Structured pigeonhole principle, search problems and hard tautologies

We consider exponentially large finite relational structures (with the universe {0, 1}) whose basic relations are computed by polynomial size (n) circuits. We study behaviour of such structures when pulled back by P/polymaps to a bigger or to a smaller universe. In particular, we prove that: 1. If there exists a P/poly map g : {0, 1} → {0, 1} , 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 tournament is hard for the proof system. 2. The search problemWPHP, decoding RSAor finding a collision in a hashing function can be reduced to finding a size m homogeneous subgraph in a size 2 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. The weak pigeonhole principle (WPHP) is the statement that no f : {0, 1} → {0, 1} can be injective if m > n. The dual weak pigeonhole principle (dWPHP) is the statement that no g : {0, 1} → {0, 1} can be surjective if n < m. We study the proof complexity of WPHP and dWPHP forP/polymaps f and g. Some information is known. For example, it is a necessary condition for a family of functions to be strongly collision-free that bounded arithmetic theoryS 2 does not prove WPHP for functions in the family, cf. [11]. Or if RSA were to be secure then WPHP for the modular exponentiation cannot be proved in S 2 either, cf. [17]. In these results S 2 can be augmented by the true ∀Π b 1-theory ofN ; in particular, by the statements stating the soundness of all propositional proof systems. Consequently we cannot expect to derive hardness results for particular proof systems by appealing to witnessing theorems in bounded arithmetic as such results would automatically apply to all proof systems; we will get some hardness results for particular search problems instead. But we find a link between these search problems and the proof complexity of particular tautologies expressing a Ramsey property of a graph. The main concept used in this link are the implicit proofs of implicit formulas, cf. [15, 13]. Received February 14, 2004; revised December 15, 2004. The paper was written while the author was a member of the Institute for Advanced Study in Princeton, supported by the NSF grant DMS-0111298. On leave from the Mathematical Institute of the Academy of Sciences of the Czech Republic and from the Institute for Theoretical Computer Science of the Charles University, partially supported by grant # A 101 94 01 of the Academy of Sciences, by grant #201/05/0124 of the GA CR, and by project LN00A056 of The Ministry of Education of the Czech Republic. A part of this work was done while visiting the Mathematical Institute at Oxford, supported by the EPSRC. c © 2005, Association for Symbolic Logic 0022-4812/05/7002-0016/$2.20