We investigate the question of whether one can characterize complexity classes in terms of efficient reducibility to the set of Kolmogorov-random strings RK. We show that this question is dependent on the choice of universal machine in the definition of Kolmogorov complexity. We show for a broad class of reductions that the sets reducible to RK have very low computational complexity. Further, w...

We continue an investigation of resource-bounded Kolmogorov complexity and derandomization techniques begun in [2, 3]. We introduce nondeterministic time-bounded Kolmogorov complexity measures (KNt and KNT) and examine the properties of these measures using constructions of hitting set generators for nondeterministic circuits [22, 26]. We observe that KNt bears many similarities to the nondeter...

We have used the Kolmogorov complexities and the Kolmogorov complexity spectrum to quantify the randomness degree in river flow time series of seven rivers with different regimes in Bosnia and Herzegovina, representing their different type of courses, for the period 1965–1986. In particular, we have examined: (i) the Neretva, Bosnia and the Drina (mountain and lowland parts), (ii) the Miljacka ...

We study the computably enumerable sets in terms of the: (a) Kolmogorov complexity of their initial segments; (b) Kolmogorov complexity of finite programs when they are used as oracles. We present an extended discussion of the existing research on this topic, along with recent developments and open problems. Besides this survey, our main original result is the following characterization of the ...

This article discusses a variety of mathematically-inspired approaches to the study of complexity and their application to the study and understanding of complex systems. These approaches have their origin in the study of limitative results in logic, mathematical linguistics, and the theory of computation, based upon the work of Godel, Turing, Chomsky, and Kolmogorov, among many others. The top...

Complexity distortion theory (CDT) is a mathematical framework providing a unifying perspective on media representation. The key component of this theory is the substitution of the decoder in Shannon’s classical communication model with a universal Turing machine. Using this model, the mathematical framework for examining the efficiency of coding schemes is the algorithmic or Kolmogorov complex...

Computable versions of Kolmogorov complexity have been used in the context of pattern discovery [1]. However, these complexity measures do not take the psychological dimension of pattern discovery into account. We propose a method for pattern discovery based on a version of Kolmogorov complexity where computations are restricted to a cognitive model with limited computational resources. The pot...

Heilbronn’s triangle problem asks for the least ∆ such that n points lying in the unit disc necessarily contain a triangle of area at most ∆. Heilbronn initially conjectured ∆ = O(1/n). As a result of concerted mathematical effort it is currently known that there are positive constants c and C such that c logn/n ≤ ∆ ≤ C/n for every constant ǫ > 0. We resolve Heilbronn’s problem in the expected ...

In this paper we answer the following well-known open question in computable model theory. Does there exist a computable not א0-categorical saturated structure with a unique computable isomorphism type? Our answer is affirmative and uses a construction based on Kolmogorov complexity. With a variation of this construction, we also provide an example of an א1-categorical but not א0-categorical sa...

D. Krieger and J. Shallit have proved that every real number greater than 1 is a critical exponent of some sequence [1]. We show how this result can be derived from some general statements about sequences whose subsequences have (almost) maximal Kolmogorov complexity. In this way one can also construct a sequence that has no “approximate” fractional powers with exponent that exceeds a given val...