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Informationen zum Autor Jan Krají¿ek is a Professor of Mathematical Logic at Charles University in Prague. He is currently also affiliated with the Academy of Sciences of the Czech Republic. Klappentext A model-theoretic approach to bounded arithmetic and propositional proof complexity. Zusammenfassung Aimed at researchers and graduate students in mathematics and theoretical computer science! who are interested in logical approaches to fundamental problems of computational complexity theory! and of proof complexity in particular. Inhaltsverzeichnis Preface; Acknowledgements; Introduction; Part I. Basics: 1. The definition of the models; 2. Measure on ß; 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 I¿0, I¿0(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....
