On the methods of mechanical non-theorems (latest version)

3. The finite automata of the class are as versatile as finite automata with fixed alphabets, at least
in regards to deciding the theory of automatic structures such as Presburger arithmetic.
The first two properties are needed to obtain a basic elementary class of automata in which reachability
is first-order. The third property is sufficient for a semi-decision procedure, based on an
automaton, to recognise some infinity axiom. The latter is because Presburger arithmetic contains
infinity axioms. To fulfil the second property we will simply postulate that the relations of which we
need to take the transitive closure are transitive. This makes expressing reachability trivial. What
remains then is to make sure that we have not lost computational power, i.e., we need to show
that those of the automata that fulfil the transitivity postulates suffice for deciding the theories of
automatic structures.
We introduce and show some results on two classes with the three properties. Firstly the class
of transitive automata, made to be a simple and transparent example of such a class. Automata of
the second class, called PTPS automata, consist of transitive automata where one is able to express
various refinements of the reachability relation in first-order language. We are in particular able
to give first-order definitions of constructions on automata, that correspond to logical connectives,
quantifiers and substitutions, internally in a finite PTPS automaton.
2.1.1 Outline of paper
In section 1 we repeat well known definitions and results to make the present paper fairly self
contained. In section 2 we introduce n-fold vector-spaces. These are vector-spaces with two extra
operators. Whether this particular subclass of groups with operators has been described before is
unknown to the present author. The n-fold vector-spaces serve as alphabets for all the automata
introduced in the present paper.
In section 3, n-tape p-automata which have n-fold vector-spaces as alphabets are introduced.
These are believed to be new. The rest of the automata described in the present paper are variations
over these. We define what it means for an automaton, with a possibly abstract n-fold vector-space
as alphabet, to recognise an n-ary relation on natural numbers.
In section 4 a subclass of n-tape p-automata is introduced. They are called transitive automata.
We provide a finite set of first-order axioms for transitive automata, and show that in finite transitive
automata the reachability relation is first-order definable, quite trivially. Less trivial is the fact that
finite transitive n-tape p-automata are as versatile as finite automata in general when it comes to
their use as parts of decision procedures for theories of automatic structures.
In section 5 the definitions and proof techniques of section 4 are elaborated upon and we
introduce PTPS automata. They are also definable by a finite set of first-order formula. In PTPS
automata we are able to express, in first-order language, various forms of reachability between states.
For example we are able to express that one state is reachable from another using a tape whose second
track represents the number 0.
In section 6 we look at how PTPS automata, used to decide a given sentence about an automatic
structure, are in relationship to one another. We show that the relationship between an automaton
for a formula and the automata for its sub-formulae is first-order definable over finite structures.
Finally we see how this can be used for proving consistency computationally.
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