Logical Foundations Of Proof Complexity

This page intentionally left blank Logical Foundations of Proof Complexity This book treats bounded arithmetic and propositional proof complexity from the point of view of computational complexity. The first seven chapters include the necessary logical background for the material and are suitable for a graduate course. Associated with each of many complexity classes are both a two-sorted predicate calculus theory, with induction restricted to concepts in the class, and a propositional proof system. The complexity classes range from AC0 for the weakest theory up to the polynomial hierarchy. Each bounded theorem in a theory translates into a family of (quantified) propositional tautologies with polynomial size proofs in the corresponding proof system. The theory proves the soundness of the associated proof system. The result is a uniform treatment of many systems in the literature, including Buss’s theories for the polynomial hierarchy and many disparate systems for complexity classes such as AC0 , AC0 (m), TC0 , NC1 , L, NL, NC, and P. Stephen Cook is a professor at the University of Toronto. He is author of many research papers, including his famous 1971 paper “The Complexity of Theorem Proving Procedures,” and the 1982 recipient of the Turing Award. He was awarded a Steacie Fellowship in 1977 and a Killam Research Fellowship in 1982 and received the CRM/Fields Institute Prize in 1999. He is a Fellow of the Royal Society of London and the 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. Phuong Nguyen is a postdoctoral researcher at McGill University. He received his MSc and PhD degrees from University of Toronto in 2004 and 2008, respectively. ˇ He has been awarded postdoctoral fellowships by the Eduard Cech Center for Algebra and Geometry (the Czech Republic) and by the Natural Sciences and Engineering Research Council of Canada (NSERC). PERSPECTIVES IN LOGIC The Perspectives in Logic series publishes substantial, high-quality books whose central theme lies in any area or aspect of logic. Books that present new material not now available in book form are particularly welcome. The series ranges from introductory texts suitable for beginning graduate courses to specialized monographs at the frontiers of research. Each book offers an illuminating perspective for its intended audience. The series has its origins in the old Perspectives in Mathematical Logic series edited by the -Group for “Mathematische Logik” of the Heidelberger Akademie der Wissenchaften, whose beginnings date back to the 1960s. The Association for Symbolic Logic has assumed editorial responsibility for the series and changed its name to reflect its interest in books that span the full range of disciplines in which logic plays an important role. Pavel Pudlak, Managing Editor Mathematical Institute of the Academy of Sciences of the Czech Republic Editorial Board Michael Benedikt Department of Computing Science, University of Oxford Michael Glanzberg Department of Philosophy, University of California, Davis Carl G. Jockusch, Jr. Department of Mathematics, University of Illinois