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 [4, 3]. In this paper, we investigate improving this upper bound from PSPACE to PSPACE ∩ P/poly. More precisely, 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 of Peano arithmetic, then BPP ⊆ C ⊆ PSPACE ∩ P/poly. We conjecture that C is equal to P, and discuss the possibility this might be an avenue for trying to prove the equality of BPP and P. ∗Supported in part by NSF Grants CCF-0830133, CCF-0832787, and CCF-1064785. †Supported in part by the [European Community’s] Seventh Framework Programme [FP7/20072013] under grant agreement n◦ 238381. ‡Supported in part NSF Grant CCF-1004956 with the DIMACS REU Program. §Supported in part by the National Basic Research Program of China Grant 2011CBA00300, 2011CBA00301, the National Natural Science Foundation of China Grant 61033001, 61061130540, 61073174