Forcing with Random Variables and Proof Complexity

Jan Krajíček

doi:10.1017/CBO9781139107211

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Allender, Eric and Das, Bireswar 2014. Mathematical Foundations of Computer Science 2014. Vol. 8635, Issue. , p. 25.

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Beckmann, Arnold and Buss, Samuel R. 2014. Improved witnessing and local improvement principles for second-order bounded arithmetic. ACM Transactions on Computational Logic, Vol. 15, Issue. 1, p. 1.

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Beyersdorff, Olaf Chew, Leroy Mahajan, Meena and Shukla, Anil 2015. Automata, Languages, and Programming. Vol. 9134, Issue. , p. 180.

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Atserias, Albert and Müller, Moritz 2015. Partially definable forcing and bounded arithmetic. Archive for Mathematical Logic, Vol. 54, Issue. 1-2, p. 1.

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Bydžovský, Jan and Müller, Moritz 2020. Polynomial time ultrapowers and the consistency of circuit lower bounds. Archive for Mathematical Logic, Vol. 59, Issue. 1-2, p. 127.

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Khaniki, Erfan 2024. Jump Operators, Interactive Proofs and Proof Complexity Generators. p. 573.

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Pich, Ján 2025. Learning algorithms from circuit lower bounds. computational complexity, Vol. 34, Issue. 1,

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Jan Krajíček, Charles University, Prague

Publisher:: Cambridge University Press
Online publication date:: November 2011
Print publication year:: 2010
Online ISBN:: 9781139107211
DOI:: https://doi.org/10.1017/CBO9781139107211

Subjects:: Mathematics (general), Mathematics, Logic, Categories and Sets
Series:: London Mathematical Society Lecture Note Series (382)

£56.00

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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.

"Jan Krajíček is the leading expert on these problems and in this book he provides a new approach to builing models of bounded arithmetic which combines methods and techniques from model theory, forcing and computational complexity. Personally, I find Krajíček's approach a highly stimulating collage of ideas. I recommend this book strongly to anyone interested in logical approaches to fundamental problems in complexity theory."
Soren M. Riis for Mathematical Reviews

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Forcing with Random Variables and Proof Complexity

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