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

Proof: (If): For tapes of length 1 we prove this as follows. By the definition of ∼ t there is an x s.t.
δ(0, x) = q. Since h is an isomorphism this case of the lemma follows by letting B 0 = h −1 (x).
For longer tapes we combine this with repeated use of the property of lemma 2.5.10.2.
(Only if): This follows by repeated use of the property of lemma 2.5.10.1.
qed
2.7 The automata for a formula and its sub-formulae
Here we investigate how PTPS automata used in Büchis procedure for deciding a first-order sentence
about the structure (N, +, | p ) relate. We are in particular interested in the relationship between
the automaton associated with a formula and the automata associated with its sub-formulae.
We provide a sense in which this relationship is first-order.
The fact that PTPS automata are multi-automata is now put to use. We switch to using a set
of first-order formulae as index-set for the terminal states, as this is convenient when we set up a
correspondence between automata and relations defined by formulae. The following proposition
states that concrete one-sorted projection multi-automata are as versatile as finite sets of classical
automata.
Definition Let W = (V, δ, T ) be an n-tape p-automaton. Let h be an isomorphism from
M p (n, m) to V. Then L(W, h) ⊆ N n denotes is the relation recongised by (W, h).
Proposition 2.7.1 There is an effective mapping from the set of finite sets X of classical finite n-
tape p-automata (one- or two-sorted) to finite concrete PTPS automata, (W, h), such that for each
(W ′ , h ′ ) ∈ X there exists a component (W φ , h) of (W, h) such that L(W ′ , h ′ ) = L(W φ , h).
Proof: Let X be a finite set of classical finite n-tape p-automata. We now describe a procedure
to build a one-sorted n-tape p-multi-automaton out of X. By proposition 2.4.8 there exists a
recognition-invariant mapping, f from the classical automata (one- or two-sorted) to the finite
two-sorted concrete automata K 2,con . By lemma 2.5.2 we are able to concatenate the elements of
f(X) into a finite two-sorted n-tape p-multi automaton, which by proposition 2.6.3 we can turn
into a one-sorted PTPS automaton with the desired property.
qed
Part of the procedure to decide a given n-variable formula, φ, in the language of (N, +, | p ), is
to associate one concrete n-tape p-automaton (W ′ ψ, h ′ ) with each sub-formula ψ of φ. By the just
proven proposition 2.7.1 there is for all φ in the language of (N, +, | p ), a concrete PTPS automaton
(W, h) such that for each sub-formula ψ of φ there is a component (W ψ , h) that recognises the
interpretation of ψ in (N, +, | p ). The components of (W, h) that recognise interpretations for
formulae and immediate sub-formulae are related as follows.
Proposition 2.7.2 Let W = (V, δ, {T φ } φ∈Γ ) be an PTPS automaton. Let h be an isomorphism from
M p (n, m) to V. Let R be the set of states reachable from the initial state, which since W is projectively
transitive is the set of states q defined by the formula ∃x[δ(0, x) = q]. Consider the operations ˆπ i
∼, ⊲ˆσ
and ⋄ˆσ as defined in the language of PTPS automata.
1. Negation: Let {φ, ¬φ} ⊆ Γ. Then
W |= ∀q ∈ R[T ¬φ (q) ↔ ¬T φ (q)]
iff
(W ¬φ , h) recognises the complement of the relation recognised by (W φ , h).
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