196 research outputs found
Scaling Qualitative Probability
There are different approaches to qualitative probability, which includes
subjective probability. We developed a representation of qualitative
probability based on relational systems, which allows modeling uncertainty by
probability structures and is more coherent than existing approaches. This
setting makes it possible proving that any comparative probability is induced
by some probability structure (Theorem 2.1), that classical probability is a
probability structure (Theorem 2.2) and that inflated, i.e., larger than 1,
probability is also a probability structure (Theorem 2.3). In addition, we
study representation of probability structures by classical probability
Mathematical Knowledge and the Role of an Observer: Ontological and epistemological aspects
As David Berlinski writes (1997), the existence and nature of mathematics is
a more compelling and far deeper problem than any of the problems raised by
mathematics itself. Here we analyze the essence of mathematics making the main
emphasis on mathematics as an advanced system of knowledge. This knowledge
consists of structures and represents structures, existence of which depends on
observers in a nonstandard way. Structural nature of mathematics explains its
reasonable effectiveness
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
Extended Probabilities: Mathematical Foundations
There are important problems in physics related to the concept of
probability. One of these problems is related to negative probabilities used in
physics from 1930s. In spite of many demonstrations of usefulness of negative
probabilities, physicists looked at them with suspicion trying to avoid this
new concept in their theories. The main question that has bothered physicists
is mathematical grounding and interpretation of negative probabilities. In this
paper, we introduce extended probability as a probability function, which can
take both positive and negative values. Defining extended probabilities in an
axiomatic way, we show that classical probability is a positive section of
extended probability
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
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
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
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
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
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)
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