Kolmogorov Complexity, Circuits, and the Strength of Formal Theories of Arithmetic
Can complexity classes be characterized in terms of efficient reducibility to the (undecidable) set of Kolmogorov-random strings? Although this might seem improbable, a series of papers has recently provided evidence that this may be the case. In particular, it is known that there is a class of problems C defined in terms of polynomial-time truth- table reducibility to RK (the set of Kolmogorov-random strings) that lies between BPP and PSPACE (5, 4). The results in this paper were obtained, as part of an investigation of whether this upper bound can be improved, to show BPP C PSPACE\ P/poly: ( ) In fact, we conjecture that C = BPP = P, and we close this paper with a discussion of the possibility this might be an avenue for trying to prove the equality of BPP and P. In this paper, we present a collection of true statements in the language of arithmetic, (each provable in ZF) and show that if these statements can be proved in certain extensions