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

A way of defining f is for ξ ∈ X to let [f(ξ)](t) = {(δ(q, h(A)), δ(q ′ , h(B))) : (q, q ′ ) ∈
ξ(t) ∧ A, B ∈ M p (n, m) ∧ t(A) = t(B)}.
By lemma 2.1.4 we obtain a k such that f k (α) = f 2·k (α). Using this k consider the k-
outreach (W ′ , h ′ ) of (W, h). With each t ∈ P L(M p (n, m)) associate t ′ ∈ P L(M p (n, k · m))
by the equation t ′ (A 0 ⌢ · · · ⌢A k−1 ) = t(A 0 ) ⌢ · · · ⌢t(A k−1 ). Note that this association sets up
a bijection between P L(M p (n, m)) and P L(M p (n, k · m)), since both are sets of operations
on n rows. We now have what we need to show that (W ′ , h ′ ) fulfils the requirement of being a
projection automaton. So letting δ ′ be as in the definition of k-outreach we have:
{(δ ′ (q, A), δ ′ (q ′ , B)) : q ∼ t′
q ′ ∧ A, B ∈ M p (n, k · m) ∧ t ′ (A) = t ′ (B)}
= {(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 ′ ) =
t ′ (B ′ )} by definition of δ ′ and ∼
t′
= {(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 ∈
M p (n, m)
∧t(A 0 ) = t(B 0 ) ∧ · · · ∧ t(A 2k−1 ) = t(B 2k−1 )} by definition of t ′
= [f 2·k (α)](t) by design of f
= [f k (α)](t) by choosing the k of lemma 2.1.4
= {(q, q ′ ) : q ∼ t′
q ′ } again by definition of ∼
t′
The left and right side of this sequence of equations is what we require of a projection automaton.
qed
As before the k-outreach of a finite automaton is a finite equivalent automaton. Hence with each
finite automaton, we may associate an equivalent finite automaton that is a projection automaton.
By examining the proof of the previous proposition we gather the following.
Corollary 2.5.7 For k ∈ N the k-outreach of a projection automaton is again a projection automaton.
2.5.5 Substitution automata
For each linear transformation in SL(M p (n, m)) we now define a relation on pairs of states of
a given automaton which for example we can use as follows. From the fact that q and q ′ are
in relation, we may infer that there exist two symbols, one whose rows are a permutation of the
other symbols rows, which from the initial state we can use to reach q and q ′ respectively. This
property turns out to be useful when comparing the automata that recognise the interpretations of
for example the atomic formulas R(v 0 , v 1 ) and R(v 1 , v 0 ).
Definition Let (V, K, δ, ι, . . .) be a two-sorted n-tape p-multi-automaton. Let h be an n-fold
linear transformation from M p (n, m) onto V. Let ˆσ ∈ SL(M p (n, m)). Then ⊲ˆσ ⊆ K × K is
defined as follows.
q ⊲ˆσ q ′ if
there exists a symbol A ∈ M p (n, m) such that
δ ∗ h(ι, A) = q and
60