Forcing with Random Variables and Proof Complexity
Forcing with Random Variables and Proof Complexity

Author: Jan Krajíček

Publisher: Cambridge University Press

Published: 2010-12-23

Total Pages: 265

ISBN-13: 1139493922

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This book introduces a new approach to building models of bounded arithmetic, with techniques drawn from recent results in computational complexity. Propositional proof systems and bounded arithmetics are closely related. In particular, proving lower bounds on the lengths of proofs in propositional proof systems is equivalent to constructing certain extensions of models of bounded arithmetic. This offers a clean and coherent framework for thinking about lower bounds for proof lengths, and it has proved quite successful in the past. This book outlines a brand new method for constructing models of bounded arithmetic, thus for proving independence results and establishing lower bounds for proof lengths. The models are built from random variables defined on a sample space which is a non-standard finite set and sampled by functions of some restricted computational complexity. It will appeal to anyone interested in logical approaches to fundamental problems in complexity theory.

Forcing with Random Variables and Proof Complexity
Forcing with Random Variables and Proof Complexity

Author: Jan Krajíček

Publisher:

Published: 2011

Total Pages: 247

ISBN-13: 9781139123082

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This book introduces a new approach to building models of bounded arithmetic, with techniques drawn from recent results in computational complexity. Propositional proof systems and bounded arithmetics are closely related. In particular, proving lower bounds on the lengths of proofs in propositional proof systems is equivalent to constructing certain extensions of models of bounded arithmetic. This offers a clean and coherent framework for thinking about lower bounds for proof lengths, and it has proved quite successful in the past. This book outlines a brand new method for constructing models of bounded arithmetic, thus for proving independence results and establishing lower bounds for proof lengths. The models are built from random variables defined on a sample space which is a non-standard finite set and sampled by functions of some restricted computational complexity. It will appeal to anyone interested in logical approaches to fundamental problems in complexity theory.

Logical Approaches to Computational Barriers
Logical Approaches to Computational Barriers

Author: Arnold Beckmann

Publisher: Springer Science & Business Media

Published: 2006-06-26

Total Pages: 623

ISBN-13: 3540354662

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This book constitutes the refereed proceedings of the Second International Conference on Computability in Europe, CiE 2006, held in Swansea, UK, June/July 2006. The book presents 31 revised full papers together with 30 invited papers, including papers corresponding to 8 plenary talks and 6 special sessions on proofs and computation, computable analysis, challenges in complexity, foundations of programming, mathematical models of computers and hypercomputers, and Gödel centenary: Gödel's legacy for computability.

Proof Complexity
Proof Complexity

Author: Jan Krajíček

Publisher: Cambridge University Press

Published: 2019-03-28

Total Pages: 533

ISBN-13: 1108416845

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Offers a self-contained work presenting basic ideas, classical results, current state of the art and possible future directions in proof complexity.

Surveys in Combinatorics 2017
Surveys in Combinatorics 2017

Author: Anders Claesson

Publisher: Cambridge University Press

Published: 2017-06-30

Total Pages: 451

ISBN-13: 1108350356

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This volume contains nine survey articles which provide expanded accounts of plenary seminars given at the British Combinatorial Conference at the University of Strathclyde in July 2017. This biennial conference is a well-established international event attracting speakers from around the world. Written by internationally recognised experts in the field, these articles represent a timely snapshot of the state of the art in the different areas of combinatorics. Topics covered include the robustness of graph properties, the spt-function of Andrews, switching techniques for edge decompositions of graphs, monotone cellular automata, and applications of relative entropy in additive combinatorics. The book will be useful to researchers and advanced graduate students, primarily in mathematics but also in computer science and statistics.

Asymptotic Analysis in General Relativity
Asymptotic Analysis in General Relativity

Author: Thierry Daudé

Publisher: Cambridge University Press

Published: 2018-01-11

Total Pages: 381

ISBN-13: 1108500781

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This volume compiles notes from four mini courses given at the summer school on asymptotic analysis in general relativity, held at the Institut Fourier in Grenoble, France. It contains an up-to-date panorama of modern techniques in the asymptotic analysis of classical and quantum fields in general relativity. Accessible to graduate students, these notes gather results that were not previously available in textbooks or monographs and will be of wider interest to researchers in general relativity. The topics of these mini courses are: the geometry of black hole spacetimes; an introduction to quantum field theory on curved spacetimes; conformal geometry and tractor calculus; and microlocal analysis for wave propagation.

Polynomials and the mod 2 Steenrod Algebra
Polynomials and the mod 2 Steenrod Algebra

Author: Grant Walker

Publisher: Cambridge University Press

Published: 2018

Total Pages: 371

ISBN-13: 1108414486

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The first of two volumes covering the Steenrod algebra and its various applications. Suitable as a graduate text.

Polynomials and the mod 2 Steenrod Algebra
Polynomials and the mod 2 Steenrod Algebra

Author: Grant Walker (Mathematician)

Publisher: Cambridge University Press

Published: 2018

Total Pages: 381

ISBN-13: 1108414451

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This is the first book to link the mod 2 Steenrod algebra, a classical object of study in algebraic topology, with modular representations of matrix groups over the field F of two elements. The link is provided through a detailed study of Peterson's 'hit problem' concerning the action of the Steenrod algebra on polynomials, which remains unsolved except in special cases. The topics range from decompositions of integers as sums of 'powers of 2 minus 1', to Hopf algebras and the Steinberg representation of GL(n,F). Volume 1 develops the structure of the Steenrod algebra from an algebraic viewpoint and can be used as a graduate-level textbook. Volume 2 broadens the discussion to include modular representations of matrix groups.

Polynomials and the mod 2 Steenrod Algebra: Volume 2, Representations of GL (n,F2)
Polynomials and the mod 2 Steenrod Algebra: Volume 2, Representations of GL (n,F2)

Author: Grant Walker

Publisher: Cambridge University Press

Published: 2017-11-09

Total Pages: 381

ISBN-13: 1108355927

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This is the first book to link the mod 2 Steenrod algebra, a classical object of study in algebraic topology, with modular representations of matrix groups over the field F of two elements. The link is provided through a detailed study of Peterson's `hit problem' concerning the action of the Steenrod algebra on polynomials, which remains unsolved except in special cases. The topics range from decompositions of integers as sums of 'powers of 2 minus 1', to Hopf algebras and the Steinberg representation of GL(n, F). Volume 1 develops the structure of the Steenrod algebra from an algebraic viewpoint and can be used as a graduate-level textbook. Volume 2 broadens the discussion to include modular representations of matrix groups.