Computability and Logic

PROBLEMS 135
the decision problem for implication (Theorem 11.2, or equivalently Theorem
11.4) implies the unsolvability of the decision problem for validity.
11.2 The decision problem for satisfiability is the problem of devising an effective
procedure that, applied to any finite set of sentences, would in a finite amount
of time enable one to determine whether or not it is satisfiable. Show that
the unsolvability of the decision problem for implication (Theorem 11.2, or
equivalently Theorem 11.4) implies the unsolvability of the decision problem
for satisfiability.
The next several problems pertain specifically to section 11.1.
11.3 Show that
and
together imply
11.4 Show that
and
together imply
∀w∀v(Twv↔∃y(Rwy & Syv))
∀u∀v∀y((Suv & Syv) → u = y)
∀u∀v∀w((Twv & Suv) → Rwu).
∀x(∼Ax →∼∃t(Bt & Rtx))
∼∃x(Cx & Ax)
∼∃t∃x(Bt & Cx & Rtx).
11.5 The foregoing two problems state (in slightly simplified form in the case of
the second one) two facts about implication that were used in the proof of
Theorem 11.2. Where?
11.6 The operating interval for a Turing machine’s computation beginning with
input n consists of the numbers 0 through n together with the number of any
time at which the machine has not (yet) halted, and of any square the machine
visits during the course of its computations. Show that if the machine eventually
halts, then the operating interval is the set of numbers between some a ≤ 0
and some b ≥ 0, and that if the machine never halts, then the operating interval
consists either of all integers, or of all integers ≥a for some a ≤ 0.
11.7 A set of sentences Ɣ finitely implies a sentence D if D is true in every interpretation
with a finite domain in which every sentence in Ɣ is true. Trakhtenbrot’s
theorem states that the decision problem for finite logical implication is unsolvable.
Prove this theorem, assuming Turing’s thesis.
The remaining problems pertain specifically to section 11.2.
11.8 Add to the theory Ɣ in the proof of Theorem 11.4 the sentence
∀x 0 ≠ x ′ & ∀x∀y(x ′ = y ′ → x = y).
Show that m ≠ n is then implied by Ɣ for all natural numbers m ≠ n, where
m is the usual numeral for m.