A unification-theoretic method for investigating the k-provability problem

Annals of Pure and Applied Logic 51 (3):173-214 (1991)
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Abstract

The k-provability for an axiomatic system A is to determine, given an integer k 1 and a formula in the language of A, whether or not there is a proof of in A containing at most k lines. In this paper we develop a unification-theoretic method for investigating the k-provability problem for Parikh systems, which are first-order axiomatic systems that contain a finite number of axiom schemata and a finite number of rules of inference. We show that the k-provability problem for a Parikh system reduces to a unification problem that is essentially the unification problem for second-order terms. By solving various subproblems of this unification problem , we solve the k- probability problem for a variety of Parikh systems, including several formulations of Peano arithmetic. Our method for investigating the k-provability problem employs algorithms that compute and characterize unifiers. We give some examples of how these algorithms can be used to solve complexity problems other than the k-provability problem

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Citations of this work

The Undecidability of K-Provability.Samuel R. Buss - 1991 - Annals of Pure and Applied Logic 53 (1):75-102.
Algorithmic Structuring of Cut-Free Proofs.Matthias Baaz & Richard Zach - 1993 - In Egon Börger, Hans Kleine Büning, Gerhard Jäger, Simone Martini & Michael M. Richter (eds.), Computer Science Logic. CSL’92, San Miniato, Italy. Selected Papers. Berlin: Springer. pp. 29–42.
Bounded Arithmetic, Proof Complexity and Two Papers of Parikh.Samuel R. Buss - 1999 - Annals of Pure and Applied Logic 96 (1-3):43-55.

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References found in this work

A Machine-Oriented Logic Based on the Resolution Principle.J. A. Robinson - 1966 - Journal of Symbolic Logic 31 (3):515-516.
A Simplified Formalization of Predicate Logic with Identity.Alfred Tarski - 1964 - Archive for Mathematical Logic 7 (1-2):61-79.
On Me Number of Steps in Proofs.Jan Krajíèek - 1989 - Annals of Pure and Applied Logic 41 (2):153-178.

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