## Journal of Symbolic Logic

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

### Structured pigeonhole principle, search problems and hard tautologies

#### Abstract

We consider exponentially large finite relational structures (with the universe {0,1}^{n}) whose basic relations are computed by polynomial size (n^{O(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 2^{n} 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 2^{2m} 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 1 July 2005

**Permanent link to this document**

https://projecteuclid.org/euclid.jsl/1120224731

**Digital Object Identifier**

doi:10.2178/jsl/1120224731

**Mathematical Reviews number (MathSciNet)**

MR2140049

**Zentralblatt MATH identifier**

1089.03049

#### Citation

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