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

δ ∗ h(ι, ˆσ(A)) = q ′
Also ⋄ˆσ ⊆ K × K is defined by
q ′ ⋄ˆσ q ′′ if
there exists a q ∈ K such that
q ⊲ˆσ q ′ and
q ⊲ˆσ q ′′
Note that if q ⊲ˆσ q ′ then the states q and q ′ are h-reachable from the initial state in one step.
Definition Let W = (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. Then the concrete automaton (W, h) is said
to be a substitution automaton or to have substitutions if
for each ˆσ ∈ SL(M p (n, m))
{(q, q ′ ) : q ⊲ˆσ q ′ } = {(δ ∗ h(q, A), δ ∗ h(q ′ , ˆσ(A)) : q ⊲ˆσ q ′ ∧ A ∈ M p (n, m)}
So in an automaton which has substitutions, if q ⊲ˆσ q ′ then the states q and q ′ are h-reachable from
the initial state in one step, moreover they are h-reachable from the initial state in exactly two steps,
and so forth.
Lemma 2.5.8 Let W = (V, . . .) be a finite two-sorted n-tape p-multi-automaton. Let h be an n-fold
linear transformation from M p (n, m) onto V. Then there exists a k ∈ N such that the k-outreach
(W ′ , h ′ ) of (W, h) has substitutions.
Proof: We use lemma 2.1.4, so consider X = P(K × K) SL(Mp(n,m)) , the set of mappings
from SL(M p (n, m)) to the set of binary relations on states. The set X is finite since by lemma
2.5.3, the set SL(M p (n, m)) is finite. Define the mapping α ∈ X by α(t) = {(ι, ι)}. We
now define a function f : X → X that, when beginning with α and iterating, we can use to
keep track of pairs of states that are reachable from the initial state by a pair of tapes of the form
(A 0 , . . . , A i−1 , ˆσ(A 0 ), . . . , ˆσ(A i−1 )).
This is to say that for ˆσ ∈ SL(M p (n, m)) we have
(q, q ′ ) ∈ [f i (α)](ˆσ)
iff
there exist A 0 , . . . , A i−1 ∈ M p (n, m) such that
δ ∗ h(ι, A 0 ⌢ · · · ⌢A i−1 ) = q
δ ∗ h(ι, ˆσ(A 0 ) ⌢ · · · ⌢ˆσ(A i−1 )) = q ′
A way of defining such an f is for ξ ∈ X to let
[f(ξ)](ˆσ) = {(δ(q, h(A)), δ(q ′ , h(ˆσ(A)))) : (q, q ′ ) ∈ ξ(ˆσ) ∧ A ∈ M p (n, m)}.
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