Published online by Cambridge University Press: 12 March 2014
R. Parikh has shown that in the predicate calculus without function symbols it is possible to decide whether or not a given formula A is provable in a Hilbert-style proof of k lines. He has also shown that for any formula A(x 1, …, xn ) of arithmetic in which addition and multiplication are represented by three-place predicates, {(a 1, …, an ): A(a 1, …, an ) is provable from the axioms of arithmetic in k lines} is definable in (N,′, +,0). See [2].
In §1 of this paper it is shown that if S is a definable set of n-tuples in (N,′, +, 0), then there is a formula A(x 1, …, xn ) and a number k so that (a 1 …, a n) ∈ S if and only if A(a 1 …, an ) can be proved in a proof of complexity k from the axioms of arithmetic. The result does not depend on which formalization of arithmetic is used or which (reasonable) measure of proof complexity. In particular, this gives a converse to Parikh's result.
In §II, a measure of proof complexity is given which is based on counting the quantifier steps in semantic tableaux. The idea behind this measure is that A is k provable if in the attempt to construct a counterexample one meets insurmountable difficulties in the definition of the appropriate Skolem functions over sets of cardinality ≤ k. A method is given for deciding whether or not a sentence A in the full predicate calculus is k provable. The question for the full predicate calculus and Hilbert-style systems of proof remains open.
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