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

Similarly we define two relations on automata that relate to substitution of variables for variables.
Definition Let ˆσ ∈ SL(V), i.e., ˆσ is a linear transformation that swaps and overwrites columns
according to some mapping on track indices. Then,
1. ⊳ˆσ ⊆ V × V is defined by q ⊳ˆσ q ′ is there exists an A ∈ V such that δ(0, ˆσ(A)) = q and
δ(0, A) = q ′ ,
2. ⋄ˆσ ⊆ V × V is defined by q ⋄ˆσ q ′ if there exists a q ′′ ∈ V such that q ⊳ˆσ q ′′ and q ′ ⊳ˆσ q ′′ .
As was seen in A.Rognes [Rog11] the elements of P L(V) and SL(V) are definable using the
operations of automata of PTPS-signature, more specifically the operations π, r and +. It follows
that for each t and σ the relations t ∼, E i , ⋄ˆσ , ⊳ˆσ are definable as well. We use this observation to
shorten the definition of PTPS-automata now.
Definition A concrete PTPS-automaton is a W = (V, δ, {T j } j∈J ) of concrete PTPS-signature such
that, For each j ∈ J, each t ∈ P L(V) and ˆσ ∈ SL(V) the following axioms hold:
1. W |= T j (q) ↔ T j (δ(q, 0)), to ensure well definedness of relations recognised by automata,
2. W |= ∀qAB∃C(δ(δ(q, t(A)), t(B)) = δ(q, t(C))), i.e., a state reachable by a tape of the
form t(A) ⌢ t(B) is also reachable by a tape of the form t(C),
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
B ⌢ B ′ are tapes that are the same except on the i-th row then δ ∗ (0, A ⌢ A ′ ) ˆπ i
∼ δ ∗ (0, B ⌢ B ′ ),
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
t = ˆπ i and δ ∗ (0, A ⌢ A ′ ) ˆπ i
∼ q ′ then there exists a tape B ⌢ B ′ that is the same as A ⌢ A ′
except possibly on the i-th row such that δ ∗ (0, B ⌢ B ′ ) = q ′ see A.Rognes [Rog11],
5. W |= ∀qq ′ A(q ′ ⊳ˆσ q → δ(q ′ , ˆσ(A)) ⊳ˆσ δ(q, A)), this is similar to 3,
6. W |= ∀qq ′ A(q ′ ⊳ˆσ δ(q, A) → ∃q ′′ (q ′′ ⊳ˆσ q ∧ δ(q ′′ , ˆσ(A)) = q ′ )) this is similar to 4.
The first axiom schema here is to ensure that we can extend the definition of recognition from finite
tapes to infinite tapes with finite support, and thus tuples of natural numbers, as follows.
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
δ ∗ (0, A) ∈ T j .
Note that the definition works regardless of which of the many possible p-nary expansions A for
a given tuple of numbers we have chosen, i.e., regardless of how many 0’s there are at the most
significant end of A.
It is known that with each definable relation of an automatic structure, we can associate a
classical finite n-tape p-automaton that recognises exactly the tapes that represent tuples of the
definable relation, see, J.R. Büchi [B¨60]. We recall one of the main results from A.Rognes [Rog11],
namely Proposition 7.2. The result allows us to merge finite sets of classical automata into one
finite concrete PTPS-automaton.
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