Limits on the computational power of random strings

Abstract

How powerful is the set of random strings? What can one say about a set A that is efficiently reducible to R, the set of Kolmogorov-random strings? We present the first upper bound on the class of computable sets in P R and NP R . The two most widely-studied notions of Kolmogorov complexity are the "plain" complexity C (x) and "prefix" complexity K (x); this gives rise to two common ways to define the set of random strings "R": R C and R K . (Of course, each different choice of universal Turing machine U in the definition of C and K yields another variant R CU or R KU .) Previous work on the power of "R" (for any of these variants) has shown:Since these inclusions hold irrespective of low-level details of how "R" is defined, and since BPP, PSPACE and NEXP are all in 0 1 (the class of decidable languages), we have, e.g.:Our main contribution is to present the first upper bounds on the complexity of sets that are efficiently reducible to R KU . We show:Hence, in particular, PSPACE is sandwiched between the class of sets polynomial-time Turing-and truth-table-reducible to R. As a side-product, we obtain new insight into the limits of techniques for derandomization from uniform hardness assumptions.

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

“…• Then in Section 4 we prove that TTRT ⊆ PSPACE/α(n), which is a non-trivial adaptation of the techniques from [AFG13]. In Section 5 we show how to get an analogous result without the super-constant advice term.…”

“…That is, we succeed in obtaining a P/poly upper bound (of the sort that we were unable to obtain for DTTR in [ADF + 13]), and we "nearly" obtain a PSPACE upper bound (analogous to the PSPACE upper bound that was obtained for DTTR in [AFG13]). We believe that this indicates that TTRT is "closer" to BPP than it is to PSPACE.…”

“…The proof of this theorem is patterned closely on related arguments in [AFG13], although a number of complications arise in the time-bounded case. Although we aim to make the presentation here self-contained, [AFG13] is a good primer and a source of additional intuition for the proof.…”

“…Although we aim to make the presentation here self-contained, [AFG13] is a good primer and a source of additional intuition for the proof. Also, one can refer to the conference version of this paper [ABFL12] for a presentation that is not self-contained but emphasizes the differences between the proof in the time-bounded case and the unbounded case.…”

“…In a preceding paper [AFG13], it was shown that the class of decidable sets that are polynomial-time truth-table reducible to R K U for every U is contained in PSPACE. That is, although P R K U tt contains arbitrarily complex decidable sets, an extremely complex set can only be there because of characteristics of R K U that are fragile with respect to the choice of U .…”