Classical Set Theory Research Articles

Feature selection for multiset-valued data based on fuzzy conditional information entropy using iterative model and matrix operation

Multiset-valued data is a powerful tool to deal with missing feature values. Classical rough set theory is sensitive to noise in classification learning due to the stringent condition of equivalence relation. Thus, a class of fuzzy relations can be introduced to describe the similarity between samples in multiset-valued data. However, these fuzzy relations have deficiencies when they are used in the computation of fuzzy conditional information entropy. This paper proposes a new model for multiset-valued data by introducing two variable parameters: one controls the similarity between samples, the other dominates the distance between feature values. This model employs the iterative computation strategy to define fuzzy conditional information entropy that is computing by matrix. The notions of three fuzzy information entropies in a multiset-valued decision information system (MVDIS) are put forward. For measuring any added feature, the significance based on fuzzy conditional information entropy is introduced. Moreover, fuzzy conditional information entropy iterative model (FCIEI-model) is presented. Finally, feature selection in an MVDIS based on FCIEI-model and matrix operation is researched, and the relevant algorithm (denoted as FCIEIM-MO algorithm) is proposed. The experimental results demonstrate that FCIEIM-MO algorithm possesses good robustness and is superior to some feature selection algorithms. The experimental results demonstrate that FCIEIM-MO algorithm possesses good robustness and exhibit better performance compared with some feature selection algorithms.

A novel strategy to identify the species-specific peptide biomarkers in Pheretima aspergillum (E. Perrier) based on enzymatic digestion followed by LC-MS/MS methods

Guang Dilong [P. aspergillum (E. Perrier)], is an animal-derived traditional Chinese medicine made from the dried body of Pheretima aspergillum (E. Perrier) (TCM). Due to its widely application and high medical values, preparations of P. aspergillum (E. Perrier) may be adulterated by four other species, including three crucial Pheretima species [P. vulgaris (Chen), P. pectinifera (Mkhaeken), and P. guillemi (Michaelsen)] and one considerable adulteration [Metaphire magna (Chen)]. This study developed a novel and effective strategy for analyzing and authenticating Guang Dilong based on enzymatic digestion of protein. The nanoLC-MS/MS technique used to evaluate complete peptidomics profiles of trypsin-digested samples, resulting in the identification of species-specific peptide biomarkers in P. aspergillum (E. Perrier). The significance of different samples and peptides in the target species set was then investigated using mathematical set theory. Consequently, seven peptides were chosen as prospective biomarkers. Finally, five specific peptide biomarkers for differentiating Guang Dilong with other species were confirmed and validated using UFLC-MS/MS and MRM mode. The suggested technique may also be beneficial in evaluating the quality of other animal-derived goods for safety issues in order to avoid misidentification.

Quantum Circuit Components for Cognitive Decision-Making

This paper demonstrates that some non-classical models of human decision-making can be run successfully as circuits on quantum computers. Since the 1960s, many observed cognitive behaviors have been shown to violate rules based on classical probability and set theory. For example, the order in which questions are posed in a survey affects whether participants answer 'yes' or 'no', so the population that answers 'yes' to both questions cannot be modeled as the intersection of two fixed sets. It can, however, be modeled as a sequence of projections carried out in different orders. This and other examples have been described successfully using quantum probability, which relies on comparing angles between subspaces rather than volumes between subsets. Now in the early 2020s, quantum computers have reached the point where some of these quantum cognitive models can be implemented and investigated on quantum hardware, by representing the mental states in qubit registers, and the cognitive operations and decisions using different gates and measurements. This paper develops such quantum circuit representations for quantum cognitive models, focusing particularly on modeling order effects and decision-making under uncertainty. The claim is not that the human brain uses qubits and quantum circuits explicitly (just like the use of Boolean set theory does not require the brain to be using classical bits), but that the mathematics shared between quantum cognition and quantum computing motivates the exploration of quantum computers for cognition modeling. Key quantum properties include superposition, entanglement, and collapse, as these mathematical elements provide a common language between cognitive models, quantum hardware, and circuit implementations.

Research on internal control optimization of Financial Sharing Center Based on fuzzy mathematics theory and quality function deployment theory

Through scientific theoretical methods, we take the internal control optimization of the Financial Sharing Center of H company as the research object. Firstly, we introduce the Financial Sharing Center and the development background and research significance of internal control under this mode, sort out the existing international research and related concepts, analyze the problems existing in the internal control stage of the Financial Sharing Center, and analyze the problems one by one from the five elements of internal control. What is more innovative is that we use the quality function deployment theory in the field of system science, combined with the intuitionistic fuzzy set theory, G1 method and entropy method in fuzzy mathematics to evaluate the five elements affecting the internal control optimization of the Financial Sharing Center of H company, and give the priority of optimization in theory. Finally, according to the implementation conditions of the Financial Sharing Center, this paper puts forward relevant countermeasures and suggestions to optimize the internal control of the financial sharing mode of H company, which can also provide experience for other enterprises that are building the Financial Sharing Center.

A Review of Neutrosophic Linear Programming Problems Under Uncertain Environments

This review article focuses on the integration of Neutrosophic Set Theory and Extended Fuzzy Set Theory in the context of Linear Programming (LP) problems. Neutrosophic set theory deals with uncertain, imprecise, and indeterminate information, while Extended Fuzzy Set Theory extends the classical fuzzy set theory to handle more complex and nuanced membership degrees. The combination of these two frameworks provides a powerful toolset for modeling and solving LP problems in environments where uncertainty and ambiguity are prevalent. This review aims to analyze and summarize the existing literature on Neutrosophic Linear Programming problems in extended fuzzy environments, exploring the theoretical foundations and practical applications. The review article seeks to contribute to the understanding of these integrated approaches and their potential for addressing decision-making problems under complex and uncertain conditions.

Attribute Reduction for Information Systems via Strength of Rules and Similarity Matrix

An information system is a type of knowledge representation, and attribute reduction is crucial in big data, machine learning, data mining, and intelligent systems. There are several ways for solving attribute reduction problems, but they all require a common categorization. The selection of features in most scientific studies is a challenge for the researcher. When working with huge datasets, selecting all available attributes is not an option because it frequently complicates the study and decreases performance. On the other side, neglecting some attributes might jeopardize data accuracy. In this case, rough set theory provides a useful approach for identifying superfluous attributes that may be ignored without sacrificing any significant information; nonetheless, investigating all available combinations of attributes will result in some problems. Furthermore, because attribute reduction is primarily a mathematical issue, technical progress in reduction is dependent on the advancement of mathematical models. Because the focus of this study is on the mathematical side of attribute reduction, we propose some methods to make a reduction for information systems according to classical rough set theory, the strength of rules and similarity matrix, we applied our proposed methods to several examples and calculate the reduction for each case. These methods expand the options of attribute reductions for researchers.

Lady Chandos and the Humanity Function: Reading the Postscript to J. M. Coetzee’s "Elizabeth Costello" with Lacan and Badiou

Badiou has said that his entire philosophical project stems from the need to “update” the traditional philosophical categories of Truth, Being, the Infinite and the Universal in the wake of the 19th Century German mathematician Georg Cantor’s explication of transfinite set theory. In his essay, “What is Love?”, he provides an account of one of the ways in which a post-Cantorian reconfiguration of the ontological status of the category of Woman might operate. This is given in the form of a postscript subtitled “The Feminine Position and Humanity” wherein he gives Lacan’s conception of supplementary feminine jouissance “an extra turn of the screw”. This essay draws on Badiou’s reconfiguration of the philosophical category of Woman to examine the implications of another postscript, that which abruptly concludes the 2003 novel Elizabeth Costello by J. M. Coetzee. It reads Coetzee’s invention of the “Letter of Elizabeth, Lady Chandos” with recourse to Lacanian psychoanalytic theory and on through the ontological reconfiguration of the philosophical category of Woman demonstrated by Badiou. It argues that Coetzee assembles, through the intervention of the postscript, a situation that “makes it possible to argue”, along with Badiou, that, for the woman position, love knots together the four types of truths – Politics, Art, Science, and Love (2008, 196).

Lady Chandos and the Humanity Function: Reading the Postscript to J. M. Coetzee´s Elizabeth Costello with Lacan and Badiou

Badiou has said that his entire philosophical project stems from the need to “update” the traditional philosophical categories of Truth, Being, the Infinite and the Universal in the wake of the 19th Century German mathematician Georg Cantor’s explication of transfinite set theory. In his essay, “What is Love?”, he provides an account of one of the ways in which a post-Cantorian reconfiguration of the ontological status of the category of Woman might operate. This is given in the form of a postscript subtitled “The Feminine Position and Humanity” wherein he gives Lacan’s conception of supplementary feminine jouissance “an extra turn of the screw”. This essay draws on Badiou’s reconfiguration of the philosophical category of Woman to examine the implications of another postscript, that which abruptly concludes the 2003 novel Elizabeth Costello by J. M. Coetzee. It reads Coetzee’s invention of the “Letter of Elizabeth, Lady Chandos” with recourse to Lacanian psychoanalytic theory and on through the ontological reconfiguration of the philosophical category of Woman demonstrated by Badiou. It argues that Coetzee assembles, through the intervention of the postscript, a situation that “makes it possible to argue”, along with Badiou, that, for the woman position, love knots together the four types of truths – Politics, Art, Science, and Love (2008, 196).

Information-theoretic partially labeled heterogeneous feature selection based on neighborhood rough sets

With the rapid increase of large-scale, real-world datasets, it becomes critical to address the problem of partially labeled heterogeneous feature selection (i.e., some samples, which own numerical and categorical features, have no labels). Existing solutions typically adopt linear correlations between features. In this paper, three different monotonic uncertainty measures are defined on equivalence classes and neighborhood classes to study the partially labeled heterogeneous feature selection by exploring the nonlinear correlations. First, consistent entropy and monotonic neighborhood entropy, based on classical rough set theory and neighborhood rough set theory, are proposed to construct a uniform measure for feature selection in heterogeneous datasets. Furthermore, a maximal neighborhood entropy strategy is developed by considering the inconsistency of neighborhood classes described by the features and partial labels. Finally, two feature selection algorithms are presented by three novel monotonic uncertainty measures. The comparative experiments demonstrate the effectiveness and superiority of the newly proposed feature selection measures.

DC BOOST CONVERTER USING A FUZZY CONTROL SYSTEM, SIMULATED IN SIMULINK

Decisive for the operation of voltage, current and power converters is the choice of control method. Control of complex dynamic systems in conditions of insufficient or fuzzy information requires the involvement of non-standard approaches during the construction of the control system. There are two possible ways out of the traditional framework of linear control. The first, is to develop more accurate nonlinear models on which high-performance controller design can be based. However, when choosing this way, one has to develop very complex control algorithms, and, consequently, the controller will perform complex mathematical calculations, which can have a negative impact on the control speed and response. Another way is to use a neural network controller based on fuzzy logic. Fuzzy logic is a generalization of mathematical logic and set theory, which is a function of the object belonging to a set and can take any value in the range [0; 1], not only 0 or 1. In the case of fuzzy logic we don't need simulation, because the whole work on designing of the controller is reduced to the transformation of the rules included in the fuzzy logic into the algorithm of automatic control.

Constructing Initial Algebras Using Inflationary Iteration

An old theorem of Ad\'amek constructs initial algebras for sufficiently cocontinuous endofunctors via transfinite iteration over ordinals in classical set theory. We prove a new version that works in constructive logic, using "inflationary" iteration over a notion of size that abstracts from limit ordinals just their transitive, directed and well-founded properties. Borrowing from Taylor's constructive treatment of ordinals, we show that sizes exist with upper bounds for any given signature of indexes. From this it follows that there is a rich class of endofunctors to which the new theorem applies, provided one admits a weak form of choice (WISC) due to Streicher, Moerdijk, van den Berg and Palmgren, and which is known to hold in the internal constructive logic of many kinds of topos.

Towards Verifying UML Class Diagram and Formalizing Generalization/Specialization Relationship with Mathematical Set Theory

The Unified Modeling Language (UML) is considered the de facto standard for object-oriented software model development. This makes it appropriate to be used in academia courses at both the graduate and undergraduate levels of education. Some challenges to using the UML is academia are its large number of model concepts and the imprecise semantic of some of these concepts. These challenges are daunting for students who are being introduced to the UML. One approach that can be taken in teaching UML towards addressing these concerns is to limit the number of UML concepts taught and recognize that students may not be able to develop correct UML system models. This approach leads to research work that develop a limited set of UML model concepts that are fewer in number and have more precise semantics. In this paper, we present a new approach to resolve an aspect of this problem by simplifying the generalization/specialization semantics of the class diagram through the application of mathematical formality to usage of these class diagram concepts. Along with that, we discuss the progress of research in the area of verification of UML class models. This research work derives a core set of concepts suitable for graduate and undergraduate comprehension of UML modeling and defines more precise semantics for those modeling concepts. The applicable mathematical principles applied in this work are from the domains of set theory and predicate logic. This approach is particularly relevant for the pedagogy of software engineering and the development of software systems that require a high level of reliability.

Set-theoretic relations for metasets

ABSTRACTThe paper introduces partial set-theoretic relations for metasets and investigates their properties. Metaset is a concept of imprecise set designed to represent vague notions in Artificial Intelligence, whose idea stems from Boolean-valued techniques in classical set theory. The basic set relations are extended to functions valued in a Boolean algebra or unit interval. Important classical set theory axioms: extensionality and comprehension are formulated for metasets. The latter enables formal definitions of collections with blurred boundaries by using set-theory formulae. This facilitates representing and reasoning about imprecise data.

Sets, Properties and Truth Values: A Category-Theoretic Approach to Zermelo’s Axiom of Separation

In 1908 the German mathematician Ernst Zermelo gave an axiomatization of the concept of set. His axioms remain at the core of what became to be known as Zermelo-Fraenkel set theory. There were two axioms that received diverse criticisms at the time: the axiom of choice and the axiom of separation. This paper centers around one question this latter axiom raised. The main purpose is to show how this question might be solved with the aid of another, more recent mathematical theory of sets which, like Zermelo’s, has numerous philosophical underpinnings. Keywords: properties of sets, foundations of mathematics, axiom of separation, subobject classifier, truth values

A parallel rule-based approach to compute rough approximations of dominance based rough set theory

In many datasets, conditional attributes and decision classes are preference-ordered, however, the classical Rough Set Theory (RST) does not consider the preference order between the values of the attributes. An extension of RST known as a Dominance-based Rough Set Approach (DRSA) provides dominance relation in this regard. In DRSA, data analysis mainly depends on the calculations of lower and upper approximations and these two measures are computationally utilizing many resources i.e., time and memory, due to the consideration of preference order. In this paper, we have proposed a parallel technique for calculating DRSA approximation sets. The proposed method directly computes approximations by following heuristic rules without calculating dominance positive or negative relations. The proposed parallel approach is then compared with the conventional method of calculation of DRSA approximations and a recent another technique of parallel processing using ten UCI publicly available datasets. Results validated the efficiency and effectiveness of the proposed model. An average reduction of 83% was observed in execution time and 86% in memory consumption. The structural complexity of the algorithm also considerably reduced.

Soft Sets with Atoms

The theory of finitely supported structures is used for dealing with very large sets having a certain degree of symmetry. This framework generalizes the classical set theory of Zermelo-Fraenkel by allowing infinitely many basic elements with no internal structure (atoms) and by equipping classical sets with group actions of the permutation group over these basic elements. On the other hand, soft sets represent a generalization of the fuzzy sets to deal with uncertainty in a parametric manner. In this paper, we study the soft sets in the new framework of finitely supported structures, associating to any crisp set a family of atoms describing it. We prove some finiteness properties for infinite soft sets, some order properties and Tarski-like fixed point results for mappings between soft sets with atoms.

METHOD OF CONVERTING Z-NUMBER TO CLASSIC FUZZY NUMBER

Human beings have an amazing ability to make intelligent decisions based on inaccurate, unclear, or incomplete information. Forming such an opportunity is, in a sense, a difficult task. The proposed concept provides great opportunities for describing human knowledge and can be used in decision making in the process of working with fuzzy information. This article shows that the issue of converting a Z-number into a classical fuzzy number is very relevant. The article also presents a method for converting a Z-number to a fuzzy number using classical fuzzy set theory.

IoT Intrusion Detection Using Machine Learning with a Novel High Performing Feature Selection Method

The Internet of Things (IoT) ecosystem has experienced significant growth in data traffic and consequently high dimensionality. Intrusion Detection Systems (IDSs) are essential self-protective tools against various cyber-attacks. However, IoT IDS systems face significant challenges due to functional and physical diversity. These IoT characteristics make exploiting all features and attributes for IDS self-protection difficult and unrealistic. This paper proposes and implements a novel feature selection and extraction approach (i.e., our method) for anomaly-based IDS. The approach begins with using two entropy-based approaches (i.e., information gain (IG) and gain ratio (GR)) to select and extract relevant features in various ratios. Then, mathematical set theory (union and intersection) is used to extract the best features. The model framework is trained and tested on the IoT intrusion dataset 2020 (IoTID20) and NSL-KDD dataset using four machine learning algorithms: Bagging, Multilayer Perception, J48, and IBk. Our approach has resulted in 11 and 28 relevant features (out of 86) using the intersection and union, respectively, on IoTID20 and resulted 15 and 25 relevant features (out of 41) using the intersection and union, respectively, on NSL-KDD. We have further compared our approach with other state-of-the-art studies. The comparison reveals that our model is superior and competent, scoring a very high 99.98% classification accuracy.

CONSTRUCTING WADGE CLASSES

AbstractWe show that, assuming the Axiom of Determinacy, every non-selfdual Wadge class can be constructed by starting with those of level $\omega _1$ (that is, the ones that are closed under Borel preimages) and iteratively applying the operations of expansion and separated differences. The proof is essentially due to Louveau, and it yields at the same time a new proof of a theorem of Van Wesep (namely, that every non-selfdual Wadge class can be expressed as the result of a Hausdorff operation applied to the open sets). The exposition is self-contained, except for facts from classical descriptive set theory.

A study of fixed point sets based on Z-soft rough covering models

<abstract><p>Z-soft rough covering models are important generalizations of classical rough set theory to deal with uncertain, inexact and more complex real world problems. So far, the existing study describes various forms of approximation operators and their properties by means of soft neighborhoods. In this paper, we propose the notion of $ Z $-soft rough covering fixed point set (briefly, $\mathcal{Z}$-$\mathcal{SRCFP}$-set) induced by covering soft set. We study the conditions that the family of $ \mathcal{Z} $-$ \mathcal{SRCFP} $-sets become lattice structure. For any covering soft set, the $ \mathcal{Z} $-$ \mathcal{SRCFP} $-set is a complete and distributive lattice, and at the same time, it is also a double p-algebra. Furthermore, when soft neighborhood forms a partition of the universe, then $ \mathcal{Z} $-$ \mathcal{SRCFP} $-set is both a boolean lattice and a double stone algebra. Some main theoretical results are obtained and investigated with the help of examples.</p></abstract>