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 PNP[log n]. (v) If S22 proves NP⊆P/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.
"Consequences of the provability of NP⊆P/poly." J. Symbolic Logic 72 (4) 1353 - 1371, December 2007. https://doi.org/10.2178/jsl/1203350791