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June 2005 Structured pigeonhole principle, search problems and hard tautologies
Jan Krajíček
J. Symbolic Logic 70(2): 619-630 (June 2005). DOI: 10.2178/jsl/1120224731


We consider exponentially large finite relational structures (with the universe {0,1}n) whose basic relations are computed by polynomial size (nO(1)) circuits. We study behaviour of such structures when pulled back by 𝒫/poly maps to a bigger or to a smaller universe. In particular, we prove that:

1. If there exists a 𝒫/poly map g : {0,1}n → {0,1}m, n < m, iterable for a proof system then a tautology (independent of g) expressing that a particular size n set is dominating in a size 2n tournament is hard for the proof system.

2. The search problem WPHP, decoding RSA or finding a collision in a hashing function can be reduced to finding a size m homogeneous subgraph in a size 22m graph.

Further we reduce the proof complexity of a concrete tautology (expressing a Ramsey property of a graph) in strong systems to the complexity of implicit proofs of implicit formulas in weak proof systems.


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Jan Krajíček. "Structured pigeonhole principle, search problems and hard tautologies." J. Symbolic Logic 70 (2) 619 - 630, June 2005.


Published: June 2005
First available in Project Euclid: 1 July 2005

zbMATH: 1089.03049
MathSciNet: MR2140049
Digital Object Identifier: 10.2178/jsl/1120224731

Rights: Copyright © 2005 Association for Symbolic Logic


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Vol.70 • No. 2 • June 2005
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