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 (Nguyen The Phuong) received his MSc and PhD degrees from University of Toronto in 2004 and 2008 respectively. He has been awarded postdoctoral fellowships by the Eduard Cech Center for Algebra and Geometry (the Czech Republic) for 2008-9, and by the Natural Sciences and Engineering Research Council of Canada (NSERC), effective September 2009.
1. Introduction; 2. The predicate calculus and the system; 3. Peano arithmetic and its subsystems; 4. Two-sorted logic and complexity classes; 5. The theory V0 and AC0; 6. The theory V1 and polynomial time; 7. Propositional translations; 8. Theories for polynomial time and beyond; 9. Theories for small classes; 10. Proof systems and the reflection principle; 11. Computation models.