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

2.4.2 Transitive p-automata, the two-sorted case
We single out a class of concrete n-tape p-automata in which a state is reachable from another iff
it is reachable in exactly one step. This makes reachability first-order definable quite trivially. Less
trivially the class turns out to suffice for deciding Presburger arithmetic and the theories of the other
automatic structures.
For the rest of this section we fixate n, p ∈ N where p is a prime power. Let W = (V, K, δ, ι, T )
be a finite n-tape p-automaton. Moreover let m ∈ N and let h be an n-fold linear transformation
from M p (n, m) onto V.
Definition Let W and h be as above. Let k ∈ N. Then the k-outreach of (W, h) is the the
pair (W ′ , h ′ ) where W ′ = (M p (n, m · k), K, δ ′ , ι, T ). Here δ ′ is the restriction of δ ∗ h to K ×
M p (n, m · k) and h ′ : M p (n, m · k) → M p (n, m · k) is the identity mapping and thus an n-fold
linear transformation.
Note that the k-outreach (W ′ , h ′ ) of (W, h) is a finite automaton as long as W is.
Proposition 2.4.2 Let W and h be as above. Let k ∈ N. Then the k-outreach (W ′ , h ′ ) of (W, h) is
equivalent to (W, h).
Proof: Observe that δ ′ and h ′ are defined by means of δh ∗ in such a way that δh(q, ∗ A) and δ ′ ∗
h ′(q, A)
are both defined and elements of T whenever δ ′ ∗
h ′(q, A) is defined.
qed
We now define a notion of reachability that depends on an n-fold linear transformation from a
concrete n-fold vector-space to the alphabet of an automaton.
Definition Let W and h be as above. Let q, q ′ ∈ K be two states. Then q ′ is h-reachable from q if
there is a k ∈ N and a matrix A of M p (n, k · m) such that δ ∗ h(q, A) = q ′ .
We now define the notion that makes reachability first-order definable.
Definition Let W and h be as above. Then (W, h) is said to be transitive if
for all states q ∈ K and tapes A, B ∈ M p (n, m)
there exists a tape C ∈ M p (n, m) such that
δ ∗ h(q, A ⌢ B) = δ ∗ h(q, C).
Now the main result on two-sorted transitive automata.
Proposition 2.4.3 Let W and h be as above. Then there exists a k ∈ N such that the k-outreach
(W ′ , h ′ ) of (W, h) is transitive. This implies that if a state is h ′ -reachable from another, then it is
h ′ -reachable in exactly one step.
Proof: To find the sought k we shall use lemma 2.1.4. Consider therefore X = P(K) K , the set of
mappings from the set of states to the power-set of the set of states. Define α ∈ X by α(q) = {q}.
We now define a function f : X → X that, when beginning with α and iterating, we can use to
keep track of the states reachable from a given state q. This is to say that [f i (α)](q) is the set of
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