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 ...

It is shown that the 0-1 Law for recursive logics on nite structures admits an equivalent formulation in terms of Kolmogorov complexity. The new formulation opens the possibility of using the Kolmogorov complexity apparatus to easily derive various properties for nite structures satisfying a given properties. Examples that illustrate this point are provided.

There are human beings whose intellectual power exceeds that of ordinary men. In my life, in my personal experience, there were three such men, and one of them was Andrei Nikolaevich Kolmogorov. I was lucky enough to be his immediate pupil. He invited me to be his pupil at the third year of my being student at the Moscow University. This talk is my tribute, my homage to my great teacher. Andrei...

Kolmogorov Theorem on the persistence of invariant tori of real analytic Hamiltonian systems is revisited. In this paper we are mainly concerned with the lower bound on the constant of the Diophantine condition required by the theorem. From the existing proofs in the literature, this lower bound turns to be ofO(ε1/4), where ε is the size of the perturbation. In this paper, by means of careful (...

Kolmogorov complexity theory is used to tell what the algorithmic informational content of a string is. It is defined as the length of the shortest program that describes the string. We present a programming language that can be used to describe categories, functors, and natural transformations. With this in hand, we define the informational content of these categorical structures as the shorte...

An infinite binary sequence has randomness rate at least σ if, for almost every n, the Kolmogorov complexity of its prefix of length n is at least σn. It is known that for every rational σ ∈ (0, 1), on one hand, there exists sequences with randomness rate σ that can not be effectively transformed into a sequence with randomness rate higher than σ and, on the other hand, any two independent sequ...

Muchnik's theorem about simple conditional descriprion states that for all words $a$ and $b$ there exists a short program $p$ transforming $a$ to $b$ that has the least possible length and is simple conditional on $b$. This paper presents a new proof of this theorem, based on extractors. Employing the extractor technique, two new versions of Muchnik's theorem for space- and time-bounded Kolmogo...