Journal of Symbolic Logic

Consequences of the provability of NPP/poly

Stephen Cook and Jan Krajíček

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We prove the following results: (i) PV proves NPP/poly iff PV proves coNPNP/O(1). (ii) If PV proves NPP/poly then PV proves that the Polynomial Hierarchy collapses to the Boolean Hierarchy. (iii) S_21 proves NPP/poly iff S21 proves coNPNP/O(log n). (iv) If S_21 proves NPP/poly then S21 proves that the Polynomial Hierarchy collapses to PNP[log n]. (v) If S22 proves NPP/poly then S22 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.

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J. Symbolic Logic, Volume 72, Issue 4 (2007), 1353-1371.

First available in Project Euclid: 18 February 2008

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Cook, Stephen; Krajíček, Jan. Consequences of the provability of NP ⊆ P/poly. J. Symbolic Logic 72 (2007), no. 4, 1353--1371. doi:10.2178/jsl/1203350791.

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