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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
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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.

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

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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