A tight Karp-Lipton collapse result in bounded arithmetic

Abstract

Cook and Krajíček have recently obtained the following Karp-Lipton collapse result in bounded arithmetic: if the theory PV proves NP⊆ P/ poly , then the polynomial hierarchy collapses to the Boolean hierarchy, and this collapse is provable in PV . Here we show the converse implication, thus answering an open question posed by Cook and Krajíček. We obtain this result by formalizing in PV a hard/easy argument of Buhrman et al. [2003]. In addition, we continue the investigation of propositional proof systems using advice, initiated by Cook and Krajíček. In particular, we obtain several optimality results for proof systems using advice. We further show that these optimal systems are equivalent to natural extensions of Frege systems.