Consequences of the provability of NP⊆P/poly

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) S_2¹ proves NP⊆P/poly iff S₂¹ proves coNP⊆ NP/O(log n). (iv) If S_2¹ proves NP⊆P/poly then S₂¹ proves that the Polynomial Hierarchy collapses to P^NP[log n]. (v) If S₂² proves NP⊆P/poly then S₂² proves that the Polynomial Hierarchy collapses to P^NP. Motivated by these results we introduce a new concept in proof complexity: proof systems with advice, and we make some initial observations about them.