A Tight Karp-Lipton Collapse Result in Bounded Arithmetic

Cook and Krajíček [9] have obtained the following Karp-Lipton result in bounded arithmetic: if the theory proves , then collapses to , and this collapse is provable in . Here we show the converse implication, thus answering an open question from [9]. We obtain this result by formalizing in a hard/easy argument of Buhrman, Chang, and Fortnow [3]. In addition, we continue the investigation of propositional proof systems using advice, initiated by Cook and Krajíček [9]. In particular, we obtain several optimal and even p-optimal proof systems using advice. We further show that these p-optimal systems are equivalent to natural extensions of Frege systems

A Note on Universal Measures for Weak Implicit Computational Complexity

Abstract. This note is a case study for finding universal measures for weak implicit computational complexity. We will instantiate “univer-sal measures ” by “dynamic ordinals”, and “weak implicit computational complexity ” by “bounded arithmetic”. Concretely, we will describe the connection between dynamic ordinals and witness oracle Turing ma-chines for bounded arithmetic theories