />
δ ∗ (0, A) ∈ T j .<br />
Note that <strong>the</strong> definition works regardless <strong>of</strong> which <strong>of</strong> <strong>the</strong> many possible p-nary expansions A for<br />
a given tuple <strong>of</strong> numbers we have chosen, i.e., regardless <strong>of</strong> how many 0’s <strong>the</strong>re are at <strong>the</strong> most<br />
significant end <strong>of</strong> A.<br />
It is known that with each definable relation <strong>of</strong> an automatic structure, we can associate a<br />
classical finite n-tape p-automaton that recognises exactly <strong>the</strong> tapes that represent tuples <strong>of</strong> <strong>the</strong><br />
definable relation, see, J.R. Büchi [B¨60]. We recall one <strong>of</strong> <strong>the</strong> main results from A.Rognes [Rog11],<br />
namely Proposition 7.2. The result allows us to merge finite sets <strong>of</strong> classical automata into one<br />
finite concrete PTPS-automaton.<br />
88
Proposition 3.3.1 There is an effective mapping from <strong>the</strong> set <strong>of</strong> finite sets X <strong>of</strong> classical n-tape p-<br />
automata to <strong>the</strong> set <strong>of</strong> finite concrete PTPS-automata W = (V, δ, {T j } j∈J ), such that for each W ′ ∈ X<br />
<strong>the</strong>re exists a j ∈ J such that W j recognises <strong>the</strong> same relation as does W ′ .<br />
Pro<strong>of</strong>: We outline a pro<strong>of</strong>, for a full pro<strong>of</strong> see A.Rognes [Rog11]. The mapping is defined by<br />
constructing a PTPS-automaton as follows. We take <strong>the</strong> product <strong>of</strong> <strong>the</strong> automata in X and obtain<br />
a two-sorted multi-automaton Π(X). The alphabet <strong>of</strong> Π(X) is M p (n, 1). We <strong>the</strong>n replace <strong>the</strong> <strong>the</strong><br />
alphabet <strong>of</strong> Π(X) with one <strong>of</strong> <strong>the</strong> form M p (n, m) where m is chosen so as to make <strong>the</strong> resulting<br />
automaton a two-sorted PTPS-automaton, W. Finally we imbed <strong>the</strong> states <strong>of</strong> W into <strong>the</strong> alphabet<br />
so as to obtain a one-sorted PTPS-automaton.<br />
qed<br />
We also recall proposition 7.3.3 in A.Rognes [Rog11]. The proposition tells us how <strong>the</strong> components<br />
<strong>of</strong> a PTPS-automaton, that recognise interpretations for first-order formulae and immediate subformulae<br />
are related. Note that q ′ ⊳ˆσ q here means <strong>the</strong> same as q ⊲ˆσ q ′ in A.Rognes [Rog11]. We<br />
use a set <strong>of</strong> formulae Γ as index set for <strong>the</strong> sole purpose <strong>of</strong> increasing readability.<br />
Proposition 3.3.2 Let W = (V, δ, {T φ } φ∈Γ ) be a concrete PTPS automaton. Let R be <strong>the</strong> set <strong>of</strong><br />
states reachable from <strong>the</strong> initial state, which since W is projectively transitive is <strong>the</strong> set <strong>of</strong> states, q,<br />
defined by <strong>the</strong> formula ∃x[δ(0, x) = q]. Consider <strong>the</strong> operations ˆπ i<br />
∼, ⊳ˆσ and ⋄ˆσ as defined in <strong>the</strong><br />
language <strong>of</strong> PTPS automata.<br />
1. Negation: Let {φ, ¬φ} ⊆ Γ. Then<br />
W |= ∀q ∈ R[T ¬φ (q) ↔ ¬T φ (q)]<br />
iff<br />
W ¬φ recognises <strong>the</strong> complement <strong>of</strong> <strong>the</strong> relation recognised by W φ .<br />
2. Disjunction: Let {φ, ψ, φ ∨ ψ} ⊆ Γ. Then<br />
W |= ∀q ∈ R[T φ∨ψ (q) ↔ (T φ (q) ∨ T ψ (q))]<br />
iff<br />
W φ∧ψ recognises <strong>the</strong> union <strong>of</strong> <strong>the</strong> relations recognised by W φ and W ψ .<br />
3. Substituting variables for variables:<br />
Let σ : n → n and let ˆσ ∈ SL(M p (n, m)) be <strong>the</strong> operation that swaps and overwrites rows<br />
according to <strong>the</strong> mapping σ. Let {R(v 0 , . . . , v n−1 ), R(v σ(0) , . . . , v σ(n−1) )} ⊆ Γ <strong>the</strong>n<br />
W |= ∀q ∈ R[T R(vσ(0) ,...,v σ(n−1) )(q) ↔ ∃q ′ [q ′ ⊳ˆσ q ∧ T R(v0 ,...,v n−1 )(q ′ )]]<br />
∧∀q ′ q ′′ ∈ R[q ′ ⋄ˆσ q ′′ → (T R(v0 ,...,v n−1 )(q ′ ) ↔ T R(v0 ,...,v n−1 )(q ′′ ))]<br />
iff<br />
W R(vσ(0) ,...,v σ(n−1) ) recognises {(x 0 , . . . , x n−1 ) : (x σ(0) , . . . , x σ(n−1) ) ∈ L(W R(v0 ,...,v n−1 ))}<br />
where L(W R(v0 ,...,v n−1 )) is <strong>the</strong> set recognised by W R(vσ(0) ,...,v σ(n−1) ).<br />
4. Existential quantification:<br />
Let i ≤ n. Let {φ, ∃v i φ} ⊆ Γ. Then<br />
W |= ∀q ∈ R[(T ∃vi φ(q) ↔ ∃q ′ , x[q ˆπ i<br />
∼ q ′ ∧ 0 = ˆπ i (x) ∧ T φ (δ(q ′ , x))]]<br />
iff<br />
W ∃vi φ recognises <strong>the</strong> set <strong>of</strong> (x 0 , . . . , x n−1 ) ∈ N n s.t. <strong>the</strong>re exists a<br />
(y 0 , . . . , y n−1 ) ∈ N n that is equal to (x 0 , . . . , x n−1 ) except possibly in <strong>the</strong> i’th component and<br />
s.t. W φ recognises (y 0 , . . . , y n−1 ).<br />
89
3.3.2 Properly partitioned automata<br />
We introduce <strong>the</strong> notion <strong>of</strong> a properly partitioned automaton. The notion can be thought <strong>of</strong> as<br />
a normal form for PTPS-automata. Automata <strong>of</strong> this form can be turned into polyadic atomstructures<br />
fairly directly.<br />
The second axiom schema, in <strong>the</strong> definition <strong>of</strong> PTPS-automata, allows us to give a first order<br />
definition <strong>of</strong> reachability. We continue using <strong>the</strong> symbol R for <strong>the</strong> set <strong>of</strong> reachable states, for which,<br />
in <strong>the</strong> case <strong>of</strong> PTPS-automata, we can give a first-order definition as follows. The set R ⊆ V<br />
consists <strong>of</strong> <strong>the</strong> states q such that ∃A(δ(0, A) = q).<br />
Definition Let W = (V, δ, {T j } j∈J ) be a concrete PTPS-automaton. Then W is said to be<br />
properly partitioned if<br />
1. {T j } j∈J is a partition <strong>of</strong> R,<br />
2. For each j ∈ J and ˆσ ∈ SL(V) we have W |= ∀q ′ q ′′ ∈ R(q ′ ⋄ˆσ q ′′ → (T j (q ′ ) ↔ T j (q ′′ ))).<br />
We simply say properly partitioned automaton to mean a properly partitioned PTPS automaton. The<br />
reason that we may think <strong>of</strong> properly partitioned automata as being in normal form is <strong>the</strong> following<br />
proposition.<br />
Proposition 3.3.3 Let W = (V, δ, {T j } j∈J ) be a PTPS-automaton such that:<br />
1. J is finite,<br />
2. For each j ∈ J <strong>the</strong>re exists a k ∈ J such that T j ∪ T k = R<br />
3. For each σ : n → n and W j that recognises <strong>the</strong> relation R(v 0 , . . . , v n−1 ) <strong>the</strong>re exists a k ∈ J<br />
such that W k recognises R(v σ(0) , . . . , v σ(n−1) ),<br />
Then <strong>the</strong>re is a properly partitioned automaton W ′ = (V, δ, {T j} ′ j∈J ′) such that {T j} ′ j∈J ′<br />
<strong>the</strong> same boolean algebra as does {T j } j∈J .<br />
generates<br />
Pro<strong>of</strong>: The proposition follows when we let {T j} ′ j∈J ′ consist <strong>of</strong> <strong>the</strong> atoms <strong>of</strong> <strong>the</strong> boolean algebra<br />
that {T j } j∈J generates. Then each T j ′ = ⋂ {T ′ j } j∈K for some K ⊆ J. Now clearly condition 1 <strong>of</strong><br />
being properly partitioned holds. If we now assume that condition 2 <strong>of</strong> being properly partitioned<br />
does not hold <strong>the</strong>n for some ˆσ ∈ SL(V) and q ′ , q ′′ ∈ R it is <strong>the</strong> case that q ′ ⋄ˆσ q ′′ and q ′ ∈ T j ′ ′<br />
whilst q ′′ /∈ T j ′ ′. Therefore for some j ∈ K it must be <strong>the</strong> case that q′ ∈ T j whilst q ′′ /∈ T j . If<br />
we let R(v 0 , . . . , v n−1 ) denote <strong>the</strong> relation that A j recognises <strong>the</strong>n by assumption 3 <strong>of</strong> <strong>the</strong> present<br />
proposition <strong>the</strong>re is an automaton A k that recognises R(v σ(0) , . . . , v σ(n−1) ). But this is in violation<br />
with proposition 3.3.2, which states that if W k recognises R(v σ(0) , . . . , v σ(n−1) ) and W j recognises<br />
R(v 0 , . . . , v n−1 ) <strong>the</strong>n T j is closed under ⋄ˆσ .<br />
qed<br />
3.3.3 Automata as atom-structures<br />
We carefully select <strong>the</strong> binary relations that occur in <strong>the</strong> signature <strong>of</strong> an n-dimensional polyadic<br />
atom-structure amongst <strong>the</strong> relations definable on a properly partitioned automaton.<br />
Definition The three mappings ˆr, ŝ, ˆp ∈ SL(V p,n,m ) are defined as follows.<br />
90
⎡ ⎤ ⎡ ⎤<br />
A 0 A n−1<br />
A 1<br />
ˆr(<br />
⎢<br />
⎣ . ⎥<br />
) =<br />
A 0<br />
i.e. each row is shifted one down,<br />
⎢<br />
⎦ ⎣ . ⎥<br />
⎦<br />
A n−1 A n−2<br />
⎡ ⎤ ⎡ ⎤<br />
A 0 A 1<br />
A 1<br />
ŝ(<br />
⎢<br />
⎣ . ⎥<br />
) =<br />
A 1<br />
i.e. <strong>the</strong> second row is copied to <strong>the</strong> first row,<br />
⎢<br />
⎦ ⎣ . ⎥<br />
⎦<br />
A n−1 A n−1<br />
⎡ ⎤ ⎡ ⎤<br />
A 0 A 1<br />
A 1<br />
ˆp(<br />
⎢<br />
⎣ . ⎥<br />
) =<br />
A 0<br />
i.e. <strong>the</strong> first and second rows are swapped.<br />
⎢<br />
⎦ ⎣ . ⎥<br />
⎦<br />
A n−1 A n−1<br />
For <strong>the</strong> following recall <strong>the</strong> definitions <strong>of</strong> ⊳ˆσ and E i in section 3.3.1.<br />
Definition Let W = (V, δ, {T j } j∈J ) be a PTPS-automaton. Then (R, ⊳ˆr , ⊳ŝ, ⊳ˆp , {E i } i∈n ) is<br />
called <strong>the</strong> atom-structure <strong>of</strong> W.<br />
Note that <strong>the</strong> atom-structure <strong>of</strong> a PTPS-automaton coincides with <strong>the</strong> atom-structure <strong>of</strong> <strong>the</strong> properly<br />
partitioned automaton <strong>of</strong> a PTPS-automaton since <strong>the</strong> definitions <strong>of</strong> R , ⊳ˆσ and E i depend<br />
only on V and δ. Since we have a fairly ca<strong>non</strong>ical way <strong>of</strong> turning PTPS-automata into properly<br />
partitioned automata we will be concerned with <strong>the</strong> latter from now on.<br />
3.3.4 The n-homomorphism induced by an automaton<br />
We define an n-homomorphism, h, that allows us to show that <strong>the</strong> h-complex algebra <strong>of</strong> <strong>the</strong> atomstructure<br />
<strong>of</strong> a properly partitioned automaton is representable. The n-homomorphism is defined<br />
by means <strong>of</strong> <strong>the</strong> following sequence <strong>of</strong> partitions.<br />
Definition Let W = (V, δ, {T j } j∈J ) be a properly partitioned automaton, with V = (V, . . .). For<br />
each i ≤ n we define a partition <strong>of</strong> R, where <strong>the</strong> part containing a given q ∈ R is written [q] i .<br />
1. When i = n we define [q] i as follows. If q ∈ T j <strong>the</strong>n [q] n = T j .<br />
2. When i < n we let [q] i = {q ′ |q ′ ∈ R ∧ δ(q ′ , 0) = δ(q, 0)}.<br />
Note that for i, i ′ < n <strong>the</strong> partitions are <strong>the</strong> same, i.e., [q] i = [q] i ′. If for each i ≤ n we now define<br />
a mapping h i : R → P(R) by h i (q) = [q] i we may trivially turn P(R), i.e., <strong>the</strong> power-set <strong>of</strong> <strong>the</strong><br />
reachable states, into a polyadic atom-structure such that h is an n-homomorphism.<br />
Definition Let W = (V, δ, {T j } j∈J ) be a properly partitioned automaton. Then <strong>the</strong> sequence<br />
(h n , . . . , h 0 ) where for i ≤ n and q ∈ R <strong>the</strong> mapping h i (q) = [q] i is called <strong>the</strong> n-homomorphism<br />
induced by W.<br />
Definition The h-complex algebra <strong>of</strong> a properly partitioned automaton W is <strong>the</strong> h-complex algebra<br />
<strong>of</strong> W where h is <strong>the</strong> n-homomorphism induced by W.<br />
91
3.3.5 Representability <strong>of</strong> <strong>the</strong> h-complex algebra <strong>of</strong> an automaton<br />
Using proposition 3.2.2, we show that if h is <strong>the</strong> n-homomorphism induced by an automaton W,<br />
<strong>the</strong>n <strong>the</strong> h-complex algebra <strong>of</strong> W is representable. To do this we use <strong>the</strong> following three items. We<br />
use V ∗ p,n,m ⌢ O n×N which, by virtue <strong>of</strong> a by now obvious isomorphism, we think <strong>of</strong> as <strong>the</strong> atomstructure<br />
<strong>of</strong> <strong>the</strong> full n-dimensional polyadic set algebra over <strong>the</strong> natural numbers. It corresponds<br />
to U in proposition 3.2.2. We use <strong>the</strong> n-homomorphism induced by W, which corresponds to<br />
h in proposition 3.2.2. Finally we use a mapping γ which we define now. It corresponds to g in<br />
proposition 3.2.2.<br />
Definition Let W = (V, δ, {T j } j∈J ) be a properly partitioned automaton, where V = (V, . . .).<br />
Then γ : V ∗ ⌢ {0 n×N } → V is defined by<br />
1. γ(0 n×N ) = δ(0, 0),<br />
2. If A ∈ V ∗ has length greater than 0 <strong>the</strong>n γ(A ⌢ 0 n×N ) = δ(δ ∗ (0, A), 0).<br />
Note that γ provides a mapping from representatives <strong>of</strong> N n to states that is independent <strong>of</strong> which<br />
particular representative we choose, i.e. how many 0’s <strong>the</strong>re are at <strong>the</strong> most significant end.<br />
Here is a definition that shortens <strong>the</strong> upcoming pro<strong>of</strong>s.<br />
Definition For ˆσ ∈ SL(V p,n,m ) we define ˆσ ∗ : M p (n, m) ∗ → M p (n, m) ∗ as follows. If A 0 ⌢ · · · ⌢A k ∈<br />
M p (n, m) k <strong>the</strong>n ˆσ ∗ (A 0 ⌢ · · · ⌢A k−1 ) = ˆσ(A 0 ) ⌢ · · · ⌢ˆσ(A k−1 ).<br />
Now a lemma which we shortly use to prove <strong>the</strong> main result <strong>of</strong> <strong>the</strong> present paper. In <strong>the</strong> following<br />
<strong>the</strong> <strong>the</strong> expression ˆσ ∗ (A) ⌢ 0 n×N ⊳ Ṷ σ A ⌢ 0 n×N means that <strong>the</strong> n-tuples <strong>of</strong> natural numbers that<br />
ˆσ ∗ (A) and A represent are in ⊳ Ṷ σ relation, see <strong>the</strong> definition <strong>of</strong> set polyadic atom-structure.<br />
Lemma 3.3.4 Let W = (V, δ, {T j } j∈J ) be a properly partitioned automaton. Let ˆσ ∈ {ˆr, ŝ, ˆp}. Let<br />
V = (V, . . .) and let A ∈ V ∗ . Then <strong>the</strong> following two are equivalent.<br />
1. There exists a q ′ such that q ′ ∈ [q] n and such that q ′ ⊳ˆσ γ(A ⌢ 0 n×N ).<br />
2. γ(ˆσ ∗ (A) ⌢ 0 n×N ) ∈ [q] n and ˆσ ∗ (A) ⌢ 0 n×N ⊳ Ṷ σ A ⌢ 0 n×N .<br />
Pro<strong>of</strong>: (1 implies 2): Let q ′ ∈ [q] n and q ′ ⊳ˆσ γ(A ⌢ 0 n×N ). By definition <strong>of</strong> γ this means<br />
that q ′ ⊳ˆσ δ(δ ∗ (0, A), 0). By point 5 in <strong>the</strong> definition <strong>of</strong> PTPS-automaton we also have that<br />
δ(δ ∗ (0, ˆσ ∗ (A)), 0) ⊳ˆσ δ(δ ∗ (0, A), 0). Therefore by <strong>the</strong> definition <strong>of</strong> ⋄ˆσ it is <strong>the</strong> case that q ′ ⋄ˆσ<br />
δ(δ ∗ (0, ˆσ ∗ (A)), 0). Since W is properly partitioned, [q] n = T j for some j ∈ J which is closed<br />
under ⋄ˆσ , we have δ(δ ∗ (0, ˆσ ∗ (A)), 0) ∈ [q] n . By definition <strong>of</strong> γ we now have γ(ˆσ ∗ (A) ⌢ 0 n×N ) ∈<br />
[q] n . The rest <strong>of</strong> this direction follows from <strong>the</strong> definition <strong>of</strong> ⊳ Ṷ σ .<br />
(2 implies 1): Immediate when letting q ′ = γ(ˆσ ∗ (A) ⌢ 0 n×N ). qed<br />
We now show a result which toge<strong>the</strong>r with proposition 3.2.2 means that <strong>the</strong> h-complex algebra <strong>of</strong><br />
a properly partitioned automaton is representable.<br />
Proposition 3.3.5 Let W = (V, δ, {T j } j∈J ) be a properly partitioned automaton. Let V = (V, . . .)<br />
and q ∈ R. Then<br />
92
1. for ˆσ ∈ {ˆr, ŝ, ˆp} we have γ −1 (⊳ˆσ ([q] n )) = ⊳ Ṷ σ (γ −1 ([q] n ))<br />
2. and for i < n we have γ −1 (E i ([q] i+1 )) = E U i (γ −1 ([q] i+1 ))<br />
Pro<strong>of</strong>:<br />
1. Dropping <strong>the</strong> superscript U, we show that for A ∈ V ∗ we have that<br />
A ⌢ 0 n×N ∈ γ −1 (⊳ˆσ ([q] n )) iff A ⌢ 0 n×N ∈ ⊳ˆσ (γ −1 ([q] n )). So assume:<br />
A ⌢ 0 n×N ∈ γ −1 (⊳ˆσ ([q] n ))<br />
iff γ(A ⌢ 0 n×N ) ∈ ⊳ˆσ ([q] n )<br />
by definition <strong>of</strong> inverse image<br />
iff <strong>the</strong>re exists a q ′ s.t. q ′ ∈ [q] n and q ′ ⊳ˆσ γ(A ⌢ 0 n×N ) this is what lying in <strong>the</strong> ⊳ˆσ -image <strong>of</strong><br />
[q] n means<br />
iff γ(ˆσ ∗ (A) ⌢ 0 n×N ) ∈ [q] n and ˆσ ∗ (A) ⌢ 0 n×N ⊳ Ṷ σ A ⌢ 0 n×N by lemma 3.3.4<br />
iff γ(ˆσ ∗ (A) ⌢ 0 n×N ) ∈ [q] n<br />
iff ˆσ ∗ (A) ⌢ 0 n×N ∈ γ −1 ([q] n )<br />
iff A ⌢ 0 n×N ∈ ⊳ˆσ (γ −1 ([q] n ))<br />
right conjunct follows from <strong>the</strong> definition <strong>of</strong> ⊳ Ṷ σ<br />
by definition <strong>of</strong> inverse image<br />
compare <strong>the</strong> definitions <strong>of</strong> ˆσ and ⊳ Ṷ σ<br />
2. This time we show that for A ∈ V k where k ∈ N and i < n we have that A ⌢ 0 n×N ∈<br />
γ −1 (E i (δ(q, 0))) iff A ⌢ 0 n×N ∈ E i (γ −1 (δ(q, 0))). The <strong>the</strong>orem now follows for those [q] i<br />
where i < n, since <strong>the</strong>n [q] i = {q ′ |q ′ ∈ R ∧ δ(q ′ , 0) = δ(q, 0)}. The <strong>the</strong>orem also follows<br />
for [q] n since [q] n = [δ(q, 0)] n by <strong>the</strong> first axiom schema in <strong>the</strong> definition <strong>of</strong> PTPS-automata.<br />
We introduce <strong>the</strong> following notation for this pro<strong>of</strong>, if A = A 0 ⌢ · · · ⌢A k−1 <strong>the</strong>n π ∗ i (A) =<br />
ˆπ i (A 0 ) ⌢ · · · ⌢ˆπ i (A k−1 ).<br />
We prove <strong>the</strong> equivalence in two separate directions. For <strong>the</strong> first direction assume:<br />
A ⌢ 0 n×N ∈ γ −1 (E i (δ(q, 0)))<br />
<strong>the</strong>n γ(A ⌢ 0 n×N ) ∈ E i (δ(q, 0))<br />
<strong>the</strong>n δ(q, 0)E i γ(A ⌢ 0 n×N )<br />
<strong>the</strong>n δ(q, 0)E i δ(δ ∗ (0, A), 0)<br />
<strong>the</strong>n δ(δ(q, 0), 0) ˆπ i<br />
∼ δ(δ ∗ (0, A), 0)<br />
<strong>the</strong>n δ(q, 0) ˆπ i<br />
∼ δ(δ ∗ (0, A), 0)<br />
PTPS-automaton schema 2<br />
<strong>the</strong>n δ(q, 0) ˆπ i<br />
∼ δ ∗ (0, A ⌢ 0)<br />
by <strong>the</strong> definition <strong>of</strong> image <strong>of</strong> a binary relation<br />
by definition <strong>of</strong> γ<br />
since E i is defined by means <strong>of</strong> ˆπ i<br />
∼ like this<br />
since δ(δ(q, 0), 0) = δ(q, 0), by <strong>the</strong> definition <strong>of</strong><br />
by <strong>the</strong> definition <strong>of</strong> δ ∗<br />
<strong>the</strong>n <strong>the</strong>re exists a B ∈ V k+1 s.t. π ∗ i (A ⌢ 0) = π ∗ i (B) and s.t. δ ∗ (0, B) = δ(q, 0) by lemma<br />
6.8 in A. Rognes [Rog11]<br />
93
<strong>the</strong>n <strong>the</strong>re exists a B ∈ V k+1 s.t. π ∗ i (A ⌢ 0) = π ∗ i (B) and s.t. δ(δ ∗ (0, B), 0) = δ(q, 0) since<br />
δ(δ(q, 0), 0) = δ(q, 0) by <strong>the</strong> definition <strong>of</strong> PTPS-automaton schema 2<br />
<strong>the</strong>n <strong>the</strong>re exists a B ∈ V k+1 s.t. πi ∗ (A ⌢ 0) = πi ∗ (B) and s.t. γ(B ⌢ 0 n×N ) = δ(q, 0)<br />
definition <strong>of</strong> γ<br />
<strong>the</strong>n <strong>the</strong>re exists a B ∈ V k+1 s.t. B ⌢ 0 n×N E U i A ⌢ 0 n×N and γ(B ⌢ 0 n×N ) = δ(q, 0)<br />
by<br />
by<br />
definition <strong>of</strong> E U i<br />
<strong>the</strong>n <strong>the</strong>re exists a B ∈ V k+1 s.t. A ⌢ 0 n×N ∈ E U i (B ⌢ 0 n×N ) and γ(B ⌢ 0 n×N ) = δ(q, 0)<br />
<strong>the</strong>n A ⌢ 0 n×N ∈ E U i (γ −1 (δ(q, 0)))<br />
Now we prove <strong>the</strong> o<strong>the</strong>r direction <strong>of</strong> <strong>the</strong> equivalence. So assume<br />
A ⌢ 0 n×N ∈ E U i (γ −1 (δ(q, 0)))<br />
<strong>the</strong>n <strong>the</strong>re exists a B ∈ V ∗ s.t. π ∗ i (A) ⌢ 0 n×N = π ∗ i (B) ⌢ 0 n×N and s.t. γ(B ⌢ 0 n×N ) =<br />
δ(q, 0)<br />
by definition <strong>of</strong> E U i<br />
<strong>the</strong>n <strong>the</strong>re exists a B ∈ V k+1 s.t. πi ∗ (A) ⌢ 0 n×N = πi ∗ (B) ⌢ 0 n×N and s.t. γ(B ⌢ 0 n×N ) =<br />
δ(q, 0)<br />
by lemma 6.7 in A.Rognes [Rog11]<br />
<strong>the</strong>n <strong>the</strong>re exists a B ∈ V k+1 s.t. πi ∗ (A ⌢ 0) = πi ∗ (B) and s.t. δ(δ ∗ (0, B), 0) = δ(q, 0)<br />
definition <strong>of</strong> γ<br />
<strong>the</strong>n <strong>the</strong>re exists a B ∈ V k+1 s.t. πi ∗ (A ⌢ 0) = πi ∗ (B) and s.t. δ ∗ (0, B ⌢ 0) = δ(q, 0)<br />
definition <strong>of</strong> δ ∗<br />
by<br />
by<br />
<strong>the</strong>n <strong>the</strong>re exists a B ∈ V k+1 s.t. π ∗ i (A ⌢ 0 ⌢ 0) = π ∗ i (B ⌢ 0) and s.t. δ ∗ (0, B ⌢ 0) = δ(q, 0)<br />
by definition <strong>of</strong> π ∗ i<br />
<strong>the</strong>n δ ∗ (0, A ⌢ 0 ⌢ 0) ˆπ i<br />
∼ δ(q, 0)<br />
<strong>the</strong>n δ ∗ (0, A ⌢ 0 ⌢ 0) ˆπ i<br />
∼ δ(δ(q, 0), 0)<br />
PTPS-automaton schema 2<br />
<strong>the</strong>n δ(δ(q, 0), 0) ˆπ i<br />
∼ δ ∗ (0, A ⌢ 0 ⌢ 0)<br />
<strong>the</strong>n δ(q, 0)E i δ ∗ (0, A ⌢ 0 ⌢ 0)<br />
<strong>the</strong>n δ(q, 0)E i δ(δ ∗ (0, A ⌢ 0), 0)<br />
<strong>the</strong>n δ(q, 0)E i γ(A ⌢ 0 n×N )<br />
<strong>the</strong>n γ(A ⌢ 0 n×N ) ∈ E i (δ(q, 0))<br />
by lemma 6.8 in A.Rognes [Rog11]<br />
since δ(δ(q, 0), 0) = δ(q, 0) by <strong>the</strong> definition <strong>of</strong><br />
since ˆπ i<br />
∼ is symmetric by definition<br />
by definition <strong>of</strong> E i<br />
by definition <strong>of</strong> δ ∗<br />
by definition <strong>of</strong> γ<br />
by definition <strong>of</strong> image <strong>of</strong> a relation<br />
<strong>the</strong>n A ⌢ 0 n×N ∈ γ −1 (E i (δ(q, 0)))<br />
qed<br />
94
3.3.6 The main result<br />
Combining <strong>the</strong> last proposition with proposition 3.2.2 we get <strong>the</strong> following which we regard as <strong>the</strong><br />
main result.<br />
Corollary 3.3.6 Let W = (V, δ, {T j } j∈J ) be a properly partitioned automaton and H <strong>the</strong> atomstructure<br />
<strong>of</strong> W. Then H + h is representable (by suitable restrictions <strong>of</strong> γ−1 : P(V ) → P(U)).<br />
3.4 Diversity <strong>of</strong> <strong>the</strong> h-complex algebras <strong>of</strong> finite automata<br />
We have seen that <strong>the</strong> complex algebra <strong>of</strong> (<strong>the</strong> atom-structure <strong>of</strong>) a properly partitioned automaton<br />
is a dMsPs n , but so far we have no results on <strong>the</strong> diversity <strong>of</strong> dMsPs n ’s that may occur as<br />
<strong>the</strong> complex algebra <strong>of</strong> properly partitioned automata. To prove a result on diversity we use <strong>the</strong><br />
structure (N, +, | p ) where N are <strong>the</strong> natural numbers + is addition and | p (x, y) is <strong>the</strong> binary<br />
relation that is true if y is <strong>the</strong> greatest power <strong>of</strong> p such that y divides x. We shall in particular<br />
use <strong>the</strong> Büchi-Bruyère <strong>the</strong>orem, which states that an n-ary relation is first-order definable over<br />
(N, +, | p ) if and only if <strong>the</strong> relation is recognised by a finite n-tape p-automaton, see V. Bruyère<br />
et.al. [BHMV94].<br />
The result we prove on diversity is that every dMsPs n , generated by a finite set <strong>of</strong> first-order<br />
definable relations over (N, +, | p ), is embeddable in <strong>the</strong> h-complex algebra <strong>of</strong> some properly partitioned<br />
finite automaton.<br />
We also recall different well known variants <strong>of</strong> finite-dimensional polyadic algebras, and show<br />
that <strong>the</strong> result on diversity rarely carries over to <strong>the</strong>se. The algebras we recall are; <strong>the</strong> undirected<br />
many-sorted MsPs n ’s, <strong>the</strong> one-sorted Ps n ’s and polyadic equality variants <strong>of</strong> <strong>the</strong> above namely,<br />
dMsPEs n ’s, MsPEs n ’s and PEs n ’s.<br />
3.4.1 dMsPs n ’s<br />
Proposition 3.4.1 Every dMsPs n generated by a finite set <strong>of</strong> first-order definable relations over<br />
(N, +, | p ) can be embedded in <strong>the</strong> h-complex algebra <strong>of</strong> a properly partitioned finite automaton, where<br />
h is <strong>the</strong> induced n-homomorphism.<br />
Pro<strong>of</strong>: Let A = (B n , . . . , B 0 , r, p, s, c 0 , . . . , c n−1 ) be a dMsPs n that is generated by a finite set<br />
<strong>of</strong> first-order definable relations over (N, +, | p ). We now define <strong>the</strong> properly partitioned automaton<br />
<strong>the</strong> h-complex algebra <strong>of</strong> which we eventually embed A into. Let Γ be a finite set <strong>of</strong> sentences in<br />
<strong>the</strong> language <strong>of</strong> (N, +, | p ) that define a set <strong>of</strong> generators for A. By <strong>the</strong> Büchi-Bruyère <strong>the</strong>orem<br />
<strong>the</strong>re is a finite set Y <strong>of</strong> n-tape p-automata that recognises each <strong>of</strong> <strong>the</strong> generators for A. Let X be<br />
<strong>the</strong> minimal set <strong>of</strong> automata such that<br />
1. If W φ ∈ Y <strong>the</strong>n W φ ∈ X,<br />
2. If W φ ∈ X <strong>the</strong>n W ¬φ ∈ X where W ¬φ recognises <strong>the</strong> complement <strong>of</strong> <strong>the</strong> relation recognised<br />
by W φ ,<br />
3. If W φ ∈ X and W φ recognises R(v 0 , . . . , v n−1 ) and σ : n → n and W σφ recognises<br />
R(v σ(0) , . . . , v σ(n−1) ) <strong>the</strong>n W σφ ∈ X.<br />
95
By well known results from automata <strong>the</strong>ory we can, given a finite set Y <strong>of</strong> automata, effectively<br />
construct such an X. By proposition 3.3.1 <strong>the</strong>re is one finite PTPS-automaton W =<br />
(V, δ, {T j } j∈J ) that recognises each relation recognised by <strong>the</strong> elements <strong>of</strong> X. Now X was carefully<br />
designed so as to make W meet <strong>the</strong> criteria <strong>of</strong> proposition 3.3.3. Therefore <strong>the</strong>re is a properly<br />
partitioned automaton, W ′ = (V, δ, {T j} ′ j∈J ′), such that {T j} ′ j∈J ′ generates <strong>the</strong> same boolean<br />
algebra as does {T j } j∈J . We see that W ′ is finite since it has <strong>the</strong> same carrier set as W.<br />
Let now H = (R, . . .) be <strong>the</strong> atom-structure <strong>of</strong> W ′ , let h = (h n , . . . , h 0 ) be <strong>the</strong> n-homomorphism<br />
induced by W ′ and let H h<br />
+ = (B′ n, . . . , B 0, ′ r ′ , p ′ , s ′ , c ′ 0, . . . , c ′ n−1) be <strong>the</strong> h-complex<br />
algebra <strong>of</strong> H. By proposition 3.3.5 we may define an embedding f = (f n , . . . , f 0 ) from H h + to <strong>the</strong><br />
full dMsPs n over N as in <strong>the</strong> pro<strong>of</strong> <strong>of</strong> proposition 3.2.2, i.e.<br />
for i ≤ n and q ∈ R we let f i [q] hi = γ −1 [q] hi ,<br />
for Q ⊆ R let f i ( ⋃ {[q] hi |q ∈ Q}) = ⋃ {γ −1 [q] hi |q ∈ Q}.<br />
We now show that <strong>the</strong> given A is a sub-dMsPs n <strong>of</strong> <strong>the</strong> image <strong>of</strong> H h<br />
+ under f, <strong>the</strong> embedding<br />
needed to prove <strong>the</strong> present proposition is <strong>the</strong>n obtained by restricting f −1 to A. Since {T j} ′ j∈J ′<br />
generates <strong>the</strong> same boolean algebra as does {T j } j∈J and since each set recognised by an automaton<br />
<strong>of</strong> Y is <strong>of</strong> <strong>the</strong> form γ −1 (T j ) we see that <strong>the</strong> sort B n is a sub-boolean algebra <strong>of</strong> <strong>the</strong> image <strong>of</strong> B n<br />
′<br />
under f n . Therefore <strong>the</strong> finitely generated A is a sub-dMsPs n <strong>of</strong> <strong>the</strong> image <strong>of</strong> H h + under f. qed<br />
3.4.2 Ps n ’s<br />
Here we show that in <strong>the</strong> case <strong>of</strong> one-sorted polyadic algebras we are not generally able to obtain an<br />
embedding as in <strong>the</strong> case <strong>of</strong> dMsPs n . We use a trick and define one-sorted n-dimensional polyadic<br />
algebras as a sub-class <strong>of</strong> dMsPs n . It is left to <strong>the</strong> reader to verify that this definition coincides with<br />
n-dimensional (quasi) polyadic algebra as found in <strong>the</strong> literature, see e.g. L. Henkin, J.D. Monk and<br />
A. Tarski [MHT85] or I. Németi [Ném91]. These may also be compared to <strong>the</strong> quantifier algebras<br />
due to C.Pinter [Pin73], which are called substitution-cylindric algebras by I. Németi [Ném91] and<br />
H. Andréka and I. Sain and I. Németi [ASN01].<br />
Definition An n-dimensional polyadic set algebra, or Ps n for short, is an<br />
A = (B, r, p, s, c 0 , . . . , c n−1 ) such that A ′ = (B, . . . , B, r, p, s, c 0 , . . . , c n−1 ) is a dMsPs n .<br />
Moreover a mapping f from A to some Ps n is a P n -embedding if (f, . . . , f) is a dMsP n -<br />
embbeding from A ′ to some dMsPs n .<br />
Example Let (N,
Pro<strong>of</strong>: It is easy to see that <strong>the</strong> h-complex algebra <strong>of</strong> a properly partitioned finite automaton<br />
is finite. So to prove <strong>the</strong> proposition it suffices to display n, p ∈ N and an infinite Ps n that is<br />
generated by a finite set <strong>of</strong> first-order definable relations over (N, +, | p ).<br />
Let n = 3, p = 2 and consider <strong>the</strong> A ∈ Ps 3 with <strong>the</strong> following generator<br />
X = {(x 0 , x 1 , x 2 )|x 1 < x 0 }. The generator is definable by <strong>the</strong> formula ¬(∃v 2 (v 0 + v 2 = v 1 )),<br />
which says that it is not <strong>the</strong> case that v 0 ≤ v 1 . We will now by induction show that for each a ∈ N<br />
it is <strong>the</strong> case that <strong>the</strong> set {(x 0 , x 1 , x 2 )|a < x 0 } lies in A. Let N denote <strong>the</strong> full dMsPs 3 over N.<br />
To begin with {(x 0 , x 1 , x 2 )|0 < x 0 } lies in A, since c N 1 (X) = {(x 0 , x 1 , x 2 )|∃y(x 0 , y, x 2 ) ∈<br />
X)}. This set is exactly <strong>the</strong> set <strong>of</strong> triples such that <strong>the</strong> second component is greater that 0.<br />
To proceed <strong>the</strong> induction, assume that A = {(x 0 , x 1 , x 2 )|a < x 0 } lies in A. Consider<br />
c 1 (p N (A) ∩ X). Here p N (A) = {(x 0 , x 1 , x 2 )|a < x 1 } <strong>the</strong>refore p N (A) ∩ X = {(x 0 , x 1 , x 2 )|a <<br />
x 1 ∧ x 1 < x 0 } fur<strong>the</strong>rmore c 1 (p N (A) ∩ X) = {(x 0 , x 1 , x 2 )|∃y(y ∈ N ∧ a < y ∧ y < x 0 )} which<br />
<strong>of</strong> course is {(x 0 , x 1 , x 2 )|a + 1 < x 0 }.<br />
qed<br />
3.4.3 MsPs n ’s<br />
Definition Let A = (B n , . . . , B 0 , r, p, s, c − 0 , . . . , c − n−1) be a dMsPs n . An n-dimensional manysorted<br />
polyadic set algebra, or MsPs n for short, is an A ′ = (A, c + 0 , . . . , c + n−1) such that for each<br />
i < n <strong>the</strong> mapping c + i : B i → B i+1 has <strong>the</strong> property that for any dMsP n -embedding f from<br />
A to a full U ∈ dMsPs n and any i < n it is <strong>the</strong> case that c + i is a boolean embedding and<br />
that f(c + i (c − i (x))) = c U (f(x)). Moreover an n-homomorphism f from A ′ to some MsPs n<br />
is a MsP n -embedding if f is an embedding from A to some dMsPs n that preserves each <strong>of</strong><br />
c + 0 , . . . , c + n−1.<br />
Corollary 3.4.3 There exists a MsPs n generated by a finite set <strong>of</strong> first-order definable relations over<br />
(N, +, | p ) that can not be embedded in <strong>the</strong> h-complex algebra <strong>of</strong> a properly partitioned finite automaton<br />
for any n-homomorphism h.<br />
Pro<strong>of</strong>: As in <strong>the</strong> case <strong>of</strong> Ps n consider <strong>the</strong> Ps 3 generated by <strong>the</strong> set X = {(x 0 , x 1 , x 2 )|x 1 < x 0 }.<br />
Let A = (B n , . . . , B 0 , . . . , c + 0 , . . . , c + n−1) be <strong>the</strong> MsPs n generated by <strong>the</strong> set X. For each i < n<br />
<strong>the</strong> boolean algebra B i can be embedded in B n using c + 0 , . . . c + n−1. Therefore <strong>the</strong> Ps 3 generated by<br />
X is definable over <strong>the</strong> boolean algebra B n . Combining this proposition 3.4.2 we conclude that<br />
B n is infinite.<br />
qed<br />
3.4.4 Polyadic equality algebras<br />
Here we will see that <strong>the</strong> situation is <strong>the</strong> same for polyadic equality algebras as for polyadic algebras<br />
using one or many sorts. Polyadic equality algebras were considered already by P. R. Halmos. In<br />
<strong>the</strong> finite-dimensional case <strong>the</strong>y coincide with C.C. Pinters quantifier algebras with equality, see<br />
[Pin73]. The cylindric algebras <strong>of</strong> A. Tarski and <strong>the</strong> relation algebras <strong>of</strong> A. De Morgan and C.<br />
Peirce are reducts <strong>of</strong> polyadic equality algebras.<br />
Definition Let A = (B n , . . .) be ei<strong>the</strong>r a dMsPs n , a Ps n or a MsPs n . Let for i, j < n <strong>the</strong><br />
ij-diagonal d ij ∈ B n be defined by d ij = {(x 0 , . . . , x n−1 )|x i = x j }. Let A ′ = (A, {d ij } i,j
A ′ is an n-dimensional polyadic equality set algebra or a PEs n for short if A is a Ps n , likewise<br />
A ′ is a MsPEs n if A is a MsPs n ,<br />
A ′ is a dMsPEs n if A is a dMsPs n .<br />
Moreover a mapping f from A ′ to a dMsPEs n , a PEs n or a MsPEs n respectively, is a dMsPE n -<br />
, PE n - or a MsPE n -embedding if f is a dMsP n -, P n - or a MsP n embedding from A and if f<br />
preserves each <strong>of</strong> {d ij } i,j
Definition Let n, p ∈ N where p is a prime power. A dMsP n -automaton is<br />
a W = (V, π, r, δ, T ) where,<br />
V = (V, . . .) is a vector-space over F p ,<br />
π, r : V → V are called projection and rotation,<br />
δ : V × V → V is called <strong>the</strong> transition function,<br />
T ⊆ V × V is an equivalence relation called <strong>the</strong> partitioning equivalence.<br />
Ax1 If V is finite <strong>the</strong>n (V, π, r) is isomorphic to (M p (n, m), π ′ , r ′ ) for some m ∈ N depending<br />
on <strong>the</strong> size <strong>of</strong> V. See A.Rognes [Rog11] for an example <strong>of</strong> how this can be axiomatised with<br />
a first-order sentence.<br />
Ax2 W is a PTPS-automaton using <strong>the</strong> abstract vector-space, V, as alphabet.<br />
Ax3 W is properly partitioned. Since we are using an equivalence relation this can be expressed by<br />
adding <strong>the</strong> following axiom for each ˆσ ∈ SL(V). W |= ∀q ′ q ′′ ∈ R(q ′ ⋄ˆσ q ′′ → T (q ′ , q ′′ ))<br />
For <strong>the</strong> second class <strong>of</strong> automata we add equality, i.e., for each i, i ′ < n <strong>the</strong>re is a subset <strong>of</strong> states,<br />
D ii ′, consisting <strong>of</strong> those states that we run through with some tape whose row i and row i ′ are<br />
equal.<br />
Definition Let n, p ∈ N where p is a prime power. A dMsPE n -automaton is a W = (W ′ , {D ii ′} i,i ′
We now define a class <strong>of</strong> automata that can be used to define <strong>the</strong> atom-structures <strong>of</strong> certain<br />
Ps n ’s and MsPs n ’s. In this definition we abandon <strong>the</strong> equivalence relations T and revert to<br />
partitioning by an indexed family {T j } j∈J where J is a finite set. To explain <strong>the</strong> reason for this<br />
let B n be <strong>the</strong> boolean algebra generated by our partition. To make an atom-structure <strong>of</strong> a Ps n<br />
or MsPs n we need to ensure that <strong>the</strong> image <strong>of</strong> B n under each cylindrification is a sub-boolean<br />
algebra <strong>of</strong> B n . The latter appears to require an infinite union or some o<strong>the</strong>r higher-order construct,<br />
see Ax4 below.<br />
Definition Let n, p ∈ N where p is a prime power and let J be a finite set. A J-indexed P n -<br />
automaton is a W = (V, π, r, δ, {T j } j∈J ) where<br />
V = (V, . . .) is a vector-space over F p ,<br />
π, r : V → V are called projection and rotation,<br />
δ : V × V → V is called <strong>the</strong> transition function,<br />
T j ⊆ V for each j ∈ J.<br />
Ax1 If V is finite <strong>the</strong>n (V, π, r) is isomorphic to (M p (n, m), π, r) for some m ∈ N depending<br />
on <strong>the</strong> size <strong>of</strong> V.<br />
Ax2 W is a PTPS-automaton using <strong>the</strong> abstract vector-space, V, as alphabet.<br />
Ax3b W is properly partitioned.<br />
Ax4 For each i < n and j ∈ J <strong>the</strong>re exists a X ⊆ J such that E i (T j ) = ⋃ {T j ′|j ′ ∈ X}.<br />
Again we add a variant with equality, i.e., one that for each i, i ′ < n has a subset <strong>of</strong> states, D ii ′,<br />
consisting <strong>of</strong> those states that we run through with some tape whose row i and row i ′ are equal.<br />
Definition Let n, p ∈ N where p is a prime power and let J be a finite set. A J-indexed PE n -<br />
automaton is a W = (W ′ , {D ii ′} i,i ′
The following corresponds to proposition 3.5.2. This time however we have to assume that <strong>the</strong><br />
finitely generated algebra is finite since by proposition 3.4.2 a finitely generated algebra is not finite<br />
in general.<br />
Proposition 3.5.4 Let J be a finite set and let k ∈ N be <strong>the</strong> number <strong>of</strong> elements <strong>of</strong> J, <strong>the</strong>n every<br />
Ps n or PEs n generated by k disjoint first-order definable relations over (N, +, | p ) and that has 2 k<br />
elements can be embedded in <strong>the</strong> h-complex algebra <strong>of</strong> a J-indexed P n -automaton, where h is <strong>the</strong><br />
induced n-homomorphism.<br />
Pro<strong>of</strong>: This is also a variation <strong>of</strong> proposition 3.4.1. As we have seen, finitely generated Ps n ’s or<br />
PEs n ’s aren’t guaranteed to be finite so we restrict our selves to Ps n ’s and PEs n ’ whose boolean<br />
reduct is <strong>the</strong> boolean algebra generated by <strong>the</strong> partition {T j } j∈J . This algebra has 2 k elements<br />
when J has k elements.<br />
qed<br />
3.6 Concluding remarks<br />
In section 3.3.4 we introduced <strong>the</strong> h-complex algebra <strong>of</strong> a properly partitioned automaton. With<br />
corollary 3.3.6 we showed that <strong>the</strong> h-complex algebra <strong>of</strong> a properly partitioned automaton is representable.<br />
Proposition 3.4.1 states that every dMsPs n generated by a finite set <strong>of</strong> first-order definable<br />
relations over (N, +, | p ) can be embedded in <strong>the</strong> h-complex algebra <strong>of</strong> a properly partitioned<br />
finite automaton.<br />
With proposition 3.5.1 we noted that, for a fixed number <strong>of</strong> tapes, a variation <strong>of</strong> properly<br />
partitioned automata called dMsP n -automata, is basic elementary. The dMsP n -automata use<br />
an equivalence relation for partitioning and a variant <strong>of</strong> proposition 3.4.1 also holds for <strong>the</strong>se.<br />
The latter two proposition also hold for for <strong>the</strong> class <strong>of</strong> dMsPE n -automata, which differ from<br />
dMsP n -automata only in that <strong>the</strong>y have diagonals and <strong>the</strong>refore are suitable for polyadic equality<br />
algebras. By proposition 3.5.1 we also have a way <strong>of</strong> recursively enumerating <strong>the</strong> atom-structures<br />
<strong>of</strong> both finite dMsP n -automata and finite dMsPE n -automata, without explicitly computing <strong>the</strong><br />
whole complex algebra. Proposition 3.5.2 measures <strong>the</strong> extent <strong>of</strong> <strong>the</strong> representable algebras whose<br />
atom-structures are enumerated thusly.<br />
We finally introduced <strong>the</strong> variants <strong>of</strong> properly partitioned automata called J-indexed P n -<br />
automata and J-indexed PE n -automata, designed to have <strong>the</strong> property that <strong>the</strong> h-complex algebra<br />
<strong>of</strong> a finite automaton is a finite polyadic algebra or finite polyadic equality algebra respectively.<br />
In order to make <strong>the</strong>se two classes basic elementary we did not use an equivalence relation but a<br />
partition in <strong>the</strong> form <strong>of</strong> one unary relation symbol for each member <strong>of</strong> an index set J. This has<br />
<strong>the</strong> somewhat undesirable property that <strong>the</strong> finite axiomatisation only works for a fixed number<br />
<strong>of</strong> tapes and a fixed index set J. Since <strong>the</strong> members <strong>of</strong> <strong>the</strong> set J correspond to <strong>the</strong> atoms <strong>of</strong> <strong>the</strong><br />
associated complex algebra this means that only a finite number <strong>of</strong> isomorphism classes <strong>of</strong> polyadic<br />
(equality) algebras may occur as <strong>the</strong> complex algebra <strong>of</strong> J-indexed P n -automata or PE n -automata<br />
for fixed n and J.<br />
We leave it as an open problem whe<strong>the</strong>r it is possible to restrict <strong>the</strong> class <strong>of</strong> dMsP n -automata<br />
by means <strong>of</strong> a first-order sentence so as to ensure that <strong>the</strong> h-complex algebra is a finite P n whilst<br />
retaining a property similar to proposition 3.4.1.<br />
Regardless <strong>of</strong> <strong>the</strong> open problem, by proposition 3.5.3 we have a way <strong>of</strong> recursively enumerating<br />
<strong>the</strong> atom-structures <strong>of</strong> finite J-indexed P n -automata and J-indexed PE n -automata, for J ranging<br />
101
over finite sets. Again we avoid explicitly computing <strong>the</strong> whole <strong>of</strong> <strong>the</strong> complex algebras in question.<br />
Proposition 3.5.4 measures <strong>the</strong> extent <strong>of</strong> <strong>the</strong> representable algebras whose atom-structures are<br />
enumerated thusly.<br />
102
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