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Forcing with Random Variables and Proof Complexity

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Part of London Mathematical Society Lecture Note Series

  • Date Published: December 2010
  • availability: Available
  • format: Paperback
  • isbn: 9780521154338

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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.

    • A brand new approach to problems of complexity theory
    • Self-contained so readers do not need to study other material
    • Presents some of the most recent developments in proof complexity
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    Product details

    • Date Published: December 2010
    • format: Paperback
    • isbn: 9780521154338
    • length: 264 pages
    • dimensions: 228 x 152 x 15 mm
    • weight: 0.38kg
    • availability: Available
  • Table of Contents

    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.

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    Forcing with Random Variables and Proof Complexity

    Jan Krajíček

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  • Author

    Jan Krajíček, Charles University, Prague
    Jan Krajíček 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.

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