Related papers: A systematic approach on some relevant theorems th…
We reconsider some classical natural semantics of integers (namely iterators of functions, cardinals of sets, index of equivalence relations), in the perspective of Kolmogorov complexity. To each such semantics one can attach a simple…
A method of random search based on Kolmogorov complexity is proposed and applied to two search problems in group theory. The method is provably effective but not practical, so the applications involve heuristic approximations. Perhaps…
TThe problem is to identify a probability associated with a set of natural numbers, given an infinite data sequence of elements from the set. If the given sequence is drawn i.i.d. and the probability mass function involved (the target)…
This is a chapter for the forthcoming New Handbook of Mathematical Psychology, to be published by Cambridge University Press. A systematic theory of random variables and joint distributions under varying conditions is presented. This is a…
Kolmogorov's axioms of probability theory are extended to conditional probabilities among distinct (and sometimes intertwining) contexts. Formally, this amounts to row stochastic matrices whose entries characterize the conditional…
This paper provides a complete suite of axioms for a version of set theory that I call Explication. Explication borrows from the two most prominent existing systems of set theory. Explication starts with class variables. After several…
Algorithmic information theory studies description complexity and randomness and is now a well known field of theoretical computer science and mathematical logic. There are several textbooks and monographs devoted to this theory where one…
The Kolmogorov axioms for probability functions are placed in the context of signed meadows. A completeness theorem is stated and proven for the resulting equational theory of probability calculus. Elementary definitions of probability…
We introduce the notion of a probabilistic measure which takes values in hyperbolic numbers and which satisfies the system of axioms generalizing directly Kolmogorov's system of axioms. We show that this new measure verifies the usual…
The use of algorithmic information theory (Kolmogorov complexity theory) to explain the relation between mathematical probability theory and `real world' is discussed.
The word "complexity" is most often used as a meta--linguistic expression referring to certain intuitive characteristics of a natural system and/or its scientific description. These characteristics may include: sheer amount of data that…
Using the ideas of abstract algebra, we introduce the basic concepts of abstract probability theory that generalize the Kolmogorov's probability theory, possibility theory and other theories that deal with uncertainty. Based on abstract…
We consider a system of weak* closed sets of finite-dimensional distributions. We show that a corresponding system of random variables can be defined on a probability space with a probability measure determined up to some set of measures,…
While Kolmogorov complexity is the accepted absolute measure of information content of an individual finite object, a similarly absolute notion is needed for the relation between an individual data sample and an individual model summarizing…
We introduce algorithmic information theory, also known as the theory of Kolmogorov complexity. We explain the main concepts of this quantitative approach to defining `information'. We discuss the extent to which Kolmogorov's and Shannon's…
We seek random versions of some classical theorems on complex approximation by polynomials and rational functions, as well as investigate properties of random compact sets in connection to complex approximation.
Kolmogorov's setting for probability theory is given an original generalization to account for probabilities arising from Quantum Mechanics. The sample space has a central role in this presentation and random variables, i.e., observables,…
First the crucial but very confidential fact is brought into evidence that, as Kolmogorov himself repeatedly claimed, there exists no abstract theory of probabilities, simply because the factual concept of probability is itself unachieved:…
We provide a formal introduction into the classic theorems of general topology and its axiomatic foundations in set theory. Starting from ZFC, the exposition in this first part includes relation and order theory as well as a construction of…
We provide a formal introduction into the classic theorems of general topology and its axiomatic foundations in set theory. In this second part we introduce the fundamental concepts of topological spaces, convergence, and continuity, as…