Proof complexity / Jan Krajíček, Charles University, Prague.

Format:

Book

Author/Creator:

Krajíček, Jan, author.

Series:

Encyclopedia of mathematics and its applications ; v. 170.

Encyclopedia of mathematics and its applications ; 170

Language:

English

Subjects (All):

Proof theory.

Computational complexity.

Physical Description:

xiv, 516 pages ; 24 cm.

Place of Publication:

Cambridge, United Kingdom ; New York, NY : Cambridge University Press, 2019.

Summary:

"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"-- Provided by publisher.

Contents:

Concepts and problems

Frege systems

Sequent calculus

Quantifed propositional calculus

Resolution

Algebraic and geometric proof systems

Further proof systems

Basic example of the correspondence

Two worlds of bounded arithmetic

Up to EF via the <...> translation

Examples of upper bounds and p-simulations

Beyond EF via the || ... || translation

R and R-like proof systems

LKD+1/2 and combinatorial restrictions

Fd and logical restrictions

Feasible interpolation: a framework

Feasible interpolation: applications

Hard tautologies

Model theory and lower bounds

Optimality

The nature of proof complexity.

Notes:

Includes bibliographical references (pages 481-505) and index.

ISBN:

1108416845

9781108416849

OCLC:

1045390825