Set-theoretical Representation Research Articles

Особенности моделирования функционирования ВУЗов как сложных организационных систем силовых структур

The article considers an approach to solving the problem of improving the indicators characterizing the state of the educational group of a modern university, which is currently difficult to imagine without the use of systems operating on the basis of mathematical models of SED. The elements of assessing approaches to improving the effectiveness of management decision-making are considered, which cannot always be characterized quantitatively. To increase the effectiveness of the R&D, it is proposed to use methods and tools that implement a quantitative assessment of the state of the management system, which is able to justify the decision made. When modeling managerial decision-making in complex organizational systems of law enforcement agencies, a conceptual approach is proposed that is based on situational modeling and provides information to the decision-maker in this system about his current state. Based on this situational information, the decision-maker will make recommendations in the interests of situational management and ensure its adaptation to real needs. The situational approach uses a set-theoretic representation of models, which will allow combining the spatial and structural interpretation of this model. Determining the relationships between employees in complex organizational systems of law enforcement agencies will allow you to link tasks with employees based on input-output information flows using a multi-criteria assessment of alternative.

Poly-scale principle of urban environment geoinformation monitoring

The article discusses general approaches for monitoring the urban environment based on geoinformation technologies. The purpose of these studies is to use the properties of poly-scale geoinformation monitoring in modeling and forecasting the state of the urban environment. The urgency of identifying the spatio-temporal features of the development of the urban environment at different territorial levels is shown. Each territorial level carries its own level of detail. These levels are the global level, regional, local, macro, meso level and primary level. The set-theoretic representation of the principle of poly-scale modelling of objects of the urban environment is considered. Based on this approach, models and methods of geoinformation monitoring at each scale level can be represented as a composition of ordered pairs of different structural types. An example of a hierarchy of methods and models used in studies of the urban environment is given. The concept of basic composition models is introduced. On the basis of such models, a technology for constructing poly-scale compositions is presented. Such compositions make it possible to convey the computational essence of the corresponding mathematical models at each level of the poly-scale approach in the study of the urban environment. A generalized scheme for applying the principle of poly-scale decomposition of a geoinformation monitoring object has been developed. At the same time, at each level of the information space, the construction of the corresponding models of the monitoring object is provided.

Contact Join-semilattices

Contact algebra is one of the main tools in region-based theory of space. In \cite{dmvw1, dmvw2,iv,i1} it is generalized by dropping the operation Boolean complement. Furthermore we can generalize contact algebra by dropping also the operation meet. Thus we obtain structures, called contact join-semilattices (CJS) and structures, called distributive contact join-semilattices (DCJS). We obtain a set-theoretical representation theorem for CJS and a relational representation theorem for DCJS. As corollaries we get also topological representation theorems. We prove that the universal theory of CJS and of DCJS is the same and is decidable.

Геокогнитивные методы обеспечения анализа и прогнозирования социально-экономического развития территорий

The issues of using geospatial data, information and knowledge in the processes of analysis and forecasting of socio-economic development of territories are considered. It is proposed to expand the composition of a single complex of means for macroeconomic interregional intersectoral planning due to a more complete consideration of the geospatial features of the studied territory and the transition to a new model of spatial and economic policy based on the widespread use of digital geoplatforms. The article provides information on the importance of geospatial support for planning and management of territories for the sustainable development of their economy and social sphere and information on the support of this industry at the international (UN) and national levels of the most developed countries of the world. The purpose of the research is indicated—the substantiation of the methodology for productive and effective geospatial support for the solutions of analytical and forecasting tasks, which increases the level of taking into account the territorial features of the regions. A new term “geocognitive modeling of the territory” and theoretical and methodological approaches to assessing this process to ensure strategic regional planning are proposed. A number of initial conceptual provisions have been formulated, including: geodata and geoinformation are more in demand at the microlevel of the economy, and geoscience at the macrolevel of the economy. geosciences refer to certain subject-spatial heterogeneities and they can be represented as a special type of spatial objects and processed by means of gIS. A list of indicators of a separate geofragment of the territory for use in strategic planning is given. The proposed method for solving the geospatial problem of assessing the synergetic effect in the interaction of natural resources with the transport system by increasing the connectivity of territories is described. As an example, a model of cognitive indexing of geofragments is proposed and its set-theoretic representation is given. The conclusion is made about the need for a more complete use of geospatial tools in planning the development of regions, which will reduce the risks of making unreasonable managerial decisions and take into account the influence of the human factor.

ТЕОРЕТИКО-МЕТОДОЛОГИЧЕСКОЕ ПРЕДСТАВЛЕНИЕ ПРЯМОГО ПЕРЕХОДА ОТ ГЕОИНФОРМАЦИИ К ГЕОЗНАНИЯМ

The article discusses new approaches to the creation and use of geoscience and the associated geo-cognitive space. It uses the formal-logical tools for the set-theoretic representation of the studied pro-cess of transition from geoinformation to geoscience. It introduces and defines new concepts for a generalized set-theoretic representation of the process of converting geoinformation about the territory and objects located on it into geoscience, focused on solving specific spatial problems using the geoin-formation model of the area or the map. The article presents an enlarged technological scheme for cre-ating a geo-cognitive map. It also presents one of the possible solutions to the problem of forming geo-spatial knowledge and creating a geocognitive map on this basis by a direct transition from geoinfor-mation to knowledge.

RESOURCE-INDEPENDENT REPRESENTATION OF SORTING NETWORKS IN Set@l PROGRAMMING LANGUAGE

The development of scalable parallel programs for high-performance computer systems with reconfigurable and hybrid architectures is an important problem in the field of modern supercomputer engineering. Resource-independent applied software in the high-level COLAMO programming language describes only the limited number of the algorithm implementations for some configurations due to the inductive principle of the information graph forming. However, the creation of efficient resource-independent parallel software requires programming tool that supports both induction and reduction of calculations. To solve the problem, we propose the application of the Set@l programming language based on the aspect-oriented programming paradigm and set-theoretical code representation. According to the aspect-oriented concept, Set@l describes an algorithm for the problem solution and its parallelizing and other implementation features as separated program modules – source code and various aspects. The programming language being considered has much in common with traditional set-theory-based languages such as SETL, SETLX and SETL2, but Set@l introduces the essential classification of collections by parallelism, definiteness, orderliness and other criteria. In contrast to the high-level COLAMO programming language, Set@l specifies the full information graph of a computational problem in the absolutely parallel form and describes the reduction principles for all configurations of a computer system as independent aspects of a program. In this paper, we demonstrate that the aspect-oriented sorting program in the Set@l language defines different types of quadratic sorting networks and various methods of their partition and reduction.

Verification of functional models of chemical manufacturing

A generalized algorithm for the verification of functional models and the rules for the verification of diagrams related by levels of detail were developed in this paper. The algorithm is based on the analysis of a tree which describes the decompose relations in functional diagrams. At each step of the algorithm, a pair consisting of a parent diagram and a functional diagram is selected, and the correlation of the arrows and their roles is checked for both. The formalization of the verification rules was based on the set-theoretic representation of functional diagrams in the form of labeled oriented graphs. The rules make it possible to map the position and roles of the arrows associated with the detailed function block of the parent diagram to the arrows of the child diagram. The following rules for each of the possible arrow roles were established: “input”, “output”, “control”, “mechanism”. The use of the logic programming language PROLOG was proposed for the implementation of the algorithm. A knowledge base structure comprised of 3 interrelated predicates to describe the tree of diagrams, nodes and edges of the graphs was suggested. A query to check the verification rules was formed, and methods of binding variables and fixing roles were considered. The analysis and verification of a fragment of a functional model for the production of vinyl acetate from ethylene was conducted as an example. The functional diagrams for the processes “Condensate separation” and “Vinyl acetate isolation” connected by a decompose relation were developed, their set-theoretic models were constructed, and the use of rules for the verification of each type of arrow were considered.

Symmetry arguments against regular probability: A reply to recent objections

A probability distribution is regular if it does not assign probability zero to any possible event. While some hold that probabilities should always be regular, three counter-arguments have been posed based on examples where, if regularity holds, then perfectly similar events must have different probabilities. Howson (2017) and Benci et al. (2018) have raised technical objections to these symmetry arguments, but we see here that their objections fail. Howson says that Williamson’s (2007) “isomorphic” events are not in fact isomorphic, but Howson is speaking of set-theoretic representations of events in a probability model. While those sets are not isomorphic, Williamson’s physical events are, in the relevant sense. Benci et al. claim that all three arguments rest on a conflation of different models, but they do not. They are founded on the premise that similar events should have the same probability in the same model, or in one case, on the assumption that a single rotation-invariant distribution is possible. Having failed to refute the symmetry arguments on such technical grounds, one could deny their implicit premises, which is a heavy cost, or adopt varying degrees of instrumentalism or pluralism about regularity, but that would not serve the project of accurately modelling chances.

Optimization of Code Constructions of Binary Sequences Class on a Basis of Set-Theoretical Representation

In this paper there are solved two related problems: the development of the method and correspondent algorithm of code binary sequences synthesis with property of minimal level of side lobes of their aperiodic autocorrelation functions. The method is based on discrete representation of autocorrelation function in form of equations system considered on a set of integer numbers, set-theoretical representation of components of the sequence, their integer-number transformations, mutual properties and relations. There are obtained the expressions defining the dependences of absolute value of the sequence elements on a sum of levels of side lobes of aperiodic and periodic correlation functions; there are defined the necessary conditions of existence binary code sequences ; there are developed the mathematical models of binary code sequences components transformations. The efficiency criterion of each proposed algorithm is the relation of amount of all possible variants of binary code sequences of definite size l to amount of defined by proposed algorithms; the expressions for these quantities are obtained. The efficiency of proposed method and algorithms is verified by simulation results and it increases with increase of size l.

Model of e-learning with electronic educational resources of new generation

Purpose of the article: improving of scientific and methodical base of the theory of the е-learning of variability. Methods used: conceptual and logical modeling of the е-learning of variability process with electronic educational resource of new generation and system analysis of the interconnection of the studied subject area, methods, didactics approaches and information and communication technologies means. Results: the formalization complex model of the е-learning of variability with electronic educational resource of new generation is developed, conditionally decomposed into three basic components: the formalization model of the course in the form of the thesaurusclassifier (“Author of e-resource”), the model of learning as management (“Coordination. Consultation. Control”), the learning model with the thesaurus-classifier (“Student”). Model “Author of e-resource” allows the student to achieve completeness, high degree of didactic elaboration and structuring of the studied material in triples of variants: modules of education information, practical task and control tasks; the result of the student’s (author’s of e-resource) activity is the thesaurus-classifier. Model of learning as management is based on the principle of personal orientation of learning in computer environment and determines the logic of interaction between the lecturer and the student when determining the triple of variants individually for each student; organization of a dialogue between the lecturer and the student for consulting purposes; personal control of the student’s success (report generation) and iterative search for the concept of the class assignment in the thesaurus-classifier before acquiring the required level of training. Model “Student” makes it possible to concretize the learning tasks in relation to the personality of the student and to the training level achieved; the assumption of the lecturer about the level of training of a particular student is formed using the reporting data of model as management. Theoretical provisions of variation e-learning, including its base tasks (teaching, learning, management), and also order, formallogical and program means of development and most important requirements for an effective electronic educational resource are grounded; it is shown that the formalization of educational resources is possible on the basis of the structural-mathematical approach using the set-theoretical representations, that allow defining the features of the educational process and the role of participants in the language of sets and mapping between them, and also to receive “pedagogical diagnostics” of the course and learning outcomes, using the intellectual potential of each trainee both personally and in the classroom. An example of practical implementation of model of variation elearning with electronic educational resource of new generation is given. Conclusion: The developed theoretical provisions and basis components for the model of variation e-learning with electronic educational resource of new generation allow the student to realize functional mastering of knowledge, abilities and skills of trainees on the base of variation approach, and also to choose a personal student-centered way of “finding concept (meaning)” of the given tasks in the process of motivated learning based on formalized information thesaurus-classifier of electronic educational resource.

Моделювання логічної підсистеми маршрутизації залізничної станції на основі функціональної ознаки

Currently, relevant questions remain formalizing objects gridiron of railway stations, their connections and functional properties. Their decision is connected with the extension of automation capability for software configuration management systems at railway stations, as well as studies of their system properties, in particular - the safety of use and operational reliability. Traditional approaches based on the set-theoretical, graphic and matrix-analytic representation gridiron stations do not allow to generate the necessary logical relationship between the technological objects. As a result, the key issues of functioning of station automation systems related to the formation and implementation of routes remain unresolved in terms of their formalization. That is why the article describes how to perform such formalization by modelling the railway station routing subsystem on the basis of a functional feature of a single route. Results of the study can be used to automate the configuration software, as well as for further research of the system properties of electric centralization of switches and signals

Composite method of reliability research for hierarchical multilayer routing systems

The paper deals with the idea of a research method for hierarchical multilayer routing systems. The method represents a composition of methods of graph theories, reliability, probabilities, etc. These methods are applied to the solution of different private analysis and optimization tasks and are systemically connected and coordinated with each other through uniform set-theoretic representation of the object of research. The hierarchical multilayer routing systems are considered as infrastructure facilities (gas and oil pipelines, automobile and railway networks, systems of power supply and communication) with distribution of material resources, energy or information with the use of hierarchically nested functions of routing. For descriptive reasons theoretical constructions are considered on the example of task solution of probability determination for up state of specific infocommunication system. The author showed the possibility of constructive combination of graph representation of structure of the object of research and a logic probable analysis method of its reliability indices through uniform set-theoretic representation of its elements and processes proceeding in them.

A set-theoretical representation for weakly idempotent lattices and interlaced weakly idempotent bilattices

We give set-theoretical characterizations both for weakly idempotent lattices and interlaced weakly idempotent bilattices. In particular, we obtain a set-theoretical representation for interlaced bilattices and distributive bilattices (without bounds).

Rough Sets, Kernel Set, and Spatiotemporal Outlier Detection

Nowadays, the high availability of data gathered from wireless sensor networks and telecommunication systems has drawn the attention of researchers on the problem of extracting knowledge from spatiotemporal data. Detecting outliers which are grossly different from or inconsistent with the remaining spatiotemporal data set is a major challenge in real-world knowledge discovery and data mining applications. In this paper, we deal with the outlier detection problem in spatiotemporal data and describe a rough set approach that finds the top outliers in an unlabeled spatiotemporal data set. The proposed method, called Rough Outlier Set Extraction (ROSE), relies on a rough set theoretic representation of the outlier set using the rough set approximations, i.e., lower and upper approximations. We have also introduced a new set, named Kernel Set, that is a subset of the original data set, which is able to describe the original data set both in terms of data structure and of obtained results. Experimental results on real-world data sets demonstrate the superiority of ROSE, both in terms of some quantitative indices and outliers detected, over those obtained by various rough fuzzy clustering algorithms and by the state-of-the-art outlier detection methods. It is also demonstrated that the kernel set is able to detect the same outliers set but with less computational time.

Modeling Skill Dependence in Probabilistic Competence Structures

Abstract In knowledge space theory (KST) a competence structure is a set theoretical representation of the dependencies among a given set of skills. A probabilistic model for skill dependence is proposed which respects a precise correspondence requirement between set theoretical and probabilistic representations of skill dependence. An empirical application on integer subtraction problems at the primary school shows that the proposed model fits the data pretty well. Moreover, in a comparison with an unrestricted skill based version of the basic local independence model (BLIM), the proposed model fits better than this last, indicating that the restrictions implied by the correspondence requirement are not too strong.

Oriented matroid systems

The domination invariant has played an important part in reliability theory. While most of the work in this field has been restricted to various types of network system models, many of the results can be generalized to much wider families of systems associated with matroids. Previous papers have explored the relation between undirected network systems and matroids. In this paper the main focus is on directed network systems and their relation to oriented matroids. An oriented matroid is a special type of matroid where the circuits are signed sets. Using these signed sets one can e.g., obtain a set theoretic representation of the direction of the edges of a directed network system. Classical results for directed network systems include the fact that the signed domination is either +1 or −1 if the network is acyclic, and zero otherwise. It turns out that these results can be generalized to systems derived from oriented matroids. Several classes of systems for which the generalized results hold will be discussed. These include oriented versions of k -out-of- n systems and a certain class of systems associated with matrices.

Kolmogorov complexity and set theoretical representations of integers

AbstractWe 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 representation of integers that we suitably effectivize in order to develop an associated Kolmogorov theory. Such effectivizations are particular instances of a general notion of “self‐enumerated system” that we introduce in this paper. Our main result asserts that, with such effectivizations, Kolmogorov theory allows to quantitatively distinguish the underlying semantics. We characterize the families obtained by such effectivizations and prove that the associated Kolmogorov complexities constitute a hierarchy which coincides with that of Kolmogorov complexities defined via jump oracles and/or infinite computations (cf. [6]). This contrasts with the well‐known fact that usual Kolmogorov complexity does not depend (up to a constant) on the chosen arithmetic representation of integers, let it be in any base n ≥ 2 or in unary. Also, in a conceptual point of view, our result can be seen as a mean to measure the degree of abstraction of these diverse semantics. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

PSE: An Object-Oriented Representation for Modeling and Managing Part-of Relationships

Part-of relationships are needed to model structures of composite objects. Especially, composite objects, known as physical assemblies, are very popular in several applications. The existing object-oriented representations of physical assemblies and query languages based on them have several disadvantages. For example, they suppose that the user knows the structure of a physical assembly completely. This is not a realistic requirement in those cases where a physical assembly consists of a huge number of parts. Likewise the user has in them to control traversals of physical assemblies, i.e. they have a low degree of declarativity. Further, they do not support intensional and combined extensional-intensional queries which would be necessary in an advanced manipulation of physical assemblies. We develop the PSE (Part-of Structure Element) representation for removing the above disadvantages. The PSE representation is a set theoretical and constructor-oriented representation for physical assemblies and it can be used in the implementation based on both the object-oriented and deductive object-oriented database (DOOD) paradigms.

RELATIONAL DEDUCTIVE OBJECT-ORIENTED MODELING (RDOOM) APPROACH FOR FINDING, REPRESENTING AND INTEGRATING APPLICATION-SPECIFIC CONCEPTS

Next generation information systems (NGISs) have to support the manipulation of data-oriented, behavioral and deductive aspects of application domains. Many modeling methods, e.g. UML and other object-oriented modeling methods offer primitives for modeling data-oriented and behavioral aspects but they do not support the modeling of deductive aspects. In addition, NGISs may be implemented by several separate software/database systems that are based on different paradigms. Therefore it is not appropriate to use such a modeling method which is based on one paradigm. In NGISs it is essential to integrate the value-oriented approach with the object-oriented approach. We develop a relational deductive object-oriented modeling (RDOOM) approach for NGISs. Our goal is to combine into one modeling method, the navigation power of the relational model, the modeling power of the object-orientation and the inference power of the deductive (logical) framework. It is obvious that in NGISs complex and large specifications have to be embedded in application-specific concepts and structures that are defined beforehand for users to facilitate their query formulation. The detection and specification of application-specific concepts and structures means a new challenge for analysis methods. We show that on the basis of the primitives of RDOOM it is possible to represent this kind of application-specific information in a natural way. It is also necessary that the appropriateness and adequacy of application-specific concepts and structures can be tested before their expensive design and implementation phases. For the definition of these concepts and structures a diagrammatic representation typical of many modeling methods is not sufficient. Rather, a precise and executable representation is needed. Especially the complex and large derivation rules behind deductive concepts cannot be expressed precisely with diagrammatic representations. We develop set-theoretical representations for our primitives. The precise representation of the result of systems analysis also gives a more substantial starting point for the design and implementation of NGISs.

Comparing classification tree structures: A special case of comparingq-ary relations II

Comparing q-ary relations on a set of elementary objects is one of the most fundamental problems of classification and combinatorial data analysis. In this paper the specific comparison task that involves classification tree structures (binary or not) is considered in this context. Two mathematical representations are proposed. One is defined in terms of a weighted binary relation; the second uses a 4-ary relation. The most classical approaches to tree comparison are discussed in the context of a set theoretic representation of these relations. Formal and combinatorial computing aspects of a construction method for a very general family of association coefficients between relations are presented. The main purpose of this article is to specify the components of this construction, based on a permutational procedure, when the structures to be compared are classification trees.