Reductions to the Set of Random Strings: The Resource-Bounded Case

Abstract

Abstract. This paper is motivated by a conjecture [All12, ADF + 13] that BPP can be characterized in terms of polynomial-time nonadaptive reductions to the set of Kolmogorovrandom strings. In this paper we show that an approach laid out in [ADF + 13] 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.

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“…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 of Peano arithmetic (PA), then (*) holds. Although it was subsequently proved that infinitely many of these statements are, in fact, independent of those extensions of PA [1], we present these results in the hope that related ideas may yet contribute to a proof of C = BPP, and because this work did serve as a springboard for subsequent work in the area [1].…”

“…Kolmogorov complexity provides a mathematically precise definition of the set R of "random" strings. Actually, it provides at least two distinct but closely-related notions of randomness that we will need to discuss here, one defined in terms of the prefix Kolmogorov complexity function K, and one defined in terms of the plain Kolmogorov complexity function C. 1 This yields the two sets that lie at the center of this paper: R K = {x : K(x) ≥ |x|} and R C = {x : C(x) ≥ |x|}. When it is not important to distinguish between K and C we will simply refer to R.…”

“…Thus, in particular, we have BPP ⊆ C K ⊆ PSPACE ⊆ P R . 1 We provide a brief introduction to some basic notions of Kolmogorov complexity in Section 2. For a more comprehensive introduction to the topic, we refer the reader to standard texts such as [13,6].…”

“…3 At the time that the results in this paper were originally submitted for publication, we believed that a plausible approach to prove Conjecture 1.2 would be to show that proofs of Ψ A (n,j,k) exist in these extensions of PA. However, it is now known that infinitely many of the statements Ψ A (n,j,k) are independent of these extensions of PA [1]. In light of these developments, the reasons for publishing these results are:…”

“…• They served as a springboard for the subsequent work of [1], including results having a flavor similar to Conjecture 1.2, but in the context of Kolmogorov complexity with (large) time bounds,…”