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