Hostname: page-component-6bb9c88b65-fsdjw Total loading time: 0 Render date: 2025-07-28T00:57:06.566Z Has data issue: false hasContentIssue false
  • English
  • Français

Consequences of the provability of NP ⊆ P/poly

Published online by Cambridge University Press:  12 March 2014

Stephen Cook  and
Jan Krajíček
Stephen Cook
Affiliation:
University of Toronto, Department of Computer Science, Toronto, M5S 3G4, Canada. E-mail: sacook@cs.toronto.edu Academy of Sciences, Mathematical Institute, Zitna 25, Prague CZ-115 67, Czech Republic
Jan Krajíček
Affiliation:
Academy of Sciences, Mathematical Institute, Zitna 25, Prague CZ-115 67, Czech Republic Charles University, Faculty of Mathematics and Physics, Sokolovská 83, 186 75 Prague, Czech Republic. E-mail: krajicek@math.cas.cz
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove the following results: (i) PV proves NP ⊆ P/poly iff PV proves coNP ⊆ NP/O(1). (ii) If PV proves NP ⊆ P/poly then PV proves that the Polynomial Hierarchy collapses to the Boolean Hierarchy, (iii) proves NP ⊆ P/poly iff proves coNP ⊆ NP/O(log n). (iv) If proves NP ⊆ P/poly then proves that the Polynomial Hierarchy collapses to PNP[log n]. (v) If proves NP ⊆ P/poly then proves that the Polynomial Hierarchy collapses to PNP.

Motivated by these results we introduce a new concept in proof complexity: proof systems with advice, and we make some initial observations about them.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 4 , December 2007 , pp. 1353 - 1371
Copyright
Copyright © Association for Symbolic Logic 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

[1]Beigel, Richard, Bounded queries to SAT and the Boolean hierarchy, Theoretical Computer Science, vol. 84 (1991), no. 2, pp. 199–223.CrossRefGoogle Scholar
[2]Buss, Samuel, Bounded Arithmetic, Bibliopolis, 1986.Google Scholar
[3]Buss, Samuel, Axiomatizations and conservation results for fragments of bounded arithmetic, Logic and Computation, Proceedings of a Workshop held at Carnegie Mellon University, Contemporary Mathematics, vol. 106, American Mathematical Society, 1990, pp. 57–84.Google Scholar
[4]Buss, Samuel, Relating the bounded arithmetic and polynomial time hierarchies, Annals of Pure and Applied Logic, vol. 75 (1995), pp. 67–77.CrossRefGoogle Scholar
[5]Buss, Samuel, First–order proof theory of arithmetic, Handbook of Proof Theory (Buss, Samuel, editor), Elsevier, 1998, Available on line at www.math.ucsd.edu/~sbuss/ResearchWeb/HandbookProofTheory/, pp. 79–147.CrossRefGoogle Scholar
[6]Buss, Samuel and Hay, Louise, On truth-table reducibility to SAT, Information and Computation, vol. 91 (1991), no. 1, pp. 86–102.CrossRefGoogle Scholar
[7]Buss, Samuel, Krajíček, Jan, and Takeuti, Gaisi, On provably total functions in bounded arithmetic theories , and , Arithmetic, Proof Theory and Computational Complexity (Clote, Peter and Krajíček, Jan, editors), Oxford University Press, 1993, pp. 116–161.Google Scholar
[8]Cai, Jin-Yi, ⊆ ZPPNP, Proceedings of the IEEE Symposium on Foundations of Computer Science (FOCS), 2001, pp. 620–628.CrossRefGoogle Scholar
[9]Cook, Stephen, Feasibly constructive proofs and the propositional calculus, Proceedings of the ACM Symposium on Theory Of Computing (STOC), 1975, pp. 83–97.Google Scholar
[10]Cook, Stephen, Theories for complexity classes and their propositional translations, Complexity of Computations and Proofs (Krajíček, Jan, editor), Quaderni di Matematica, 2005, pp. 175–227.Google Scholar
[11]Cook, Stephen and Nguyen, Phuong, Foundations of proof complexity: Bounded arithmetic and propositional translations, unpublished manuscript http://www.cs.toronto.edu/~sacook/, 2006.Google Scholar
[12]Cook, Stephen and Thapen, Neil, The strength of replacement in weak arithmetic, ACM Transactions on Computational Logic, vol. 7 (2006), no. 4, pp. 749–764.CrossRefGoogle Scholar
[13]Karp, R. M. and Lipton, R. J., Some connections between nonuniform and uniform complexity classes, Proceedings of the ACM Symposium on Theory Of Computing (STOC), 1980, pp. 302–309.Google Scholar
[14]Karp, R. M. and Lipton, R. J., Turing machines that take advice, Enseignement Mathematique, vol. 30 (1982), pp. 255–273.Google Scholar
[15]Krajíček, Jan, Fragments of bounded arithmetic and bounded query classes, Transactions of the American Mathematical Society, vol. 338 (1993), no. 2, pp. 587–598.CrossRefGoogle Scholar
[16]Krajíček, Jan, Bounded Arithmetic, Propositional Logic and Computational Complexity, Cambridge University Press, 1995.Google Scholar
[17]Krajíček, Jan, Pudlák, Pavel, and Takeuti, Gaisi, Bounded arithmetic and the polynomial hierarchy, Annals of Pure and Applied Logic, vol. 52 (1991), pp. 143–153.CrossRefGoogle Scholar
[18]Pollett, Chris, Structure and definability in general bounded arithmetic theories, Annals of Pure and Applied Logic, vol. 100 (1999), pp. 189–245.CrossRefGoogle Scholar
[19]Zambella, D., Notes on polynomially bounded arithmetic, this Journal, vol. 61 (1996), no. 3, pp. 942–966.Google Scholar