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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.

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

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