Proof complexity is a rich subject drawing on methods from logic, combinatorics, algebra and computer science. This self-contained book presents the basic concepts, classical results, current state of the art and possible future directions in the field. It stresses a view of proof complexity as a whole entity rather than a collection of various topics held together loosely by a few notions, and it favors more generalizable statements. Lower bounds for lengths of proofs, often regarded as the key issue in proof complexity, are of course covered in detail. However, upper bounds are not neglected: this book also explores the relations between bounded arithmetic theories and proof systems and how they can be used to prove upper bounds on lengths of proofs and simulations among proof systems. It goes on to discuss topics that transcend specific proof systems, allowing for deeper understanding of the fundamental problems of the subject.

‘… the book has very rich content and its bibliographical material includes all previous books and survey articles related to proof complexity.’

Anahit Artashes Chubaryan Source: MathSciNet

'This book is in my view an excellent reference manual for a fundamental topic in mathematical logic and theoretical computer science.'

Jaap van Oosten Source: Boekbesprekingen