{"product_id":"forcing-with-random-variables-and-proof-complexity-paperback-9780521154338","title":"Forcing with Random Variables and Proof Complexity (Paperback) 9780521154338","description":"\u003cfont face=\"Georgia\"\u003e\r\n\u003cp\u003e\u003cfont size=\"6\"\u003eForcing with Random Variables and Proof Complexity\u003c\/font\u003e\u003cbr\u003e\r\n\r\n\r\n\u003c\/p\u003e\n\u003cp\u003e\u003cem\u003eA model-theoretic approach to bounded arithmetic and propositional proof complexity.\u003c\/em\u003e\u003c\/p\u003e\r\n\r\n\r\n\u003cp\u003e\u003cfont size=\"4\"\u003eJan Krají?ek (Author)\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003e9780521154338, Cambridge University Press\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003ePaperback, published 23 December 2010\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003e264 pages\u003cbr\u003e22.8 x 15.2 x 1.5 cm, 0.38 kg\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\r\n\r\n\u003cp align=\"justify\"\u003e\u003cem\u003e\u003cfont size=\"3\"\u003e\"Jan Krají?ek 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 Krají?ek'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.\"  \u003cbr\u003eSoren M. Riis for Mathematical Reviews\u003c\/font\u003e\u003c\/em\u003e\u003c\/p\u003e\r\n\r\n\u003cp align=\"justify\"\u003e\u003cstrong\u003e\u003cfont size=\"3\"\u003eThis 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.\u003c\/font\u003e\u003c\/strong\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003ePreface\u003cbr\u003e Acknowledgements\u003cbr\u003e Introduction\u003cbr\u003e Part I. Basics: 1. The definition of the models\u003cbr\u003e 2. Measure on ?\u003cbr\u003e 3. Witnessing quantifiers\u003cbr\u003e 4. The truth in N and the validity in K(F)\u003cbr\u003e Part II. Second Order Structures: 5. Structures K(F,G)\u003cbr\u003e Part III. AC0 World: 6. Theories I?0, I?0(R) and V10\u003cbr\u003e 7. Shallow Boolean decision tree model\u003cbr\u003e 8. Open comprehension and open induction\u003cbr\u003e 9. Comprehension and induction via quantifier elimination: a general reduction\u003cbr\u003e 10. Skolem functions, switching lemma, and the tree model\u003cbr\u003e 11. Quantifier elimination in K(Ftree,Gtree)\u003cbr\u003e 12. Witnessing, independence and definability in V10\u003cbr\u003e Part IV. AC0(2) World: 13. Theory Q2V10\u003cbr\u003e 14. Algebraic model\u003cbr\u003e 15. Quantifier elimination and the interpretation of Q2\u003cbr\u003e 16. Witnessing and independence in Q2V10\u003cbr\u003e Part V. Towards Proof Complexity: 17. Propositional proof systems\u003cbr\u003e 18. An approach to lengths-of-proofs lower bounds\u003cbr\u003e 19. PHP principle\u003cbr\u003e Part VI. Proof Complexity of Fd and Fd(+): 20. A shallow PHP model\u003cbr\u003e 21. Model K(Fphp,Gphp) of V10\u003cbr\u003e 22. Algebraic PHP model?\u003cbr\u003e Part VII. Polynomial-Time and Higher Worlds: 23. Relevant theories\u003cbr\u003e 24. Witnessing and conditional independence results\u003cbr\u003e 25. Pseudorandom sets and a Löwenheim–Skolem phenomenon\u003cbr\u003e 26. Sampling with oracles\u003cbr\u003e Part VIII. Proof Complexity of EF and Beyond: 27. Fundamental problems in proof complexity\u003cbr\u003e 28. Theories for EF and stronger proof systems\u003cbr\u003e 29. Proof complexity generators: definitions and facts\u003cbr\u003e 30. Proof complexity generators: conjectures\u003cbr\u003e 31. The local witness model\u003cbr\u003e Appendix. Non-standard models and the ultrapower construction\u003cbr\u003e Standard notation, conventions and list of symbols\u003cbr\u003e References\u003cbr\u003e Name index\u003cbr\u003e Subject index.\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003eSubject Areas: Mathematical theory of computation [\u003ca title=\"See our other books on Mathematical theory of computation\" href=\"https:\/\/freshlyprintedbooks.co.uk\/search?q=%22Mathematical%20theory%20of%20computation%20%5BUYA%5D%22\"\u003eUYA\u003c\/a\u003e], Mathematical logic [\u003ca title=\"See our other books on Mathematical logic\" href=\"https:\/\/freshlyprintedbooks.co.uk\/search?q=%22Mathematical%20logic%20%5BPBCD%5D%22\"\u003ePBCD\u003c\/a\u003e]\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\r\n\u003c\/font\u003e","brand":"Cambridge University Press","offers":[{"title":"Default Title","offer_id":46006719906072,"sku":"9780521154338","price":48.99,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0730\/2037\/5320\/products\/9780521154338i_8a8c5fad-61f5-49d3-8340-fff2593dc702.jpg?v=1691372704","url":"https:\/\/freshlyprintedbooks.co.uk\/products\/forcing-with-random-variables-and-proof-complexity-paperback-9780521154338","provider":"Freshly Printed Books","version":"1.0","type":"link"}