arXiv subjects

Mark Burgin

Publications and source records attributed to Mark Burgin.

35 records · Page 2Linked to original sources

arXiv2008

Algorithmic Problem Complexity

People solve different problems and know that some of them are simple, some are complex and some insoluble. The main goal of this work is to develop a mathematical theory of algorithmic complexity for problems. This theory is aimed at determination of computer abilities in solving different problems and estimation of resources that computers need to do this. Here we build the part of this theory related to static measures of algorithms. At first, we consider problems for finite words and study algorithmic complexity of such problems, building optimal complexity measures. Then we consider problems for such infinite objects as functions and study algorithmic complexity of these problems, also building optimal complexity measures. In the second part of the work, complexity of algorithmic problems, such as the halting problem for Turing machines, is measured by the classes of automata that are necessary to solve this problem. To classify different problems with respect to their complexity, inductive Turing machines, which extend possibilities of Turing machines, are used. A hierarchy of inductive Turing machines generates an inductive hierarchy of algorithmic problems. Here we specifically consider algorithmic problems related to Turing machines and inductive Turing machines, and find a place for these problems in the inductive hierarchy of algorithmic problems.

cs.CC↗

arXiv2008

Fuzzy Statistical Limits

Statistical limits are defined relaxing conditions on conventional convergence. The main idea of the statistical convergence of a sequence l is that the majority of elements from l converge and we do not care what is going on with other elements. At the same time, it is known that sequences that come from real life sources, such as measurement and computation, do not allow, in a general case, to test whether they converge or statistically converge in the strict mathematical sense. To overcome these limitations, fuzzy convergence was introduced earlier in the context of neoclassical analysis and fuzzy statistical convergence is introduced and studied in this paper. We find relations between fuzzy statistical convergence of a sequence and fuzzy statistical convergence of its subsequences (Theorem 2.1), as well as between fuzzy statistical convergence of a sequence and conventional convergence of its subsequences (Theorem 2.2). It is demonstrated what operations with fuzzy statistical limits are induced by operations on sequences (Theorem 2.3) and how fuzzy statistical limits of different sequences influence one another (Theorem 2.4). In Section 3, relations between fuzzy statistical convergence and fuzzy convergence of statistical characteristics, such as the mean (average) and standard deviation, are studied (Theorems 3.1 and 3.2).

math.CA↗

arXiv2007

Superrecursive Features of Interactive Computation

Functioning and interaction of distributed devices and concurrent algorithms are analyzed in the context of the theory of algorithms. Our main concern here is how and under what conditions algorithmic interactive devices can be more powerful than the recursive models of computation, such as Turing machines. Realization of such a higher computing power makes these systems superrecursive. We find here five sources for superrecursiveness in interaction. In addition, we prove that when all of these sources are excluded, the algorithmic interactive system in question is able to perform only recursive computations. These results provide computer scientists with necessary and sufficient conditions for achieving superrecursiveness by algorithmic interactive devices.

cs.DC↗

arXiv2007

Languages, Algorithms, Procedures, Calculi, and Metalogic

Logicians study and apply a multiplicity of various logical systems. Consequently, there is necessity to build foundations and common grounds for all these systems. This is done in metalogic. Like metamathematics studies formalized mathematical theories, metalogic studies theories in logic, or logics. The discipline of logic has been developed with the aim to model and study human thinking and reasoning. A more realistic understanding relates logic only to reasoning. Reasoning is a mental and verbal activity. Any activity is based on actions and operations organized by procedures and algorithms. That is why procedures and algorithms are basic for the development of different logics, their study and application. In this work, we study structures used in logic with the aim to reflect goals of logic as a discipline. The main structure is a logical calculus, which pivotal for the contemporary logic. Logical calculus is considered here as a two-tier construction with a logical language on the first level and the calculus itself on the second level. A system of interdependencies that exists between properties of deductive calculi and utilized by these calculi algorithms are explicated and studied. Here only syntactic parts of logics, namely, deductive logical calculi and corresponding languages are considered. Semantic and pragmatic parts are studied elsewhere.

math.LO↗

arXiv2006

Fuzzy Limits of Functions

The goal of this work is to introduce and study fuzzy limits of functions. Two approaches to fuzzy limits of a function are considered. One is based on the concept of a fuzzy limit of a sequence, while another generalizes the conventional epsilon-delta definition. It is demonstrated that these constructions are equivalent. Different properties of fuzzy limits of functions are found. Properties of conventional limits are easily deduced from properties of fuzzy limits. In the second part of this work, the concept of fuzzy limits of a function is extended to provide means to define nontrivial continuity of functions on discrete sets. In addition, fuzzy limits of functions are introduced and studied.

math.CA↗

arXiv2006

Statistical Convergence and Convergence in Statistics

Statistical convergence was introduced in connection with problems of series summation. The main idea of the statistical convergence of a sequence l is that the majority of elements from l converge and we do not care what is going on with other elements. We show (Section 2) that being mathematically formalized the concept of statistical convergence is directly connected to convergence of such statistical characteristics as the mean and standard deviation. At the same time, it known that sequences that come from real life sources, such as measurement and computation, do not allow, in a general case, to test whether they converge or statistically converge in the strict mathematical sense. To overcome limitations induced by vagueness and uncertainty of real life data, neoclassical analysis has been developed. It extends the scope and results of the classical mathematical analysis by applying fuzzy logic to conventional mathematical objects, such as functions, sequences, and series. The goal of this work is the further development of neoclassical analysis. This allows us to reflect and model vagueness and uncertainty of our knowledge, which results from imprecision of measurement and inaccuracy of computation. In the context on the theory of fuzzy limits, we develop the structure of statistical fuzzy convergence and study its properties.

math.GM↗

arXiv2006

Evolutionary Optimization in an Algorithmic Setting

Evolutionary processes proved very useful for solving optimization problems. In this work, we build a formalization of the notion of cooperation and competition of multiple systems working toward a common optimization goal of the population using evolutionary computation techniques. It is justified that evolutionary algorithms are more expressive than conventional recursive algorithms. Three subclasses of evolutionary algorithms are proposed here: bounded finite, unbounded finite and infinite types. Some results on completeness, optimality and search decidability for the above classes are presented. A natural extension of Evolutionary Turing Machine model developed in this paper allows one to mathematically represent and study properties of cooperation and competition in a population of optimized species.

cs.NE↗

arXiv2005

Mathematical Models in Schema Theory

In this paper, a mathematical schema theory is developed. This theory has three roots: brain theory schemas, grid automata, and block-shemas. In Section 2 of this paper, elements of the theory of grid automata necessary for the mathematical schema theory are presented. In Section 3, elements of brain theory necessary for the mathematical schema theory are presented. In Section 4, other types of schemas are considered. In Section 5, the mathematical schema theory is developed. The achieved level of schema representation allows one to model by mathematical tools virtually any type of schemas considered before, including schemas in neurophisiology, psychology, computer science, Internet technology, databases, logic, and mathematics.

cs.AI↗

arXiv2004

Hardware-Oriented Group Solutions for Hard Problems

Group and individual solutions are considered for hard problems such as satisfiability problem. Time-space trade-off in a structured active memory provides means to achieve lower time complexity for solutions of these problems.

cs.CC↗

arXiv2004

Axiomatic Theory of Algorithms: Computability and Decidability in Algorithmic Classes

Axiomatic approach has demonstrated its power in mathematics. The main goal of this preprint is to show that axiomatic methods are also very efficient for computer science. It is possible to apply these methods to many problems in computer science. Here the main modes of computer functioning and program execution are described, formalized, and studied in an axiomatic context. The emphasis is on three principal modes: computation, decision, and acceptation. Now the prevalent mode for computers is computation. Problems of artificial intelligence involve decision mode, while communication functions of computer demand accepting mode. The main goal of this preprint is to study properties of these modes and relations between them. These problems are closely related to such fundamental concepts of computer science and technology as computability, decidability, and acceptability. In other words, we are concerned with the question what computers and software systems can do working in this or that mode. Consequently, results of this preprint allow one to achieve higher understanding of computations and in such a way, to find some basic properties of computers and their applications. Classes of algorithms, which model different kinds of computers and software, are compared with respect to their computing, accepting or deciding power. Operations with algorithms and machines are introduced. Examples show how to apply axiomatic results to different classes of algorithms and machines in order to enhance their performance.

math.LO↗

arXiv2004

Unified Foundations for Mathematics

There are different meanings of foundation of mathematics: philosophical, logical, and mathematical. Here foundations are considered as a theory that provides means (concepts, structures, methods etc.) for the development of whole mathematics. Set theory has been for a long time the most popular foundation. However, it was not been able to win completely over its rivals: logic, the theory of algorithms, and theory of categories. Moreover, practical applications of mathematics and its inner problems caused creation of different generalization of sets: multisets, fuzzy sets, rough sets etc. Thus, we encounter a problem: Is it possible to find the most fundamental structure in mathematics? The situation is similar to the quest of physics for the most fundamental "brick" of nature and for a grand unified theory of nature. It is demonstrated that in contrast to physics, which is still in search for a unified theory, in mathematics such a theory exists. It is the theory of named sets.

math.LO↗

arXiv2002

The Rise and Fall of the Church-Turing Thesis

The essay consists of three parts. In the first part, it is explained how theory of algorithms and computations evaluates the contemporary situation with computers and global networks. In the second part, it is demonstrated what new perspectives this theory opens through its new direction that is called theory of super-recursive algorithms. These algorithms have much higher computing power than conventional algorithmic schemes. In the third part, we explicate how realization of what this theory suggests might influence life of people in future. It is demonstrated that now the theory is far ahead computing practice and practice has to catch up with the theory. We conclude with a comparison of different approaches to the development of information technology.

cs.CC↗

arXiv2002

Elements of the System Theory of Time

In the paper, elements of the system theory of time are presented, mathematical models for time are constructed, and various properties are deduced from the main principles of the system theory of time. This theory is a far-reaching development of the special relativity theory. One of the main principles of the special relativity theory is that two physical systems that are moving relative to each other have different times and it is necessary to use a correspondence between clocks in these systems to coordinate their times. Such correspondence is established by means of electromagnetic signals. In accordance with this principle, it is postulated in the system theory of time that each system has its own time. In some cases, two systems have the same time. In other cases, times of systems are coordinated or correlated. However, there are systems in which times are independent from one another.

physics.gen-ph↗

arXiv2001

Representation of Uncertainty for Limit Processes

Many mathematical models utilize limit processes. Continuous functions and the calculus, differential equations and topology, all are based on limits and continuity. However, when we perform measurements and computations, we can achieve only approximate results. In some cases, this discrepancy between theoretical schemes and practical actions changes drastically outcomes of a research and decision-making resulting in uncertainty of knowledge. In the paper, a mathematical approach to such kind of uncertainty, which emerges in computation and measurement, is suggested on the base of the concept of a fuzzy limit. A mathematical technique is developed for differential models with uncertainty. To take into account the intrinsic uncertainty of a model, it is suggested to use fuzzy derivatives instead of conventional derivatives of functions in this model.

cs.AI↗

arXiv2001

Distributed Computing for Localized and Multilayer Visualizations

The aim of this paper is to develop an approach to visualizations that benefits from distributed computing. Three schemes of process distribution are considered: parallel, pipeline, and expanding pipeline computations. Expanding pipeline structure synthesizes the advantages and traits of both parallel and pipeline computations. In expanding pipeline computing, a novel approach presented in this paper, a multiplicity of processes are concurrently developed in parallel and knotted processor pipelines. The theoretical foundations for expanding pipeline computing as a computational process are in the domains of alternating Turing machines, molecular computing, and E-machines. Expanding pipeline computing constitutes the development of the conventional pipeline architecture aimed at utilization of implicit parallel structures existing in algorithms. Such structures appear in various kinds of visualization. Image deriving and processing is a field that provides diverse opportunities for utilization of the advantages of distributed computing. The most relevant to the distributed architecture is stratified visualization with its two cases based on data localization and layer separation. Visualization is treated here as a special case of simulation. The conceptual approach to distributed computing developed in this paper have been applied to visualization in a computer support system, which is utilized in radiology and namely, for the noninvasive treatment of brain aneurysms.

cs.DC↗

arXiv2001

Multivariant Branching Prediction, Reflection, and Retrospection

In branching simulation, a novel approach to simulation presented in this paper, a multiplicity of plausible scenarios are concurrently developed and implemented. In conventional simulations of complex systems, there arise from time to time uncertainties as to which of two or more alternatives are more likely to be pursued by the system being simulated. Under these conditions the simulationist makes a judicious choice of one of these alternatives and embeds this choice in the simulation model. By contrast, in the branching approach, two or more of such alternatives (or branches) are included in the model and implemented for concurrent computer solution. The theoretical foundations for branching simulation as a computational process are in the domains of alternating Turing machines, molecular computing, and E-machines. Branching simulations constitute the development of diagrams of scenarios representing significant, alternative flows of events. Logical means for interpretation and investigation of the branching simulation and prediction are provided by the logical theories of possible worlds, which have been formalized by the construction of logical varieties. Under certain conditions, the branching approach can considerably enhance the efficiency of computer simulations and provide more complete insights into the interpretation of predictions based on simulations. As an example, the concepts developed in this paper have been applied to a simulation task that plays an important role in radiology - the noninvasive treatment of brain aneurysms.

cs.CE↗

arXiv2001

Diophantine and Non-Diophantine Arithmetics: Operations with Numbers in Science and Everyday Life

Science and mathematics help people better to understand world, eliminating different fallacies and misconceptions. One of such misconception is related to arithmetic, which is so important both for science and everyday life. People think that their counting is governed by the rules of the conventional arithmetic and that other kinds of arithmetic do not exist and cannot exist. It is demonstrated in this paper that this popular image of the situation with integer numbers is incorrect. In many situations, we have to utilize different rules of counting and operating. This is a consequence of the existing diversity in nature and society and to represent correctly this diversity people have to utilize different arithmetics. To distinct them, we call the conventional arithmetic Diophantine, while other arithmetics are called non-Diophantine. Theory of non-Diophantine arithmetics is developed in the book of the author "Non-Diophantine arithmetics or is it possible that 2 + 2 is not equal to 4." In this work, some properties of non-Diophantine arithmetics are considered, as well as their connections to numerical computations and contemporary physics are explained.

math.GM↗