Skip to main content
Springer Nature Link
Log in

Quantified propositional calculus and a second-order theory for NC1

  • Published:
Archive for Mathematical Logic Aims and scope Submit manuscript

Abstract

Let H be a proof system for quantified propositional calculus (QPC). We define the Σq j -witnessing problem for H to be: given a prenex Σq j -formula A, an H-proof of A, and a truth assignment to the free variables in A, find a witness for the outermost existential quantifiers in A. We point out that the Σq 1 -witnessing problems for the systems G* 1 and G1 are complete for polynomial time and PLS (polynomial local search), respectively.

We introduce and study the systems G* 0 and G0, in which cuts are restricted to quantifier-free formulas, and prove that the Σq j -witnessing problem for each is complete for NC1. Our proof involves proving a polynomial time version of Gentzen’s midsequent theorem for G* 0 and proving that G0-proofs are TC0-recognizable. We also introduce QPC systems for TC0 and prove witnessing theorems for them.

We introduce a finitely axiomatizable second-order system VNC1 of bounded arithmetic which we prove isomorphic to Arai’s first order theory AID + Σb 0 -CA for uniform NC1. We describe simple translations of VNC1 proofs of all bounded theorems to polynomial size families of G* 0 proofs. From this and the above theorem we get alternative proofs of the NC1 witnessing theorems for VNC1 and AID.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles and news from researchers in related subjects, suggested using machine learning.

References

  1. Arai, T.: A bounded arithmetic AID for Frege systems. Ann. Pure Appl. Logic 103, 155–199 (2000)

    Article  Google Scholar 

  2. Barrington, D.A.M., Immerman, N., Straubing, H.: On Uniformity within NC1. J. Comput. Syst. Sci. 41, 274–306 (1990)

    Article  Google Scholar 

  3. Buss, : Bounded Arithmetic. Bibliopolis, 1986

  4. Buss, : The Boolean formula value problem is in ALOGTIME. Proceedings of the 19th Annual ACM Symposium on Theory of Computing (STOC’87). 1987, pp. 123–131

  5. Buss, S.: Axiomatizations and conservation results for fragments of bounded arithmetic. In: Logic and Computation, Proceedings of a Workshop held at Carnegie Mellon University. AMS, 1990, pp. 57–84

  6. Buss, S.: Propositional consistency proofs. Ann. Pure Appl. Logic 52, 3–29 (1991)

    Article  Google Scholar 

  7. Buss, S.: Algorithms for Boolean formula evaluation and for tree-contraction. In: P. Clote, J. Krajicek (eds.), Proof Theory, Complexity, and Arithmetic. Oxford University Press, 1993, pp. 95–115

  8. Buss, S.: First-order proof theory of arithmetic. In: S. Buss (ed.), Handbook of Proof Theory. Elsevier, 1998, pp. 79–147. Available on line at http://www.math.ucsd.edu/~sbuss/ResearchWeb/

  9. Buss, : An introduction to proof theory. In: S. Buss (ed.), Handbook of Proof Theory. Elsevier, 1998, pp. 1–78. Available on line at http://www.math.ucsd.edu/~sbuss/ResearchWeb/

  10. Buss, S.: 2003. Personnal communication

  11. Buss, S., Clote, P.: Cutting planes, connectivity, and threshold logic. Arch. Math. Logic 35, 33–62 (1996)

    Google Scholar 

  12. Buss, S., Krajicek, J.: An application of Boolean complexity to separation problems in bounded arithmetic. The Proceedings of the London Mathematical Society 60 (3), 1–21 (1994)

    Google Scholar 

  13. Chiari, M., Krajicek, J.: Witnessing functions in bounded arithmetic and search problems. The J. Symbolic Logic 63, 1095–1115 (1998)

    Google Scholar 

  14. Clote, P.: ALOGTIME and a conjecture of S.A. Cook. Ann. Math. Art. Intell. 6, 57–106 (1990). Extended abstract in Proc. 13th IEEE Symposium on Logic in Computer Science, 1990

    Article  Google Scholar 

  15. Clote, P. Sequential, machine-independent characterizations of the parallel complexity classes AlogTIME,ACk,NCk and NC. In: S. Buss, P. Scott (eds), Feasible Mathematics. Birkhauser, 1990, pp. 49–69

  16. Cook, S.: Csc 2429 course notes: Proof complexity and bounded arithmetic, 2002. Available from the web at http://www.cs.toronto.edu/~sacook/csc2429h/+

  17. Cook, : Theories for complexity classes and their propositional translations. In: J. Krajicek (ed.), Complexity of Computations and Proofs. Quaderni di Matematica, 2003, pp. 175–227

  18. Cook, S., Kolokolova, A.: A second-order system for polytime reasoning based on Grädel’s theorem. Ann. Pure Appl. Logic 124, 193–231 (2003)

    Article  Google Scholar 

  19. Cook, S., Thapen, N.: The strength of replacement in weak arithmetic. ACM Transactions on Computational Logic. 2005, pp. 1–16 (in Press)

  20. Cook, S.A., Reckhow, R.A.: The relative efficiency of propositional proof systems. J. Symbolic Logic 44 (1), 36–50 (1977)

    Google Scholar 

  21. Immerman, N.: Descriptive Complexity. Springer, 1999

  22. Johnson, D.S.: A catalog of complexity classes. In: J. van Leewen (ed.), Handbook of Theoretical Computer Science. Elsevier Science Publishers, 1990, pp. 67–161

  23. Johnson, D.S., Papadimitriou, C.H., Yannakakis, M.: How easy is local search? J. Comput. Syst. Sci. 37, 79–100 (1988)

    Article  Google Scholar 

  24. Krajíček, J.: Exponentiation and second-order bounded arithmetic. Ann. Pure Appl. Logic 48, 261–276 (1990)

    Article  Google Scholar 

  25. Krajicek, J.: Fragments of Bounded Arithmetic and Bounded Query Classes. Transactions of the American Mathematical Society 338 (2), 587–598 (1993)

    Google Scholar 

  26. Krajíček, J.: Bounded Arithmetic, Propositional Logic and Computational Complexity. Cambridge University Press, 1995

  27. Krajíček, J., Pudlák, P.: Quantified propositional calculi and fragments of bounded arithmetic. Zeitschrift f. Mathematische Logik u. Grundlagen d. Mathematik 36, 29–46 (1990)

    Google Scholar 

  28. Morioka, : Logical approaches to the complexity of search problems: Proof complexity, quantified propositional calculus, and bounded arithmetic. PhD thesis, Department of Computer Science, University of Toronto, 2005

  29. Nguyen, P.: Proving that VNC1 is finitely axiomatizable. unpublished note, 2004

  30. Pitassi, T.: Using hardness to prove Frege lower bounds, 2002. A seminar at the Fields Institute for Research in Mathematical Sciences, Toronto, Canada

  31. Pollett, C.: A propositional proof systems for Ri2. In: P. Beame, S. Buss (eds.), Proof Complexity and Feasible Arithmetics, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, The American Mathematical Society, 1997, pp. 253–278

  32. Pollett, C.: Structure and Definability in General Bounded Arithmetic Theories. Ann. Pure Appl. Logic 100, 189–245 (1999)

    Article  Google Scholar 

  33. Razborov, A.A.: An equivalence between second order bounded domain bounded arithmetic and first order bounded arithmetic. In: P. Clote, J. Krajicek (eds.), Arithmetic, Proof Theory and Computational Complexity. Oxford University Press, 1993, pp. 247–77

  34. Skelley, A.: Personal communication., 2002

  35. Takeuti, G.: Proof Theory. Elsevier Science Publisers, second edition, 1987

  36. Takeuti, G.: RSUV isomorphism. In: P. Clote, J. Krajicek (eds.), Arithmetic, Proof Theory and Computational Complexity. Oxford University Press, 1993, pp. 364–86

  37. Zambella, D.: Notes on polynomially bounded arithmetic. J. Symbolic Logic 61 (3), 942–966 (1996)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Toronto, 10 King’s College Road, Toronto, Ontario, M5S 3G4, Canada

    Stephen Cook

  2. Hitachi Systems Development Laboratory,  , 292 Yoshida-cho, Totsuka-ku, Yokohama-city, 244-0817, Japan

    Tsuyoshi Morioka

Authors
  1. Stephen Cook
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Tsuyoshi Morioka
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Stephen Cook.

Additional information

This research was carried while this author was a student at the University of Toronto.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cook, S., Morioka, T. Quantified propositional calculus and a second-order theory for NC1. Arch. Math. Logic 44, 711–749 (2005). https://doi.org/10.1007/s00153-005-0282-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00153-005-0282-2

Key words or phrases