Axiomatic Approach Research Articles

Ontology-based Axiomatic System Design Semantic Modeling and Reasoning Rules

Ontology-based Axiomatic System Design Semantic Modeling and Reasoning Rules

A tensor rank theory and maximum full rank subtensors

A matrix always has a full rank submatrix such that the rank of this matrix is equal to the rank of that submatrix. This property is one of the corner stones of the matrix rank theory. We call this property the max-full-rank-submatrix property. Tensor ranks play a crucial role in low rank tensor approximation, tensor completion and tensor recovery. However, their theory is still not matured yet. Can we set an axiom system for tensor ranks? Can we extend the max-full-rank-submatrix property to tensors? We explore these in this paper. We first propose some axioms for tensor rank functions. Then we introduce proper tensor rank functions. The CP rank is a tensor rank function, but is not proper. There are two proper tensor rank functions, the max-Tucker rank and the submax-Tucker rank, which are associated with the Tucker decomposition. We define a partial order among tensor rank functions and show that there exists a unique smallest tensor rank function. We introduce the full rank tensor concept, and define the max-full-rank-subtensor property. We show the max-Tucker tensor rank function and the smallest tensor rank function have this property. We define the closure for an arbitrary proper tensor rank function, and show that it is still a proper tensor rank function and has the max-full-rank-subtensor property. An application of the submax-Tucker rank is also presented.

An Approximated Shapley Value for Partially Defined Cooperative Games

In the classical approach to cooperative games it is assumed that the worth of every coalition is known. However, in the real world problems there may be situations in which the amount of information is limited and consequently the worth of some coalitions are unknown. The games corresponding to those problems are called partially defined cooperative games (PDGs in short). The Shapley values for PDGs proposed so far need so much time to calculate them. Since the Shapley value plays an important role in applications of cooperative game theory to real world problems, we should consider methods to obtain the Shapley value or its approximation efficiently. We propose an approximated Shapley value which can be obtained efficiently. An axiom system of the proposed value is also shown.

Fuzzy Axiomatic Approach to Blue-green Infrastructure Strategy Selection: A Sustainability Perspective

Flood management is a global problem that has created immense contributions from researchers and practitioners, especially those in developing countries. These people often seek ways to minimise the aftermath of a flood. Recently, they are making a case for sustainable solutions to flood management. This study, therefore, contributes a sustainability model that addresses the problem of blue-green technology selection to the current discussion on flood management. It coupled the techno-economic, social, and environmental impact of a blue-green technology using the unique attributions of three multi-criteria decision-making tools: best-worst method, fuzzy axiomatic method and VIKOR; its performance was investigated with qualitative data sets that were obtained from experts. The outcomes of the investigation showed that techno-economic criteria contributed about 88.18% to the ranking of blue-green technology. The most and least suitable blue-green technologies for a community in Nigeria are Rainwater and floodwater harvesting and Retention lake, respectively. With these results, the proposed model will aid decision-makers strategic and tactical criteria that can be used to evaluate blue-green technology selection.

«GeOrigaMetry» An Approach to the Accessibility of Geometry for Blind People

Making geometry accessible for blind people, apart from the formal aspects, can pose some difficulties, especially in terms of accessibility to figures. To deal with this problem this article focuses on paper folding where both Euclidean and origami axiomatic systems are used simultaneously. In the first case, with a ruler and compass, we can solve quadratic problems in a plane. In addition, the axioms of origami allow us to address unanswered questions with classical geometry methods, which involve cubic equations, such as the trisection of an angle. An experiment with INJA (National Institute for Blind Youth, Paris) students and other blind people will take place so that we can see the possibilities offered by this method, which brings a ludic, but rigorous approach to these complex and frequently off-putting issues. We believe that this dynamic pedagogical approach can increase interest and motivation, encourage tactile stimulation and facilitate the development of specific structures of brain plasticity. The article is written in a linear way, accessible to blind people; figures are provided to facilitate understanding for "visually impaired" people, who are not used to following a geometric concept without pictures. Finally, it should be noted that the method is particularly suitable in an inclusive education context.

Information, signs, knowledge and intelligence

Understanding the special role of information in scientific knowledge formation is simultaneously accompanied by a high level of uncertainty regarding the nature of both information and knowledge. There are many approaches to information that can be constructively applied in some special areas of activity but remain completely useless in others. Against the background of such “informational diversity”, many researchers more often had the idea of the need for a universal explanation of the information phenomenon. The need to solve this problem is intuitively substantiated by the fact that the formation of information and its application create the foundation for many processes of self-organization and control in systems of various nature: chemical, biological, social, technical, and others. From the same point of view, the concept of information is often referred to as the primary one when explaining other also little understood phenomena: signs, semantics, knowledge, psyche, intelligence, mind, consciousness, self-awareness, mathematics, probability, and others. The universal concept of information and concepts derived from it can become a set of tools for formalized analysis from common conceptual positions of information processes in various scientific fields. The article presents the results of research, within the framework of which a universal information concept was obtained using a set of basic statements (axiomatic approach) and means of terminological, graphic, mathematical formalization (means of overcoming uncertainties). The information phenomenon is presented as a special result of the interaction of physical objects: the property of an object acquired during interaction contains the characteristics of another object. This approach to explaining the nature of information (attributive-transfer nature of information, ATNI), made it possible to determine the components of the process of its formation: informational impact, information carrier, essence, and semantics of information. With their help, derived concepts are defined: information environment and its characteristics; cybernetic system, the formalized model of its controlled behavior and security criteria; signs and their relationship with information; knowledge and intelligence of the cybernetic system. The applicability of the proposed approach to the analysis of processes in information environments of various natures: protein, neural, computer, has been tested.

MODEL OF SEMANTIC INTEGRATION OF INFORMATION SYSTEMS PROPERTIES IN RELAY DATABASE REENGINEERING PROBLEMS

The subject of research is methods of semantic integration of subject areas of heterogeneous information systems and distributed databases. This class of systems are created on the basis of database technologies, are widespread and used in all areas of economic activity. Such systems are characterized by high complexity of design, maintenance and modification. The purpose of the research is the development of a subject area model based on the semantic properties and relationships of data elements; development of effective technology of integration of information resources of heterogeneous computer systems on the basis of technology of management of database systems; research and formalization of classes of inhomogeneity of data structures aimed at solving the problem of determining the types of information objects. Development of tools for the design and maintenance of application problems for the integration of heterogeneous information systems and distributed databases. Results: the analysis of existing methods and models of integration of subject areas on the basis of semantic properties and connections of data elements is carried out; developed an effective mathematical model, technology and algorithm for semantic integration of information resources of heterogeneous computer systems for relational databases; investigated and formalized classes of inhomogeneity of data structures aimed at solving the problem of determining the types of information objects; the model is developed and means of the logical description of properties of information objects for definition of border of the considered subject area are investigated. Conclusion: the article considers a formalized infographic model of the subject area, which focuses on the semantic relationships between information objects of databases. An axiomatic approach to the description of the subject area is formulated, which allows to consider the problem of modeling the relations of the elements of the subject area in the form of a set of rules that determine the existence of data elements. On the basis of the analysis of structures and models of databases of information systems the general approach to construction of the universal technology focused on the decision of problems of management of heterogeneous information resources of computer systems is defined.

A Behavioural Theory of Recursive Algorithms

“What is an algorithm?” is a fundamental question of computer science. Gurevich’s behavioural theory of sequential algorithms (aka the sequential ASM thesis) gives a partial answer by defining (non-deterministic) sequential algorithms axiomatically, without referring to a particular machine model or programming language, and showing that they are captured by (nondeterministic) sequential Abstract State Machines (nd-seq ASMs). However, recursive algorithms such as mergesort are not covered by this theory, as has been pointed out by Moschovakis, who had independently developed a different framework to mathematically characterize the concept of (in particular recursive) algorithm. In this article we propose an axiomatic definition of the notion of sequential recursive algorithm which extends Gurevich’s axioms for sequential algorithms by a Recursion Postulate and allows us to prove that sequential recursive algorithms are captured by recursive Abstract State Machines, an extension of nd-seq ASMs by a CALL rule. Applying this recursive ASM thesis yields a characterization of sequential recursive algorithms as finitely composed concurrent algorithms all of whose concurrent runs are partial-order runs.

Identificação de lacunas no processo de aprendizagem dos conteúdos de geometria no ensino médio pelo método de Van Hiele

Este artigo tem por objetivo classificar o nível de conhecimento sobre geometria euclidiana em relação ao modelo de Van Hiele. Percebe-se que os estudantes, em geral, não conseguem relacionar sistemas axiomáticos, apresentando assim dificuldades em sistematizar o pensamento geométrico, bem como estabelecer conexões gráficas entre a matemática e outras áreas do conhecimento. Dessa forma, a investigação, caracteriza-se por ser uma abordagem quali-quantitativo com uma base de dados coletados em escolas da rede pública estadual, com estudantes do terceiro ano do Ensino Médio, da cidade de Bagé/RS. Como procedimento metodológico, para a realização desta pesquisa, aplicou-se questionários com questões objetivas e discursivas direcionados a um grupo composto de 280 estudantes. Com base nos resultados, conclui-se que o material desenvolvido na investigação serviu de apoio aos professores em suas práticas pedagógicas que busquem contemplar e atender características dos níveis de pensamento geométrico.

Algorithmic definitions for KLM-style defeasible disjunctive Datalog

Datalog is a declarative logic programming language that uses classical logical reasoning as its basic form of reasoning. Defeasible reasoning is a form of non-classical reasoning that is able to deal with exceptions to general assertions in a formal manner. The KLM approach to defeasible reasoning is an axiomatic approach based on the concept of plausible inference. Since Datalog uses classical reasoning, it is currently not able to handle defeasible implications and exceptions. We aim to extend the expressivity of Datalog by incorporating KLM-style defeasible reasoning into classical Datalog. We present a systematic approach for extending the KLM properties and a well-known form of defeasible entailment: Rational Closure. We conclude by exploring Datalog extensions of less conservative forms of defeasible entailment: Relevant and Lexicographic Closure. We provide algorithmic definitions for these forms of defeasible entailment and prove that the definitions are LM-rational.

New similarity measures for single-valued neutrosophic sets with applications in pattern recognition and medical diagnosis problems

The single-valued neutrosophic set (SVNS) is a well-known model for handling uncertain and indeterminate information. Information measures such as distance measures, similarity measures and entropy measures are very useful tools to be used in many applications such as multi-criteria decision making (MCDM), medical diagnosis, pattern recognition and clustering problems. A lot of such information measures have been proposed for the SVNS model. However, many of these measures have inherent problems that prevent them from producing reasonable or consistent results to the decision makers. In this paper, we propose several new distance and similarity measures for the SVNS model. The proposed measures have been verified and proven to comply with the axiomatic definition of the distance and similarity measure for the SVNS model. A detailed and comprehensive comparative analysis between the proposed similarity measures and other well-known existing similarity measures has been done. Based on the comparison results, it is clearly proven that the proposed similarity measures are able to overcome the shortcomings that are inherent in existing similarity measures. Finally, an extensive set of numerical examples, related to pattern recognition and medical diagnosis, is given to demonstrate the practical applicability of the proposed similarity measures. In all numerical examples, it is proven that the proposed similarity measures are able to produce accurate and reasonable results. To further verify the superiority of the suggested similarity measures, the Spearman’s rank correlation coefficient test is performed on the ranking results that were obtained from the numerical examples, and it was again proven that the proposed similarity measures produced the most consistent ranking results compared to other existing similarity measures.

Wind farm site selection from the perspective of sustainability: A novel satisfaction degree‐based fuzzy axiomatic design approach

Wind farm site selection from the perspective of sustainability: A novel satisfaction degree‐based fuzzy axiomatic design approach

Multicriteria Choice Based on Fuzzy Information

This paper proposes a new method for solving the problem of multicriteria optimization of a numerical vector function on a fuzzy set. The membership function of a fuzzy feasible set is joined to the original set of criteria that allows the original problem of multi-criteria optimization to be treated as the task of finding a suitable compromise (Pareto-optimal) solution with respect to an extended set of criteria. It is assumed that in a search for the “best” compromise solution there is only fuzzy information about the preferences of decision maker in the form of information quanta. At the first stage of the proposed method, the search for a compromise is made on the basis of an axiomatic approach, with which the Pareto set is reduced. The result of the reduction is a fuzzy set with a membership function, which is determined on the basis of the used fuzzy information. At the second stage, the obtained membership function is added to the extended set of criteria, after which the scalarization procedure based on the idea of goal programming is used to solve the formed multicriteria problem.

Normalization by second order graphs: A visual alternative to simplify systems

This issue stems from the need for tools to analyze and make decisions around complex systems, where they apply the rules for linearly dependent sets, with the purpose of providing a visual tool, which serves to support complexity reduction processes. Two great precedents are Armstrong's Axioms, which has been applied from its publication to the present for database normalization, the other is set theory, a fundamental pillar of the Structured Query Language; based on them, together with the second-order logic, which adds qualifiers for subsets or properties, this work has been prepared, with an explanatory metrology with a qualitative approach, in an axiomatic system. As a result, a support tool has been provided to analyze complex systems naturally, by breaking cycles and detecting patterns, without interfering with existing models; however, for large systems it can be difficult to address it in its entirety, so it is recommended to divide by subsystems. With this work a technique has been accomplished, repeatable by anyone, but with a strong theoretical foundation. This work has great utility for the normalization of relational databases and an enormous potential for application in the design of systems beyond computational systems, it is also useful for understanding dependencies by their axiomatic nature.

Pollution‐adjusted productivity analysis: The use of Malmquist and Luenberger productivity measures

This paper aims to define pollution‐adjusted Malmquist and Luenberger productivity measures through generalized pollution‐adjusted production technology. This axiomatic framework is fairly weak and does not require any convexity property for the production technology. Hence, a general context to analyse and decompose pollution‐adjusted productivity change is introduced. Decomposing productivity measures allows to determine the sources of productivity variation and is of interest to define recommendations for decision makers.

Generation and Use of Hints and Feedback in a Hilbert-Style Axiomatic Proof Tutor

This paper describes logax, an interactive tutoring tool that gives hints and feedback to a student who stepwise constructs a Hilbert-style axiomatic proof in propositional logic. logax generates proofs to calculate hints and feedback. We compare these generated proofs with expert proofs and student solutions, and conclude that the quality of the generated proofs is comparable to that of expert proofs. logax recognizes most steps that students take when constructing a proof. Even if a student diverges from the generated solution, logax still provides hints, including next steps or reachable subgoals, and feedback. With a few improvements in the design of the set of buggy rules, logax will cover about 80% of the mistakes made by students by buggy rules. The hints help students to complete the exercises.

Reverse Bayesianism: Revising Beliefs in Light of Unforeseen Events

Bayesian Updating is the dominant theory of learning in economics. The theory is silent about how individuals react to events that were previously unforeseeable or unforeseen. Recent theoretical literature has put forth axiomatic frameworks to analyze the unknown. In particular, we test if subjects update their beliefs in a way that is consistent Bayesian, which ensures that the old information is used correctly after an unforeseen event materializes. We find that participants do not systematically deviate from reverse Bayesianism, but they do not seem to expect an unknown event when this is reasonably unforeseeable, in two pre-registered experiments that entail unforeseen events. We argue that participants deviate less from the reverse Bayesian updating than from the usual Bayesian updating. We provide further evidence on the moderators of belief updating.

Development of modern models and methods of the theory of statistical hypothesis testing

Typical problems of the theory of statistical hypothesis testing are considered. All these problems belong to the same object area and are formulated in a single system of axioms and assumptions using a common linguistic thesaurus. However, different approaches are used to solve each of these problems and a unique solution method is developed. In this regard, the work proposes a unified methodological approach for formulating and solving these problems. The mathematical basis of the approach is the theory of continuous linear programming (CLP), which generalizes the known mathematical apparatus of linear programming for the continuous case. The mathematical apparatus of CLP allows passing from a two-point description of the solution of the problem in the form {0; 1} to a continuous one on the segment [0; 1]. Theorems justifying the solution of problems in terms of CLP are proved. The problems of testing a simple hypothesis against several equivalent or unequal alternatives are considered. To solve all these problems, a continuous function is introduced that specifies a randomized decision rule leading to continuous linear programming models. As a result, it becomes possible to expand the range of analytically solved problems of the theory of statistical hypothesis testing. In particular, the problem of making a decision based on the maximum power criterion with a fixed type I error probability, with a constraint on the average risk, the problem of testing a simple hypothesis against several alternatives for given type II error probabilities. The method for solving problems of statistical hypothesis testing for the case when more than one observed controlled parameter is used to identify the state is proposed

An infinitary axiomatization of dynamic topological logic

Abstract Dynamic topological logic ($\textsf{DTL}$) is a multi-modal logic that was introduced for reasoning about dynamic topological systems, i.e. structures of the form $\langle{\mathfrak{X}, f}\rangle $, where $\mathfrak{X}$ is a topological space and $f$ is a continuous function on it. The problem of finding a complete and natural axiomatization for this logic in the original tri-modal language has been open for more than one decade. In this paper, we give a natural axiomatization of $\textsf{DTL}$ and prove its strong completeness with respect to the class of all dynamic topological systems. Our proof system is infinitary in the sense that it contains an infinitary derivation rule with countably many premises and one conclusion. It should be remarked that $\textsf{DTL}$ is semantically non-compact, so no finitary proof system for this logic could be strongly complete. Moreover, we provide an infinitary axiomatic system for the logic ${\textsf{DTL}}_{\mathcal{A}}$, i.e. the $\textsf{DTL}$ of Alexandrov spaces, and show that it is strongly complete with respect to the class of all dynamical systems based on Alexandrov spaces.

Törnqvist Formula for Сalculating a Consumer Price Index in Russia: Theory and Practice

The article reflects the author’s position on the adjustment of the so-called substitution bias, which affects the value of the consumer price index, currently calculated using the Laspeyres formula. The author proposes a solution to the problem of the adequacy of statistical measurements of the dynamics of consumer prices in the case when, as a result of changing cost of the consumer basket, a buyer replaces a relatively expensive product with a relatively cheaper one. This solution is based on the existing index construction methodology (axiomatic, economic and stochastic approaches). The article substantiates the use of the Törnqvist formula, which has better properties in comparison with other formulae used in the construction of superlative indices. The authors calculated the Törnqvist price index for Russia based on Rosstat methodology and data using country-level quarterly group price indices and shares of consumer spending. To evaluate the results of empirical testing Laspeyres price index was compiled using the same quarterly data as the Törnqvist index. The values of the Törnqvist price index in most cases are less than the price dynamics obtained according to the Laspeyres formula. This conclusion is proved both theoretically and empirically, and it is confirmed for Russia as well. However, due to the non-observance of the conditions of smooth trends in consumer prices, the difference between the values of the Törnqvist and Laspeyres indices is significantly larger in certain quarters than that presented in empirical studies in other countries. Consumer price index, calculated using the Törnqvist formula, in the system of indicators of price statistics in Russia can be defined as an indicator that specifies the main consumer price index. Calculation of its value is necessary for a more realistic description of the processes taking place in the consumer market.