. We investigate the distribution of nonuniform complexities in uniform complexity classes. We prove that almost every problem decidable in exponential space has essentially maximum circuit-size and space-bounded Kolmogorov complexity almost everywhere. (The circuit-size lower bound actually exceeds, and thereby strengthens, the Shannon 2 n n lower bound for almost every problem, with no computability constraint.) In exponential time complexity classes, we prove that the strongest relativizable lower bounds hold almost everywhere for almost all problems. Finally, we show that infinite pseudorandom sequences have high nonuniform complexity almost everywhere. The results are unified by a new, more powerful formulation of the underlying measure theory, based on uniform systems of density functions, and by the introduction of a new nonuniform complexity measure, the selective Kolmogorov complexity. This research was supported in part by NSF Grants CCR-8809238 and CCR-9157382 and in ...

The probability distribution P from which the history of our universe is sampled represents a theory of everything or TOE. We assume P is formally describable. Since most (uncountably many) distributions are not, this imposes a strong inductive bias. We show that P(x) is small for any universe x lacking a short description, and study the spectrum of TOEs spanned by two Ps, one reflecting the most compact constructive descriptions, the other the fastest way of computing everything. The former derives from generalizations of traditional computability, Solomonoff’s algorithmic probability, Kolmogorov complexity, and objects more random than Chaitin’s Omega, the latter from Levin’s universal search and a natural resource-oriented postulate: the cumulative prior probability of all x incomputable within time t by this optimal algorithm should be 1/t. Between both Ps we find a universal cumulatively enumerable measure that dominates traditional enumerable measures; any such CEM must assign low probability to any universe lacking a short enumerating program. We derive P-specific consequences for evolving observers, inductive reasoning, quantum physics, philosophy, and the expected duration of our universe.

Abstract We consider hypotheses about nondeterministic computation that have been studied in dif-ferent contexts and shown to have interesting consequences: * The measure hypothesis: NP does not have p-measure 0.* The pseudo-NP hypothesis: there is an NP language that can be distinguished from anyDTIME(2 nffl) language by an NP refuter. * The NP-machine hypothesis: there is an NP machine accepting 0 * for which no 2n ffl-time machine can find infinitely many accepting computations. We show that the NP-machine hypothesis is implied by each of the first two. Previously, norelationships were known among these three hypotheses. Moreover, we unify previous work by showing that several derandomizations and circuit-size lower bounds that are known to followfrom the first two hypotheses also follow from the NP-machine hypothesis. In particular, the NPmachine hypothesis becomes the weakest known uniform hardness hypothesis that derandomizesAM. We also consider UP versions of the above hypotheses as well as related immunity and scaled dimension hypotheses. 1 Introduction The following uniform hardness hypotheses are known to imply full derandomization of ArthurMerlin games (NP = AM): * The measure hypothesis: NP does not have p-measure 0 [24].

This paper presents one method of using time-bounded Kolmogorov complexity as a measure of the complexity of sets, and outlines anumber of applications of this approach to di#erent questions in complexity theory. Connections will be drawn among the following topics: NE predicates, ranking functions, pseudorandom generators, and hierarchy theorems in circuit complexity.

This paper has the following goals: -- To survey some of the recent developments in the field of derandomization. -- To introduce a new notion of time-bounded Kolmogorov complexity (KT), and show that it provides a useful tool for understanding advances in derandomization, and for putting various results in context. -- To illustrate the usefulness of KT, by answering a question that has been posed in the literature, and -- To pose some promising directions for future research.

This paper investigates the distribution and nonuniform complexity of problems that are complete or weakly complete for ESPACE under nonuniform reductions that are computed by polynomial-size circuits (P/Poly-Turing reductions and P/Poly-many-one reductions). A tight, exponential lower bound on the space-bounded Kolmogorov complexities of weakly P/PolyTuring -complete problems is established. A Small Span Theorem for P/Poly-Turing reductions in ESPACE is proven and used to show that every P/Poly-Turing degree --- including the complete degree --- has measure 0 in ESPACE. (In contrast, it is known that almost every element of ESPACE is weakly P-many-one complete.) Every weakly P/Poly-many-one-complete problem is shown to have a dense, exponential, nonuniform complexity core. More importantly, the P/Poly-many-one-complete problems are shown to be unusually simple elements of ESPACE, in the sense that they obey nontrivial upper bounds on nonuniform complexity (size of nonuniform complexit...

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

this paper appeared as #AG91a#

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.

Symmetry of information (in Kolmogorov complexity) is a concept that comes from formalizing the idea of how much information about a string y is contained in a string x. The situation is symmetric because it can be shown that the amount of information contained in the string y about the string x is almost exactly the same as that contained in x about y. In this paper we address symmetry of information in resource bounded environments. While we show that symmetry still holds in space bounded environments, it probably doesn't hold in time bounded environments. We show that if it holds for polynomial time bounds, then one-way functions cannot exist. 1 Introduction Keywords: computational complexity, Kolmogorov complexity, one-way functions. In probability theory, the phenomenon of dependence between random variables is well known. Cast in terms of classical Shannon entropy [Sha48, Sha49], the quantity of information in a random variable Y about another random variable X is I(X; Y ) =...