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On the complexity of Gödel's proof predicate

Published online by Cambridge University Press:  12 March 2014

Yijia Chen  and
Jörg Flum
Yijia Chen
Affiliation:
Basics, Department of Computer Science, Shanghai Jiaotong University, Shanghai 200030, China, E-mail: yijia.chen@cs.sjtu.edu.cn
Jörg Flum
Affiliation:
Abteilung für Mathematische Logik, Universität Freiburg, 79104 Freiburg, Germany, E-mail: joerg.flum@math.uni-freiburg.de
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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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 75 , Issue 1 , March 2010 , pp. 239 - 254
Copyright
Copyright © Association for Symbolic Logic 2010

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References

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