Luke Friedman

Reductions to the set of random strings: the resource-bounded case

This paper is motivated by a conjecture [1,5] that BPP can be characterized in terms of polynomial-time nonadaptive reductions to the set of Kolmogorov-random strings. In this paper we show that an approach laid out in [5] to settle this conjecture cannot succeed without significant alteration, but that it does bear fruit if we consider time-bounded Kolmogorov complexity instead. We show that if a set A is reducible in polynomial time to the set of time-t-bounded Kolmogorov-random strings (for all large enough time bounds t), then A is in P/poly, and that if in addition such a reduction exists for any universal Turing machine one uses in the definition of Kolmogorov complexity, then A is in PSPACE.

Reductions to the set of random strings : The resource-bounded case

This paper is motivated by a conjecture that BPP can be characterized in terms of polynomial-time nonadaptive reductions to the set of Kolmogorov-random strings. In this paper we show that an approach laid out in [Allender et al] to settle this conjecture cannot succeed without significant alteration, but that it does bear fruit if we consider time-bounded Kolmogorov complexity instead. We show that if a set A is reducible in polynomial time to the set of time-t-bounded Kolmogorov random strings (for all large enough time bounds t), then A is in P/poly, and that if in addition such a reduction exists for any universal Turing machine one uses in the definition of Kolmogorov complexity, then A is in PSPACE.

Exponential Lower Bounds for Refuting Random Formulas Using Ordered Binary Decision Diagrams

A propositional proof system based on ordered binary decision diagrams (OBDDs) was introduced by Atserias et al. in [3]. Krajicek proved exponential lower bounds for a strong variant of this system using feasible interpolation [14], and Tveretina et al. proved exponential lower bounds for restricted versions of this system for refuting formulas derived from the Pigeonhole Principle [20]. In this paper we prove the first lower bounds for refuting randomly generated unsatisfiable formulas in restricted versions of this OBDD-based proof system. In particular we consider two systems OBDD* and OBDD+; OBDD* is restricted by having a fixed, predetermined variable order for all OBDDs in its refutations, and OBDD+ is restricted by having a fixed order in which the clauses of the input formula must be processed. We show that for some constant e > 0, with high probability an OBDD* refutation of an unsatisfiable random 3-CNF formula must be of size at least 2 en , and an OBDD+ refutation of an unsatisfiable random 3-XOR formula must be of size at least 2 en .