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Structured pigeonhole principle, search problems and hard tautologies

Published online by Cambridge University Press:  12 March 2014

Jan Krajíček
Jan Krajíček*
Affiliation:
Mathematical Institute, Academy of Sciences, ŽitnÁ 25, Prague 1, CZ - 115 67 The Czech Republic, E-mail: krajicek@math.cas.cz
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Abstract

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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 70 , Issue 2 , June 2005 , pp. 619 - 630
Copyright
Copyright © Association for Symbolic Logic 2005

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References

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