Skip to content

Your Cart


You have 0 items in your cart.

Register Sign in Wishlist

Forcing with Random Variables and Proof Complexity

$70.99 (C)

Part of London Mathematical Society Lecture Note Series

  • Date Published: February 2011
  • availability: In stock
  • format: Paperback
  • isbn: 9780521154338

$ 70.99 (C)

Add to cart Add to wishlist

Other available formats:

Looking for an examination copy?

If you are interested in the title for your course we can consider offering an examination copy. To register your interest please contact providing details of the course you are teaching.

Product filter button
About the Authors
  • 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
    Read more

    Reviews & endorsements

    "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

    Customer reviews

    Not yet reviewed

    Be the first to review

    Review was not posted due to profanity


    , create a review

    (If you're not , sign out)

    Please enter the right captcha value
    Please enter a star rating.
    Your review must be a minimum of 12 words.

    How do you rate this item?


    Product details

    • Date Published: February 2011
    • format: Paperback
    • isbn: 9780521154338
    • length: 264 pages
    • dimensions: 228 x 152 x 15 mm
    • weight: 0.38kg
    • availability: In stock
  • Table of Contents

    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
    Name index
    Subject index.

  • Resources for

    Forcing with Random Variables and Proof Complexity

    Jan Krajíček

    General Resources

    Welcome to the resources site

    Here you will find free-of-charge online materials to accompany this book. The range of materials we provide across our academic and higher education titles are an integral part of the book package whether you are a student, instructor, researcher or professional.

    Find resources associated with this title

    Type Name Unlocked * Format Size

    Showing of

    Back to top

    *This title has one or more locked files and access is given only to instructors adopting the textbook for their class. We need to enforce this strictly so that solutions are not made available to students. To gain access to locked resources you either need first to sign in or register for an account.

    These resources are provided free of charge by Cambridge University Press with permission of the author of the corresponding work, but are subject to copyright. You are permitted to view, print and download these resources for your own personal use only, provided any copyright lines on the resources are not removed or altered in any way. Any other use, including but not limited to distribution of the resources in modified form, or via electronic or other media, is strictly prohibited unless you have permission from the author of the corresponding work and provided you give appropriate acknowledgement of the source.

    If you are having problems accessing these resources please email

  • 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.

Sign In

Please sign in to access your account


Not already registered? Create an account now. ×

Sorry, this resource is locked

Please register or sign in to request access. If you are having problems accessing these resources please email

Register Sign in
Please note that this file is password protected. You will be asked to input your password on the next screen.

» Proceed

You are now leaving the Cambridge University Press website. Your eBook purchase and download will be completed by our partner Please see the permission section of the catalogue page for details of the print & copy limits on our eBooks.

Continue ×

Continue ×

Continue ×

Find content that relates to you

This site uses cookies to improve your experience. Read more Close

Are you sure you want to delete your account?

This cannot be undone.


Thank you for your feedback which will help us improve our service.

If you requested a response, we will make sure to get back to you shortly.

Please fill in the required fields in your feedback submission.