<title>Granular language and its reasoning</title>

CHAPTER VII - AXIOMATIC SET THEORY

8 - BOOLEAN ALGEBRA

The use of small-scale machinery proves to be common for agricultural, transportation and other types of work on small-scale farms, peasant farms, family gardens, farmsteads, on hilly lands and inconvenient areas, where the use of traditional powerful machinery is impossible or economically impractical. Small-scale machinery as well as powerful agricultural machinery rarely passes state tests. One of the main reasons for that is a lack of indicator systems for agrotechnical, operational and technological, reliability, economic and ergonomic assessment. The necessity of determining such systems (sets, criteria, indicators) is argued to be important. (Research purpose) The research aims to identify performance assessment indicators for agricultural small-scale machinery by capitalizing on the existing indicators for powerful machines and hands-on experience of exploiting mini-machinery in small farms of Uzbekistan. (Materials and methods) The primary materials are the performance indicators for the agricultural machinery, provided by the existing state and industry standards. The study uses the methods of system analysis, logical thinking, monographic observation and set theory. (Results and discussion) As a result of ranking, several performance indicator systems were selected for the agrotechnical, operational-technological, reliability, economic and ergonomic assessment of small-scale machinery. (Conclusions) It was found out that for the general assessment of the structure parameters, the quality of the technological process and the functionality of small-scale machinery prototypes, it is reasonable to use 26 main and 14 auxiliary indicators. At the same time, a reduction in material and labor resources is expected in the process of testing.

This paper introduces an algorithm for the automatic classification of surface targets based on data from passive radar stations. The primary focus is on developing an end-to-end classification approach that minimizes the loss of potential solutions by accounting for all possible classification scenarios of radio-electronic systems (RES) at intermediate stages. To describe the parameters of radio emissions, n-dimensional vectors are utilized, where discrete and continuous parameters are modeled through intervals. This allows for the consideration of emission variability under real-world conditions. The algorithm is grounded on set theory and probabilistic methods, adapted to scenarios where narrow probabilistic distributions are unavailable. The classification process consists of two stages: identifying RES based on their radio emissions and classifying carriers using combinations of RES. The key advantage of the proposed approach is its ability to preserve all classification options and incorporate them into the decision-making process for selecting the recommended solution. Examples of the algorithm’s application illustrate its effectiveness in real-world scenarios. Computational costs are analyzed, and potential directions for further optimization are identified to enable integration into automated surface surveillance systems.

The principles of solving the multicriteria problem of retail marketing for the rational choice of the location of trade enterprises (alternatives) based on the classical method of analysis of Saaty hierarchies, using the theory of fuzzy sets, are considered. The choice of the best alternative is carried out on the basis of the arguments of the maxima (abbreviated arg max), as the arguments, which are evaluated based on the operation of intersection of fuzzy sets of criteria. This approach is very practical from the point of view of using the experience and intuition of marketing workers and requires minimal costs to solve similar problems.KeywordsCommercial establishmentsRetail marketingMulti-criteria decision-makingSaaty hierarchy analysisFuzzy logicFuzzy inference sets

Attribute reduction methods in fuzzy rough set theory: An overview, comparative experiments, and new directions

The aim of this paper is to discuss the significance of a certain set-theoretic invariant г and the relation of quotient-equivalence for separable abelian p-groups of cardinality ωl. By means of these tools we gain new insight into the abundance of such groups. Both of these tools were originally introduced for the study of almost free groups (cf. [E1]).KeywordsDirect SummandStationary SubsetPure SubgroupBasic SubgroupLimit OrdinalThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The aim of this paper is to discuss the significance of a certain set-theoretic invariant г and the relation of quotient-equivalence for separable abelian p-groups of cardinality ωl. By means of these tools we gain new insight into the abundance of such groups. Both of these tools were originally introduced for the study of almost free groups (cf. [E1]).

Purpose. The scientific paper highlights improving the efficiency of export rail transportation of grain cargoes in Ukraine by introducing shipper routing and concentration of loading at the terminal grain elevators. Methodology. According to the experience of the USA and Canada, one of the most effective ways to reduce costs in the grain to-port supply chain is a shipper routing of the rail traffic. Shipper routing for transportation of grain cargoes involves the concentration of their loading on the multiple junctions. The junctions are proposed to be selected with the use of cluster analysis methods. For the formation of the grain loading concentration areas the authors used methods of set theory and multi-criteria optimization. Findings. Based on agglomerative cluster analysis algorithm, the junctions on a network of Ukrainian railways are selected and the areas of possible concentration of grain loading are formed. DSU-algorithm allowed distinguishing the overlapping and non-overlapping areas of concentration. The problem of selecting non-overlapping areas of the grain loading concentration is formalized as the problem of multiobjective integer programming with boolean variables. The solution of this problem by a modified simplex algorithm allows selecting on the railway network of Ukraine 24 districts of possible grain loading concentration, which cover 70 stations and at minimal additional cost provide routing of about 7.5 million tons of grain per year. Originality. The originality of the work lies in the fact that the authors developed the mathematical procedure for selection of junctions and concentration areas of grain loading at the Ukrainian railway network, taking into account the economic efficiency of the process. Practicalvalue. Application of the developed method of grain loading concentration for the formation of unit trains will significantly reduce the logistics costs in the supply chain of grain to Ukrainian ports for export and, consequently, increase its competitiveness in foreign markets.

In the article, we propose a new approach to creating a clustered software structure that provides the greatest protection against gaining an attacker's control over system support for decision-making, as well as high resistance to corruption and burglary. An approach is considered to solve the problem of software information system security by clustering it into modules. The possibility of formal construction of such structures by methods of set theory and category theory is shown. It is proved that the set of modules satisfies all postulates and axioms of category theory, hence is a category. The conditions for the fractal representation of the automated information system from the level of structural modules of the system to the level of microinstructions with the identification of specific objects and morphisms at hierarchy levels are determined.

Introduction. When managing traffic at the packet level in modern telecommunication networks, it is proposed to use methods that transform a self-similar stochastic packet flow into a quasi-deterministic one. To do this, it is required to apply complex probabilistic laws of distribution of self-similar flows. From the literature, methods of balancing the network load are known, which, with the problem indicated above, contribute to increasing the efficiency of telecommunication systems. However, there is no strictly mathematical solution to find out the optimal probabilistic characteristics of the output flow, based on the input flow. The presented research is intended to fill this gap. Its objective is to create a method for determining the optimal probabilistic characteristics of the packet flow, using the minimum value of the proximity measure of the self-similar input and quasi-deterministic output flows.Materials and Methods. To solve the research problem, the parameters of the output flow distribution were selected so that the approximation function was close to 𝛿𝛿-function. The Kullback-Leibler divergence was used as a proximity measure of the input and output distributions of time intervals. Methods of set theory, metric spaces, multidimensional optimization, and teletraffic were used. The solution algorithm included minimization of the Kullback-Leibler divergence and the limit passage to 𝛿𝛿-function.Results. A probability distribution is shown — an approximation of 𝛿𝛿-function, which maintains the equality of time intervals of a quasi-deterministic output packet flow. A method for transforming a self-similar input flow into a quasideterministic output flow is presented. The Kullback–Leibler divergence was used as a measure of their proximity. The minimum of the Kullback-Leibler divergence between the input and output flows with a normal distribution was achieved in the case of equality of the mathematical expectations of these flows. Using the passage to the limit, it has been established that time interval T between packets of the quasi-deterministic output flow must be equal to the mathematical expectation of the time intervals between packets of the input self-similar flow. To obtain a quasi-deterministic flow, the passage to the limit is performed for the found value of the mathematical expectation at σ → 0.Discussion and Conclusion. The application of this method will reduce the negative impact of self-similarity of network traffic on the efficiency of the telecommunication network. The use of quasi-deterministic flows makes it possible to predict the load of network resources, which can be the basis for improving the quality of user service. Two difficulties associated with calculations and practical implementation of the solution are eliminated. Firstly, it is difficult to use the delta function as a function of the output flow distribution density. Secondly, there are no ideal deterministic flows in the operation of telecommunication networks. The proposed method has great potential in the design and optimization of communication networks.

This paper presents multi-parameter qualimetric assessment models and methods of methodological significance of the individual and collective research outcomes. The authors have substantiated that no scientometric parameters based on citation rating can fully reflect the significance of the individual and collective research outcomes. The authors believe that only those research outcomes can be deemed significant that can be applied in the educational process, namely, in the teaching content. In other words, the results of research activities (both collective and individual) should have methodological significance, i.e. be suitable for teaching content. Another point of importance highlighted by the authors is in the fact that in contrast to scientometric parameters based on citation, methodological significance of the scientific outcomes of a research group shows not only their social activity (influence on the scientific community), but the interrelation between science and education whose role is steadily increasing in the modern world. Practical significance of this study is in possibility of applying proposed criteria of the methodological significance in monitoring systems for research activities of scientific institutions (including higher educational institutions). Its theoretical significance is in further comprehension of issues connected with research efficiency boost among scientists, i.e. in its importance for the sociology of sciences. Methods: review of scientific and methodological literature, management practices in scientific and educational institutions, mathematical modeling, methods of set theory, qualimetry methods, and methods of mathematical statistics. The empirical stage of the study was carried out on the basis of higher educational institutions in the Krasnodar Territory. At this stage, the authors? assessment method for determining methodological significance of lecturing researchers? outcomes was being tested. The factual information about the outcomes of research activities was obtained from the leading scientometric system in Russia - the Russian Science Citation Index.

Suppose one wishes to construct, use, and maintain a database of facts about the real world, even though the state of that world is only partially known. In the artificial intelligence domain, this problem arises when an agent has a base set of beliefs that reflect partial knowledge about the world, and then tries to incorporate new, possibly contradictory knowledge into this set of beliefs. In the database domain, one facet of this situation is the well-known null values problem. We choose to represent such a database as a logical theory, and view the models of the theory as representing possible states of the world that are consistent with all known information. How can new information be incorporated into the database? For example, given the new information that “ b or c is true,” how can one get rid of all outdated information about b and c , add the new information, and yet in the process not disturb any other information in the database? In current-day database management systems, the difficult and tedious burden of determining exactly what to add and remove from the database is placed on the user. The goal of our research was to relieve users of that burden, by equipping the database management system with update algorithms that can automatically determine what to add and remove from the database. Under our approach, new information about the state of the world is input to the database management system as a well-formed formula that the state of the world is now known to satisfy. We have constructed database update algorithms to interpret this update formula and incorporate the new information represented by the formula into the database without further assistance from the user. In this paper we show how to embed the incomplete database and the incoming information in the language of mathematical logic, explain the semantics of our update operators, and discuss the algorithms that implement these operators.

338 BOOK REVIEWS necessary propaedeutic for what follows, and some help is needed for the student to accomplish this successfully. The expanded Companion might also touch more on matters of political philosophy and might note the polemic surrounding Popper's attack on the Republic. No Companion to the Republic, however rich in informative explanation and interpretation, can ever exhaust its great subject, and there is much need for a Companion which is as rich as possible. The Catholic University of America Washington, D.C. DOMINIC O'MEARA The Kaliim Cosmological Argument. By WILLIAM LANE CRAIG. New York: Harper & Row, 1979. Contemporary discussions of the cosmological argument generally follow the philosophical model provided by Aristotle and Aquinas, which first invokes an empirical fact about the world (contingent beings exist; there is something in motion) , seeks for a cause or explanation of that fact, notes that an infinite series of causal conditions ordered transitively cannot provide an adequate explanation of that fact, and concludes to the existence of a necessary being or first cause. William Craig helpfully reminds us that this constitutes only one general form the cosmological argument has taken, a form originated by the Arabic practitioners of falsafa. His interest lies in another form developed by practitioners of kaliim, a " whole movement within Arabic thought that might best be called Arabic scholasticism" (4). Craig summarizes the lcaliim cosmological argument, which contrary to the above is concerned with the temporal sequence of events, as follows: 1. Everything that begins to exist has a cause of its existence. 2. The universe began to exist. 3. Therefore the universe has a cause of its existence. (63) The first sixty pages of his book present the kaliim cosmological argument as defended by the 9th and 11th century Arabic philosophers al-Kindi and al-Ghiizali and the 10th century Jewish philosopher Saadia. His concern is to provide a clear presentation of the argument as originally and most forcefully developed, particularly with respect to the defense of premise 2, rather than a detailed critique of it. The remainder of the book is devoted to presenting and defending a contemporary version of the kaliim cosmological argument. With respect to the argument given above, premise 1 is taken by Craig as intuitively obvious. " The first premise is so intuitively obvious, especially when applied to the universe, that probably no one in his right mind really believes it to be false" (141). The villain of the piece is BOOK REVIEWS 889 always Hume, and in reply to Hume's critique Craig argues that " all Hume has really shown is that the [causal] principle ... is not analytic and that its denial, therefore, does not involve a contradiction or a logical absurdity." But it" seems intuitively to be really, if not logically, absurd" (145). Craig's defense of the principle is not so much a defense as an appeal. It would have been profitable to develop a critique of Hume's argument, for example, to show that Hume has confused epistemological with ontological conditions in his argument. The critical premise, however, in the argument is the second, and to this Craig devotes the bulk of his energies. He presents four arguments in its support, two of which are philosophical and two empirical. The first argument is: 4. An actual infinite cannot exist. 5. An infinite temporal regress of events is an actual infinite. 6. Therefore an infinite temporal regress of events cannot exist. (69) Craig spends most of his time defending premise 4, arguing that the actual infinite of mathematics-found especially in Cantor's system and set theory -does not describe the real world. It concerns the mathematical world and was never meant to apply to the real world. Indeed, he argues, were it so applied, absurdities of all sorts result. His argument here is painstaking and (to my mind) unexceptionable. Much less time is devoted to the defense of premise 5. Basically he contends that "the fact that the events do not exist simultaneously is wholly irrelevant to the issue at hand; the fact remains that since past events, as determinate parts of reality, are definite and distinct and can be numbered, they can be conceptually collected into a totality...

Do possible worlds really exist? Or is there one (actual) world, and infinitely many (im)possibilial Despite sounding like something out of science fiction fantasy or theoretical physics, depending on who you run with, the question of possible worlds has been debated in certain philosophical circles since the turn of the twenti eth century, ranging from set theory in mathematics to modal philosophy to semi otics.1 The link between the three disciplines lies in the ontological fiat, or performative, that brings a possible world into existence, however that existence may be defined. For Umberto Eco, possible worlds are at base cultural constructs, but, then again, so is the real world; for David Lewis, the founder of modal realism, possible worlds are just as real as the real world; and for David Hubert, one of the early developers of mathematical set theory, the set-theoretical universe, which un derpins possible worlds theory, quite simply is paradise.2 My project in this essay is to detect if (and how) the theory of possible worlds may underpin certain aspects of the narrative of drama and performance, and more specifically, the 1990 play, Worlds, by Canadian playwright John Mighton. In order to do so, I will interrogate the seminal article Eco published in 1978 on tex tual semiotics, Possible Worlds and Text Pragmatics: 'Un drame bien parisien', and consider how the theory of possible worlds is transferable to a narratology of drama and performance. In the course of this discussion, though, I also want to peel back two further layers and consider: (1) the way Eco's work depends upon and de parts from the possible worlds theories of philosophy, especially that of Lewis, and (2) the way Lewis and Eco depend upon mathematical set theory. Finally, I hope to put the whole thing back together by appraising how the play Worlds navi gates the theory(s) of possible worlds.3