This paper contains a discussion of dynamical randomness among the different methods of simulation of a fluid and its characterization by the concept of Kolmogorov-Sinai entropy per unit time. Moreover, a renormalization-group method is presented in order to construct the hydrodynamic and reactive modes of relaxation in chaotic models. The renormalization-group construction allows us to obtain ...

It is a tempting idea to use formal existence proofs as a means to precisely and verifiably express algorithmic ideas. This is clearly possible for “constructive” proofs, which are informally understood via the BrouwerHeyting-Kolmogorov interpretation (BHK-interpretation for short). This interpretation of intuitionistic (and minimal) logic explains what it means to prove a logically compound st...

It is impossible to effectively modify a string in order to increase its Kolmogorov complexity. But is it possible to construct a few strings, not longer than the input string, so that most of them have larger complexity? We show that the answer is yes. We present an algorithm that on input a string x of length n returns a list with O(n) many strings, all of length n, such that 99% of them are ...

By superimposing a group structure on a sequence of projective probability spaces of Lebesgue measure preserving (l.m.p.) automorphisms of unit interval, the paper extends the Daniel-Kolmogorov consistency theorem that enables the construction of a measurable group structure with invariant Haar probability measure on an uncountably large projective limit space. The projective limit group is the...

We introduce two abstract theorems that reduce a variety of complex exponential distributional approximation problems to the construction of couplings. These are applied to obtain rates of convergence with respect to the Wasserstein and Kolmogorov metrics for the theorem of Rényi on random sums and generalizations of it, hitting times for Markov chains, and to obtain a new rate for the classica...

We present a method for hierarchical clustering of data called mutual information clustering (MIC) algorithm. It uses mutual information (MI) as a similarity measure and exploits its grouping property: The MI between three objects X, Y, and Z is equal to the sum of the MI between X and Y , plus the MI between Z and the combined object (XY ). We use this both in the Shannon (probabilistic) versi...

We revisit fixed-parameter tractability, fixed-parameter reducibility and kernelizability. The standard formulations of their definitions are enhanced for greater correctness from a structural complexity point of view. Doing so makes clear the distinction between fixed-parameter reducibility and kernelizability. Additionally, these time based definitions are augmented with space based counterpa...

We continue an investigation into resource-bounded Kolmogorov complexity [ABK06], 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 introd...

In analogy of classical Kolmogorov complexity we define a notion of the algorithmic information in bits contained in any one of continuously many pure quantum states: quantum Kolmogorov complexity. Classical Kolmogorov complexity coincides with the new quantum Kolmogorov complexity restricted to the classical domain. Quantum Kolmogorov complexity is upper bounded and can be effectively approxim...