Abstract. Impagliazzo and Wigderson (36th FOCS, 1998) gave the first construction of pseudorandom generators from a uniform complexity assumption on EXP (namely EXP � = BPP). Unlike results in the nonuniform setting, their result does not provide a continuous trade-off between worst-case hardness and pseudorandomness, nor does it explicitly establish an average-case hardness result. In this paper: ◦ We obtain an optimal worst-case to average-case connection for EXP: if EXP � ⊆ BPTIME(t(n)), then EXP has problems that cannot be solved on a fraction 1/2 + 1/t ′ (n) of the inputs by BPTIME(t ′ (n)) algorithms, for t ′ = t Ω(1). ◦ We exhibit a PSPACE-complete self-correctible and downward self-reducible problem. This slightly simplifies and strengthens the proof of Impagliazzo and Wigderson, which used a #P-complete problem with these properties. ◦ We argue that the results of Impagliazzo and Wigderson, and the ones in this paper, cannot be proved via “black-box ” uniform reductions.

We investigate the question of whether one can characterize complexity classes (such as PSPACE or NEXP) in terms of efficient reducibility to the set of Kolmogorovrandom strings RC. This question arises because PSPACE ⊆ P RC and NEXP ⊆ NP RC, and no larger complexity classes are known to be reducible to RC in this way. We show that this question cannot be posed without explicitly dealing with issues raised by the choice of universal machine in the definition of Kolmogorov complexity. What follows is a list of some of our main results. • Although Kummer showed that, for every universal machine U there is a time bound t such that the halting problem is disjunctive truth-table reducible to RCU in time t, there is no such time bound t that suffices for every universal machine U. We also show that, for some machines U, the disjunctive reduction can be computed in as little as doubly-exponential time. • Although for every universal machine U, there are very complex sets that are ≤P dtt-reducible to RCU, it is nonetheless true that P = REC ∩ ⋂ {A: U A ≤P dtt RCU}. (A similar statement holds for parity-truth-table reductions.) This is an extended version of a paper that appeared in Proceedings of the 21 st Symposium on

In this paper, we use resource-bounded dimension theory to investigate polynomial size circuits. We show that for every i 0, P/poly has ith-order scaled p 3 -strong dimension 0. We also show that P/poly i.o.

We continue an investigation of resource-bounded Kolmogorov complexity and derandomization techniques begun in [2, 3].

We continue an investigation into resource-bounded Kolmogorov complexity [ABK + 06], which highlights the close connections between circuit complexity and Levin’s time-bounded Kolmogorov complexity measure Kt (and other measures with a similar flavor), and also exploits derandomization techniques to provide new insights regarding Kolmogorov complexity. The Kolmogorov measures that have been introduced have many advantages over other approaches to defining resource-bounded Kolmogorov complexity (such as much greater independence from the underlying choice of universal machine that is used to define the measure) [ABK + 06]. Here, we study the properties of other measures that arise naturally in this framework. The motivation for introducing yet more notions of resource-bounded Kolmogorov complexity are two-fold: • to demonstrate that other complexity measures such as branching-program size and formula size can also be discussed in terms of Kolmogorov complexity, and • to demonstrate that notions such as nondeterministic Kolmogorov complexity and distinguishing complexity [BFL02] also fit well into this framework. The main theorems that we provide using this new approach to resource-bounded Kolmogorov complexity are: • A complete set (RKNt) for NEXP/poly defined in terms of strings of high Kolmogorov complexity.

We use derandomization to show that sequences of positive pspace-dimension -- in fact, even positive # k -dimension for suitable k -- have, for many purposes, the full power of random oracles. For example, we show that, if S is any binary sequence whose # 3 -dimension is positive, then and, moreover, every BPP promise problem is P -separable. We prove analogous results at higher levels of the polynomial-time hierarchy. The dimension-almost-class of a complexity class denoted by dimalmost-C, is the class consisting of all problems A such that A for all but a Hausdor# dimension 0 set of oracles S. Our results yield several characterizations of complexity classes, such as BPP = dimalmost-P and AM = dimalmost-NP, that refine previously known results on almost-classes. They also yield results, such as Promise-BPP = almost-P-Sep = dimalmost-P-Sep, in which even the almost-class appears to be a new characterization.

Let C(x) andK(x) denote plain and prefix Kolmogorov complexity, respectively, and let RC and RK denote the sets of strings that are “random ” according to these measures; both RK and RC are undecidable. Earlier work has shown that every set in NEXP is in NP relative to both RK and RC, and that every set in BPP is polynomial-time truth-table reducible to both RK and RC [ABK06a, BFKL10]. (All of these inclusions hold, no matter which “universal ” Turing machine one uses in the definitions of C(x) andK(x).) Since each machine U gives rise to a slightly different measure CU or KU, these inclusions can be stated as: • BPP ⊆ DEC ∩ ⋂ U

We describe the properties of various notions of time-bounded Kolmogorov complexity and other connections between Kolmogorov complexity and computational complexity. 1

In this paper we show that BPP is truth-table reducible to the set of Kolmogorov random strings RK. It was previously known that PSPACE, and hence BPP is Turing-reducible to RK. The earlier proof relied on the adaptivity of the Turingreduction to find a Kolmogorov-random string of polynomial length using the set RK as oracle. Our new non-adaptive result relies on a new fundamental fact about the set RK, namely each initial segment of the characteristic sequence of RK is not compressible by recursive means. As a partial converse to our claim we show that strings of high Kolmogorov-complexity when used as advice are not much more useful than randomly chosen strings. 1

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.