Related papers: 27 Open Problems in Kolmogorov Complexity
An attempt of a new kind of complexity anthropology is considered.
Is the universe computable? If so, it may be much cheaper in terms of information requirements to compute all computable universes instead of just ours. I apply basic concepts of Kolmogorov complexity theory to the set of possible…
We extend the concept of Krylov complexity to include general unitary evolutions involving multiple generators. This generalization enables us to formulate a framework for generalized Krylov complexity, which serves as a measure of the…
This paper consists of $3$ parts. The first part only considers classical processes and introduces two different extensions of the notion of hidden Markov process. In the second part, the notion of quantum hidden process is introduced. In…
Various simplicial complexes can be associated with a graph. Box complexes form an important families of such simplicial complexes and are especially useful for providing lower bounds on the chromatic number of the graph via some of their…
The idea to find the "maximal number that can be named" can be traced back to Archimedes (see his Psammit). From the viewpoint of computation theory the natural question is "which number can be described by at most n bits"? This question…
This is a list of some problems and conjectures related to various types of algebras, that is to algebraic operads. Some comments and hints are included.
Moment problems and orthogonal polynomials, both meant in a single real variable, belong to the oldest problems in Classical Analysis. They have been developing for over a century in two parallel, mostly independent streams. During the last…
One of the central open problems to classify the computational complexity of finite-domain constraint satisfaction problems within P is to prove better algorithmic results for CSPs with a Maltsev polymorphism; we do not even know whether…
This is a survey paper on commutative Koszul algebras and Castelnuovo-Mumford regularity. We describe several techniques to establish the Koszulness of algebras. We discuss variants of the Koszul property such as strongly Koszul, absolutely…
A new syntactic characterization of problems complete via Turing reductions is presented. General canonical forms are developed in order to define such problems. One of these forms allows us to define complete problems on ordered…
We find an abundance of Cremer Julia sets of an arbitrarily high computational complexity.
Every topological space has a Kolmogorov quotient that is obtained by identifying topologically indistinguishable points, that is, points that are contained in exactly the same open sets. In this survey, we look at the relationship between…
We survey various recent results that rigorously study the complexity of learning quantum states. These include progress on quantum tomography, learning physical quantum states, alternate learning models to tomography and learning classical…
In this paper the author presents some non-conventional thoughts on the complexity of the Universe and the algorithmic reproducibility of the human brain, essentially sparked off by the notion of algorithmic complexity. We must warn that…
This is the English version of the paper: "Complejidad de los n\'umeros naturales", Gaceta de la Real Sociedad Matem\'atica Espa\~nola 3 (2000) 230--250. In this paper, several conjectures about the complexity of natural numbers are…
This paper surveys the field of quantum communication complexity. Some interesting recent results are collected concerning relations to classical communication, lower bound methods, one-way communication, and applications of quantum…
We address the question of whether it may be worthwhile to convert certain, now classical, NP-complete problems to one of a smaller number of kernel NP-complete problems. In particular, we show that Karp's classical set of 21 NP-complete…
Parabolic integro-differential Kolmogorov equations with different space-dependent operators are considered in H\"{o}lder-type spaces defined by a scalable L\'{e}vy measure. Probabilistic representations are used to prove continuity of the…
Complexity theory provides a wealth of complexity classes for analyzing the complexity of decision and counting problems. Despite the practical relevance of enumeration problems, the tools provided by complexity theory for this important…