Journal of Symbolic Logic

Structured pigeonhole principle, search problems and hard tautologies

Jan Krajíček

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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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J. Symbolic Logic, Volume 70, Issue 2 (2005), 619-630.

First available in Project Euclid: 1 July 2005

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Krajíček, Jan. Structured pigeonhole principle, search problems and hard tautologies. J. Symbolic Logic 70 (2005), no. 2, 619--630. doi:10.2178/jsl/1120224731.

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