Abstract. Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. This is article includes a broad survey of the field, and a technical exposition of some recently developed techniques for proving lower bounds on proof sizes. Contents

The pigeonhole principle asserts that there is no injective mapping from m pigeons to n holes as long as m> n. It is amazingly simple, expresses one of the most basic primitives in mathematics and Theoretical Computer Science (counting) and, for these reasons, is probably the most extensively studied combinatorial principle. In this survey we try to summarize what is known about its proof complexity, and what we would still like to prove. We also mention some applications of the pigeonhole principle to the study of efficient provability of major open problems in computational complexity, as well as some of its generalizations in the form of general matching principles.

We prove a quasi-polynomial lower bound on the size of bounded-depth Frege proofs of the pigeonhole principle PHP m n where m = (1 + 1/polylog n) n. This lower bound qualitatively matches the known quasipolynomial-size bounded-depth Frege proofs for these principles. Our technique, which uses a switching lemma argument like other lower bounds for bounded-depth Frege proofs, is novel in that the tautology to which this switching lemma is applied remains random throughout the argument.

Propositional Proof Complexity is an active area of research whose main focus is the study of the length of proofs in propositional logic. There are several motivations for such a study, the main of which is probably its connection to the P vs NP problem in Computational Complexity. The experience

This is the second annual progress report for the ALCOM-FT project, supported by the European