Forcing with Random Variables and Proof Complexity: 382 (London Mathematical Society Lecture Note Series, Series Number 382)
Cambridge University Press, 12/23/2010
EAN 9780521154338, ISBN10: 0521154332
Paperback, 264 pages, 22.9 x 15.2 x 1.5 cm
Language: English
Originally published in English
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 IÃŽâ€0, IÃŽâ€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.