Derandomization and distinguishing complexity

Abstract

We continue an investigation of resource-bounded Kolmogorov complexity and derandomization techniques begun in [E. Allender (2001), E. Allender et al., (2002)]. We introduce nondeterministic time-bounded Kolmogorov complexity measures (KNt and KNT) and examine the properties of these measures using constructions of hitting set generators for nondeterministic circuits [P. B. Miltersen et al., (1999), R. Shaltiel et al., (2001)]. We observe that KNt bears many similarities to the nondeterministic distinguishing complexity CND of [H. Buhrman et al., (2002)]. This motivates the definition of a new notion of time-bounded distinguishing complexity KDt, as an intermediate notion with connections to the class FewEXP. The set of KDt-random strings is complete for EXP under P/poly reductions. Most of the notions of resource-bounded Kolmogorov complexity discussed here and in [E. Allender (2001), E. Allender et al., (2002)] have close connections to circuit size (on different types of circuits). We extend this framework to define notions of Kolmogorov complexity KB and KF that are related to branching program size and formula size, respectively. The sets of KB- and KF-random strings lie in coNP; we show that oracle access to these sets enables one to factor Blum integers. We obtain related intractability results for approximating minimum formula size, branching program size, and circuit size. The NEXP/spl sube/NC and NEXP/spl sube/L/poly questions are shown to be equivalent to conditions about the KF and KB complexity of sets in P.