On complexity classes and algorithmically random languages

Abstract

Every class C of languages satisfying a simple topological condition is shown to have probability one if and only if it contains some language that is algorithmically random in the sense of Martin-Löf. This result is used to derive separation properties of algorithmically random oracles and to give characterizations of the complexity classes P, BPP, AM, and PH in terms of reducibility to such oracles. These characterizations lead to the following result: (i) P = NP if and only if there exists an algorithmically random set that is ≤bttP-hard for NP. (ii) P = PSPACE if and only if there exists an algorithmically random set that is ≤bttP-hard for PSPACE. (iii) The polynomial-time hierarchy collapses if and only if there exists k>0 such that some algorithmically random set is ΣkP-hard for PH. (iv) PH = PSPACE if and only if there exists a algorithmically random set that is PH-hard for PSPACE.