Uniform Proof Complexity

Abstract: Uniform proof complexity (aka bounded arithmetic) studies the computational complexity of concepts needed to prove theorems. For example the pigeonhole principle cannot be proved in a system restricted to AC0 reasoning, intuitively because the complexity class AC0 cannot count. Also no polynomial time algorithm for recognizing primes can be proved correct in a system restricted to polynomial time reasoning, unless integers can be factored in polynomial time. We give a high-level introduction to uniform proof complexity, and draw interesting consequences (joint work with Jan Krajicek) from the assumption that a polytime theory can prove that NP has polynomial size Boolean circuits.

Speaker's Bio: Stephen Cook was born in Buffalo, New York, received his BSc degree from University of Michigan in 1961, and his S.M. and PhD degrees from Harvard University in 1962 and 1966 respectively. From 1966 to 1970 he was Assistant Professor, University of California, Berkeley. He joined the faculty at the University of Toronto in 1970 as an Associate Professor, and was promoted to Professor in 1975 and University Professor in 1985. His principal research areas are computational complexity and proof complexity, with excursions into programming language semantics and parallel computation. He is the author of over 60 research papers, including his famous 1971 paper "The Complexity of Theorem Proving Procedures" which introduced the theory of NP completeness and proved that the Boolean satisfiability problem is NP complete. He is the 1982 recipient of the Turing award, and was awarded a Steacie Fellowship in 1977, a Killam Research Fellowship in 1982, and received the CRM/Fields Institute Prize in 1999. He received Computer Science teaching awards in 1989 and 1995. He is a fellow of the Royal Society of London, Royal Society of Canada, and was elected to membership in the National Academy of Sciences (United States) and the American Academy of Arts and Sciences. Twenty-six students have completed their PhD degrees under his supervision, and many of them now have prominent academic careers of their own.