Problems that are complete for exponential space are prov-ably intractable and known to be exceedingly complex in several technical respects. However, every problem decidable in exponential space is eeciently reducible to every complete problem, so each complete problem must have a highly organized structure. The authors have recently exploited this fact to prove that complete problems are, in ...

C. Calude, A. Nies, L. Staiger, and F. Stephan posed the following question about the relation between plain and prefix Kolmogorov complexities (see their paper in DLT 2008 conference proceedings): does the domain of every optimal decompressor contain the domain of some optimal prefix-free decompressor? In this paper we provide a negative answer to this question.

This paper initiates the study of sets in Euclidean space R (n ≥ 2) that are defined in terms of the dimensions of their elements. Specifically, given an interval I ⊆ [0, 1], we are interested in the connectivity properties of the set DIM consisting of all points in R whose (constructive Hausdorff) dimensions lie in the interval I. It is easy to see that the sets DIM and DIM(n−1,n] are totally ...

Recently, Moser and Tardos [MT10] came up with a constructive proof of the Lovász Local Lemma. In this paper, we give another constructive proof of the lemma, based on Kolmogorov complexity. Actually, we even improve the Local Lemma slightly.

Information distance can be defined not only between two strings but also in a finite multiset of strings of cardinality greater than two. We give an elementary proof for expressing the information distance in conditional Kolmogorov complexity. It is exact since the lower bound equals the upper bound up to a constant additive term.

We revisit the notion of computational depth and sophistication for infinite sequences and study the density of the sets of deep and sophisticated infinite sequences. Koppel defined the sophistication of an object as the length of the shortest total program that given some data as input produces it and the sum of the size of the program with the size of the data is as consice as the smallest de...

The proofs of Chaitin and Boolos for Gödel’s Incompleteness Theorem are studied from the perspectives of constructibility and Rosserizability. By Rosserization of a proof we mean that the independence of the true but unprovable sentence can be shown by assuming only the (simple) consistency of the theory. It is known that Gödel’s own proof for his incompleteness theorem is not Rosserizable, and...

Recently, Moser and Tardos [MT10] came up with a constructive proof of the Lovász Local Lemma. In this paper, we give another constructive proof of the lemma, based on Kolmogorov complexity. Actually, we even improve the Local Lemma slightly.

We investigate Kolmogorov complexity of the problem (a ! c) ^ (b ! d), deened as the minimum length of a program that given a outputs c and given b outputs d. We prove that unlike all known problems of this kind its complexity is not expressible in terms of Kolmogorov complexity of a, b, c, and d, their pairs, triples etc. This solves the problem posed in 9]. In the second part we consider the ...