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

1. If Γ has an automatic model then to each φ ∈ Γ we can assign a finite concrete twosorted
n-tape p-(multi-)automaton (W φ , h φ ) that recognises the definition of φ in the automatic
model. Using proposition 2.7.1 we can merge these into one finite two-sorted multiautomaton
(W, h) where each component is equivalent to a (W φ , h φ ). The structure W is
finite and clearly a model for Γ ′′ ∪ Γ ′ ∪ {∀q ∈ R[T φ (q)]}.
2. If Γ ′′ ∪ Γ ′ ∪ {∀q ∈ R[T φ (q)]} has a finite model W then W is a multi-automaton. To make
W concrete we use a homomomorphism provided by lemma 2.2.2.
qed
2.8 Concluding remarks
With proposition 2.4.6, corollary 2.4.9 and corollary 2.4.10 we have shown that in finite transitive
automata reachability is first-order, we have shown that the class of transitive automata is finitely
axiomatisable. Moreover we have shown that the finite and concrete representatives of the class are
versatile enough to replace automata with fixed alphabets in J.R. Büchis procedure for deciding the
theory of automatic structures. With proposition 2.6.1, corollary 2.6.4 and corollary 2.6.5 we have
proven the same for PTPS automata.
We have investigated the relationship between a PTPS automaton that recognises the interpretation
of a formula with the PTPS automata that recognise the interpretation of its sub-formulae.
With proposition 3.3.2 we have shown that this relationship is in a sense first-order. The proposition
relies on first-order definability of various variants of reachability in PTPS automata.
PTPS-automata can be used to construct a semi-desicion procedure that terminates on input
on consistent sentences only, as follows.
1. Let Γ be the set of sub-formulae of the input sentence.
2. Let Γ ′′ ∪ Γ ′ be the definition of the PTPS-automaton in corollary 2.7.3.
3. Search for a finite model for Γ ′′ ∪ Γ ′ ∪ {∀q ∈ R[T φ (q)]} and terminate if one is found.
In step 3 we use the fact that the class of PTPS-automata is basic elementary to make the procedure
terminate when the input sentence has a model definable in an automatic structure. This procedure
terminates on input on at least those infinity axioms that are true in Presburger arithmetic.
One idea of transforming a first-order sentence into a first-order description of automata and
then to search for a finite model of the transformed sentence has been proposed by N. Peltier
[Pel09]. To define reachability N. Peltiers transformation introduces the element-relation and lets
the carrier-set consist of sets of states. In contrast, the transformation outlined in the present paper
has a carrier-set whose members are symbols, some of which are also used as states. To define
reachability we allow for some flexibility in the set of symbols. N. Peltier says of his transformation
that, “The main interest of the present work is to prove that the translation is feasible from a
theoretical point of view”. The same can be said of the transformation proposed in the present
paper. To make search for satisfiable interpretations in automatic models remotely feasible from a
practical point of view the present author proposes to use the atom-strucures of finite representable
polyadic algebras. How to use PTPS-automata to compute such atom-structures is the subject of
the second paper.
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