Logical Foundations of Proof Complexity

Overview

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 result is a uniform treatment of many systems in the literature, including Buss’s ...

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Overview

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 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.

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Editorial Reviews

From the Publisher
"This authoritative volume on computational complexity of logical systems provides a sound background in logic for computer science and mathematics students. The book provides a number of exercises that offer insights into the presented material and facilitate the understanding of the concepts and results. The list of bibliographic references contains the most-representative published work in the domains of proof and computational complexity theories."
L. State, reviews.com

"This most erudite of books will stand the test of time. I will revisit it often."
George Hacken, Computing Reviews

"... very great detail and with care... The presentation of the book is most thorough and the treatment rigorous, providing the reader with the right amount of intuition and detail to follow even complicated constructions. The new book by Cook and Nguyen will become another classic for the field of bounded arithmetic and its relations to computational complexity and propositional proof complexity. It will serve as a rich source of reference for the expert and as a thorough guide to the student interested in learning about this beautiful field."
Olaf Beyersdorff, Mathematical Reviews

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Product Details

  • ISBN-13: 9781107694118
  • Publisher: Cambridge University Press
  • Publication date: 3/6/2014
  • Pages: 496
  • Product dimensions: 6.14 (w) x 9.21 (h) x 1.10 (d)

Meet the Author

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 (Nguyễn Thế Phương) received his MSc and PhD degrees from University of Toronto in 2004 and 2008 respectively. He has been awarded postdoctoral fellowships by the Eduard Čech Center for Algebra and Geometry (the Czech Republic) for 2008–2009, and by the Natural Sciences and Engineering Research Council of Canada (NSERC), effective September 2009.

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Table of Contents

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.

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