Forcing with Random Variables and Proof Complexity Krajíček Paperback

Description: Forcing with Random Variables and Proof Complexity A model-theoretic approach to bounded arithmetic and propositional proof complexity. Jan Krajíček (Author) 9780521154338, Cambridge University Press Paperback / softback, published 23 December 2010 264 pages 22.8 x 15.2 x 1.5 cm, 0.38 kg "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 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. Preface Acknowledgements Introduction Part I. Basics: 1. The definition of the models 2. Measure on β 3. Witnessing quantifiers 4. The truth in N and the validity in K(F) Part II. Second Order Structures: 5. Structures K(F,G) Part III. AC0 World: 6. Theories IΔ0, IΔ0(R) and V10 7. Shallow Boolean decision tree model 8. Open comprehension and open induction 9. Comprehension and induction via quantifier elimination: a general reduction 10. Skolem functions, switching lemma, and the tree model 11. Quantifier elimination in K(Ftree,Gtree) 12. Witnessing, independence and definability in V10 Part IV. AC0(2) World: 13. Theory Q2V10 14. Algebraic model 15. Quantifier elimination and the interpretation of Q2 16. Witnessing and independence in Q2V10 Part V. Towards Proof Complexity: 17. Propositional proof systems 18. An approach to lengths-of-proofs lower bounds 19. PHP principle Part VI. Proof Complexity of Fd and Fd(+): 20. A shallow PHP model 21. Model K(Fphp,Gphp) of V10 22. Algebraic PHP model? Part VII. Polynomial-Time and Higher Worlds: 23. Relevant theories 24. Witnessing and conditional independence results 25. Pseudorandom sets and a Löwenheim–Skolem phenomenon 26. Sampling with oracles Part VIII. Proof Complexity of EF and Beyond: 27. Fundamental problems in proof complexity 28. Theories for EF and stronger proof systems 29. Proof complexity generators: definitions and facts 30. Proof complexity generators: conjectures 31. The local witness model Appendix. Non-standard models and the ultrapower construction Standard notation, conventions and list of symbols References Name index Subject index. Subject Areas: Mathematical theory of computation [UYA], Mathematical logic [PBCD]

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