Mark Burgin

Theory of information : fundamentality, diversity and unification

How Do We Know What Information Is? Information in Society, Information in Nature, Technological Aspects of Information General Theory of Information: Information Ontology, Information Axiology, Information Typology Information, Data, and Knowledge Emotions and Information Statistical Information Theory: Information and Entropy, Information and Communication in Technical Systems, Quantum Information, Information and Problem Solving, Axiomatization of Information Measurement, Information in Physics Semantic Information Theory: The Three Dimensions of Information, Logic in Information, Knowledge from Information Algorithmic Information Theory: Information and Complexity, Recursive Algorithmic Information Theory, Inductive Algorithmic Information Theory, Relative Information Measures, Axiomatic Information Measures Pragmatic Information Theory: Economic Approach, Mission-Oriented Approach and Value of Information, Transformational Approach, Dynamics of Information: Information Flow, Operator Approach, Information Algebra.

Neoclassical analysis: fuzzy continuity and convergence

Abstract The neoclassical analysis is a field in which fuzzy continuous functions are investigated. For this purpose, new measures of continuity and discontinuity (or defects of continuity) are introduced and studied. Based on such measures, classes of fuzzy continuous functions are defined and their properties are obtained. The class of fuzzy continuous functions may be considered as a fuzzy set of continuous functions. Its support consists of all functions on some topological space X into a metric space Y and its membership function is the corresponding continuity measure. Such an expansion provides a possibility to complete some important classical results. Connections between boundedness and fuzzy continuity are investigated. Criteria of boundedness and local boundedness are obtained for functions on Eucleadean spaces. Besides, such new concepts as fuzzy convergence and fuzzy uniform convergence are introduced and investigated. Their properties and connections with fuzzy continuous functions are explicated. Some results, which are obtained here, are similar to the results of classical mathematical analysis, while others differ essentially from those that are proved in classical mathematics. In the first case classical results are consequences of the corresponding results of neoclassical analysis.

Nonlinear phenomena in spaces of algorithms

Nonlinear phenomena, which are so important in nature and society, are considered here in relation to the world of algorithms and computations. To have a mathematical model for this world, formal computability spaces are introduced. It is demonstrated that the traditional approach to algorithms, which is based on such popular models as Turing machines, results in linear subspaces of the computability space. Nonlinear phenomena appear when we go to the more powerful class of such super-recursive algorithms as inductive Turing machines. It is demonstrated how this nonlinearity imports much higher computing power of inductive Turing machines in comparison with conventional Turing machines. This provides a base to consider problems of chaos, emergent computations and infinity from the algorithmic point of view.

Algorithmic complexity of recursive and inductive algorithms

The main goal of this paper is to compare recursive algorithms such as Turing machines with such super-recursive algorithms as inductive Turing machines. This comparison is made in a general setting of dual complexity measures such as Kolmogorov or algorithmic complexity. To make adequate comparison, we reconsider the standard axiomatic approach to complexity of algorithms. The new approach allows us to achieve a more adequate representation of static system complexity in the axiomatic context. It is demonstrated that for solving many problems inductive Turing machines have much lower complexity than Turing machines and other recursive algorithms. Thus, inductive Turing machines are not only more powerful, but also more efficient than Turing machines.

Information and computation : essays on scientific and philosophical understanding of foundations of information and computation

Cybersemiotics and the Question of Knowledge (S Brier) Information Dynamics in a Categorical Setting (M Burgin) Mathematics as Biological Process (G Chaitin) Information, Computation, Measurement and Irreversibility (J Collier) From Descartes to Turing: The Computational Content of Supervenience (B Cooper) On the Algorithmic Nature of the World (J-P Delahaye & H Zenil) A Dialogue Concerning Two Possible World Systems (G Dodig-Crnkovic & V Mueller) Does Computing Embrace Self-Organization? (W Hofkirchner) Analysis of Information and Computation in Physics Explains Cognitive Paradigms: From Full Cognition to Laplace Determinism to Statistical Determinism to Modern Approach (V Kreinovich & R Araiza) Bodies - Both Informed and Transformed (B J MacLennan) Computation on Information, Meaning and Representations, an Evolutionary Approach (C Menant) Interior Grounding, Reflection, and Self-Consciousness (M Minsky) Insights into the Biological Computing (W Riofrio) Super-Recursive Features of Natural Evolvability Processes and the Models for Computational Evolution (D Roglic) A Sketch of a Modeling View of Computing (O Shagrir) Whats Information, for an Organism or Intelligent Machine? How Can a Machine or Organism Mean? (A Sloman) Inconsistent Information as a Natural Phenomenon (C N J de Vey Mestdagh & J H Hoepman)

Information Theory: a Multifaceted Model of Information

Abstract: A contradictory and paradoxical situation that currently exists in information studies can be improved by the introduction of a new information approach, which is called the general theory of information. The main achievement of the general theory of information is explication of a relevant and adequate definition of information. This theory is built as a system of two classes of principles (ontological and sociological) and their consequences. Axiological principles, which explain how to measure and evaluate information and information processes, are presented in the second section of this paper. These principles systematize and unify different approaches, existing as well as possible, to construction and utilization of information measures. Examples of such measures are given by Shannon’s quantity of information, algorithmic quantity of information or volume of information. It is demonstrated that all other known directions of information theory may be treated inside general theory of information as its particular cases.

Evolutionary Automata

Expressiveness and convergence of evolutionary computation (EC) is studied using the evolutionary automata model. It turns out that all standard classes of evolutionary automata are equally expressive when they operate in the terminal mode, i.e. in the terminal mode, evolutionary finite automata (EFA) are as expressive as evolutionary pushdown automata, evolutionary linearly bounded automata, evolutionary Turing machines or evolutionary inductive Turing machines. For example, the simplest class of evolutionary automata, EFA, can accept all recursively enumerable languages (i.e. EFA have power of Turing machines) and even more—they can accept languages that are not recursively enumerable. Due to utilization of evolutionary automata, we obtain also very simple sufficient conditions for convergence of EC.

General approach to continuity measures

The neoclassical analysis is a field where an approach to investigation of different kinds of fuzzy continuous functions is suggested. In this paper we consider such property as continuity (local and global) of functions and investigate what does it mean that a function is continuous only to some extent.

Generalized kolmogorov complexity and other dual complexity measures

We construct and analyze some dual complexity measures that indicate the time it takes to obtain the desired object. The existence of optimal dual complexity measures is established. Various relations between dual measures and complexities are determined. The relationship of these measures to program quality is demonstrated.

Cooperative combinatorial optimization : Evolutionary computation case study

This paper presents a formalization of the notion of cooperation and competition of multiple systems that work toward a common optimization goal of the population using evolutionary computation techniques. It is proved that evolutionary algorithms are more expressive than conventional recursive algorithms, such as Turing machines. Three classes of evolutionary computations are introduced and studied: bounded finite, unbounded finite, and infinite computations. Universal evolutionary algorithms are constructed. Such properties of evolutionary algorithms as completeness, optimality, and search decidability are examined. A natural extension of evolutionary Turing machine (ETM) model is proposed to properly reflect phenomena of cooperation and competition in the whole population.