The resource-bounded measures of complexity classes are shown to be robust with respect to certain changes in the underlying probability measure. Specifically, for any real number ffi ? 0, any uniformly polynomial-time computable sequence ~ fi = (fi 0 ; fi 1 ; fi 2 ; : : : ) of real numbers (biases) fi i 2 [ffi; 1 \Gamma ffi], and any complexity class C (such as P, NP, BPP, P/Poly, PH, PSPACE, etc.) that is closed under positive, polynomial-time, truth-table reductions with queries of at most linear length, it is shown that the following two conditions are equivalent. (1) C has p-measure 0 (respectively, measure 0 in E, measure 0 in E 2 ) relative to the coin-toss probability measure given by the sequence ~ fi. (2) C has p-measure 0 (respectively, measure 0 in E, measure 0 in E 2 ) relative to the uniform probability measure. The proof introduces three techniques that may be useful in other contexts, namely, (i) the transformation of an efficient martingale for one probability measu...

A real is called recursively enumerable if it is the limit of a recursive, increasing, converging sequence of rationals. Following Solovay [23] and Chaitin [10] we say that an r.e. real dominates an r.e. real if from a good approximation of from below one can compute a good approximation of from below. We shall study this relation and characterize it in terms of relations between r.e. sets. Solovay's [23]-like numbers are the maximal r.e. real numbers with respect to this order. They are random r.e. real numbers. The halting probability ofa universal self-delimiting Turing machine (Chaitin's Ω number, [9]) is also a random r.e. real. Solovay showed that any Chaitin Ω number is-like. In this paper we show that the converse implication is true as well: any Ω-like real in the unit interval is the halting probability of a universal self-delimiting Turing machine.

In the 1980's, Bennett introduced computational depth as a formal measure of the amount of computational history that is evident in an object's structure. In particular, Bennett identi ed the classes of weakly deep and strongly deep sequences, and showed that the halting problem is strongly deep. Juedes, Lathrop, and Lutz subsequently extended this result by de ning the class of weakly useful sequences, and proving that every weakly useful sequence is strongly deep. The present paper investigates re nements of Bennett's notions of weak and strong depth, called recursively weak depth (introduced by Fenner, Lutz and Mayordomo) and recursively strong depth (introduced here). It is argued that these re nements naturally capture Bennett's idea that deep objects are those which \contain internal evidence of a nontrivial causal history. " The fundamental properties of recursive computational depth are developed, and it is shown that the recursively weakly (respectively, strongly) deep sequences form a proper subclass of the class of weakly (respectively, strongly) deep sequences. The above-mentioned theorem of Juedes, Lathrop, and Lutz is then strengthened by proving that every weakly useful sequence is recursively strongly deep. It follows from these results that not every strongly deep sequence is weakly useful, thereby answering a question posed by Juedes.

A computable complexity measure analogous to computational depth is developed using the Lempel-Ziv compression algorithm. This complexity measure, which we call compression depth, is then applied to the computational output of cellular automata. We find that compression depth captures the complexity found in Wolfram Class III celluar automata, and is in good agreement with his classification scheme. We further investigate the rule space of cellular automata using Langton's parameter. 1 This research was supported in part by National Science Foundation Grant CCR-9157382, with matching funds from Rockwell, Microware Systems Corporation, and Amoco Foundation. 1 Introduction Measures of the complexities of objects are widely used in both theory and applications in order to model, predict, and classify objects. Information theory gives us several methods for measuring the information content of objects. The most widely used of these information measures, entropy and algorithmic informa...

In the context of possibly in nite computations yielding finite or infinite (binary) outputs, the space 2 6! =2 ∗ ∪ 2! appears to be one of the most fundamental spaces in Computer Science. Though underconsidered, next to 2! , this space can be viewed (Section 3.5.2) as the simplest compact space native to computer science. In this paper we study some of its properties involving topology and computability. Though 2 6! can be considered as a computable metric space in the sense of computable analysis, a direct and self-contained study, based on its peculiar properties related to words, is much illuminating. It is well known that computability for maps 2! → 2! reduces to continuity with recursive

It is shown that, for every k 0 and every xed algorithmically random language B, there is a language that is polynomial time, truth-table reducible in k + 1 queries to B but not truth-table reducible in k queries in any amount of time to any algorithmically random language C. In particular, this yields the separation P k-tt(RAND) $ P (k+1)-tt(RAND), where RAND is the set of all algorithmically random languages.