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March 2010 On the complexity of Gödel's proof predicate
Yijia Chen, Jörg Flum
J. Symbolic Logic 75(1): 239-254 (March 2010). DOI: 10.2178/jsl/1264433918

Abstract

The undecidability of first-order logic implies that there is no computable bound on the length of shortest proofs of valid sentences of first-order logic. Some valid sentences can only have quite long proofs. How hard is it to prove such “hard” valid sentences? The polynomial time tractability of this problem would imply the fixed-parameter tractability of the parameterized problem that, given a natural number n in unary as input and a first-order sentence φ as parameter, asks whether φ has a proof of length ≤ n. As the underlying classical problem has been considered by Gödel we denote this problem by p-Gödel. We show that p-Gödel is not fixed-parameter tractable if DTIME(hO(1)) ≠ NTIME(hO(1)) for all time constructible and increasing functions h. Moreover we analyze the complexity of the construction problem associated with p-Gödel.

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Yijia Chen. Jörg Flum. "On the complexity of Gödel's proof predicate." J. Symbolic Logic 75 (1) 239 - 254, March 2010. https://doi.org/10.2178/jsl/1264433918

Information

Published: March 2010
First available in Project Euclid: 25 January 2010

zbMATH: 1201.03050
MathSciNet: MR2605891
Digital Object Identifier: 10.2178/jsl/1264433918

Rights: Copyright © 2010 Association for Symbolic Logic

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Vol.75 • No. 1 • March 2010
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