Proof Complexity

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.

Introduction; Part I. Basic Concepts: 1. Concepts and problems; 2. Frege systems; 3. Sequent calculus; 4. Quantified propositional calculus; 5. Resolution; 6. Algebraic and geometric proof systems; 7. Further proof systems; Part II. Upper Bounds: 8. Basic example of the correspondence between theories and proof systems; 9. Two worlds of bounded arithmetic; 10. Up to EF via the <...> translation; 11. Examples of upper bounds and p-simulations; 12. Beyond EF via the || ... || translation; Part III. Lower Bounds: 13. R and R-like proof systems; 14. {LK}_{d + 1/2} and combinatorial restrictions; 15. F_d and logical restrictions; 16. Algebraic and geometric proof systems; 17. Feasible interpolation: a framework; 18. Feasible interpolation: applications; Part IV. Beyond Bounds: 19. Hard tautologies; 20. Model theory and lower bounds; 21. Optimality; 22. The nature of proof complexity; Bibliography; Special symbols; Index.

Jan Krajícek is Professor of Mathematical Logic in the Faculty of Mathematics and Physics at Charles University, Prague. He is a member of the Academia Europaea and of the Learned Society of the Czech Republic. He has been an invited speaker at the European Congress of Mathematicians and at the International Congresses of Logic, Methodology and Philosophy of Science.