Mathematical Logic Research Articles

On computational properties of Cauchy problems generated by accretive operators

In this paper, we provide quantitative versions of results on the asymptotic behavior of nonlinear semigroups generated by an accretive operator due to O. Nevanlinna and S. Reich as well as H.-K. Xu. These results themselves rely on a particular assumption on the underlying operator introduced by A. Pazy under the name of “convergence condition”. Based on logical techniques from “proof mining”, a subdiscipline of mathematical logic, we derive various notions of a “convergence condition with modulus” which provide quantitative information on this condition in different ways. These techniques then also facilitate the extraction of quantitative information on the convergence results of Nevanlinna and Reich as well as Xu, in particular also in the form of rates of convergence which depend on these moduli for the convergence condition.

Using Descriptive Logic in Intelligent Educational Systems

The purpose of the article is to study and solve the problems of using the basic principles of the modern formalism of descriptive (terminological) logic in intelligent learning systems that use knowledge bases of educational content. The research methods are the main methodological approaches and technological tools for the development of intelligent learning systems. In particular, systemic and comparative analyses to identify the features of intelligent learning systems, the method of expert evaluation, which involves the analysis of literature and information resources, interviews and surveys of experts in the field of information technology and mathematical logic, as well as the process of developing and testing an intelligent system. The scientific novelty of the study is the development of a system for obtaining expert knowledge for an intelligent system that combines the formation of an expert knowledge base, and the formalism for representing which is descriptive logic. The proposed approach of formalized knowledge representation using descriptive logic can be used in the development of authors’ intelligent learning systems in the field of online education and cognitive skills development. Conclusions. The scope of application of descriptive logic in intelligent systems is constantly expanding, increasing the capabilities of these systems to formally derive new knowledge, including in conditions of incomplete certainty. Descriptive logic, used in intelligent learning systems, provides ample opportunities for building a conclusion (i.e., forming an answer), in particular in situations such as forming and answering various types of queries to the system’s knowledge base (e.g., fully defined and clearly formulated, incompletely defined and vaguely formulated, completely vague, incompletely defined and clearly formulated, completely vague, completely vague and vaguely formulated), helping to specify the query. To do this, it is necessary to formulate a rigorous number system (to build an axiomatic theory). Thus, as a means of solving semantic and logical problems, descriptive logic has become a unique and powerful mechanism for representing knowledge. The successful use of descriptive logic in the author’s intelligent educational system will ensure the improvement of the quality of the educational process; monitoring of the learning process and control of its results; will facilitate the integration of this system into many modern information educational systems; will provide various consultations, advice, interpretations, explanations regarding the educational content or educational process, as well as a wide range of topics of educational content and tests of different levels.

Analysis on the behavior of the logistic fixed effort harvesting model through the difference equation under uncertainty

ABSTRACT In this article, a logistic fixed effort harvesting model is architected in an imprecise, discrete dynamical frame of mathematical logic. The fuzzy difference equation explores the philosophy behind the computational structure that represents the underflowing discrete behaviour and uncertainty associated with the modelling through discrete calculus and fuzzy decision-making mechanisms. The nonlinear fuzzy difference equations with different initial conditions and coefficients as fuzzy numbers are manifested to recognize the model. Interestingly, the fuzzy difference equations identified in this article can be imparted into a system of crisp difference equations by the characterization theorem. The equilibrium points are traced, and their corresponding stability criteria are analyzed considering different fuzzy cases. The merits and applicability of the proposed theory have been validated through numerical simulation and graphical visualization.

The application of a novel variable-order fractional calculus on rheological model for viscoelastic materials

Viscoelastic materials have become popular for novel multifunctional materials in emerging modern technologies, which have complex elastic-viscous-plastic behaviors. The variable-order fractional derivative has been gradually applied to characterize such variable memory effects of complex physical systems, while existing definitions are somewhat controversial. The new Scarpi’s variable-order fractional calculus has rigorous mathematical logic compared with the existed definitions. However, its practicality and feasibility in engineering are ambiguous due to the fewer applications up to now. In this paper, we aimed at introducing the Scarpi’s variable-order fractional derivative into the rheological applications of viscoelastic materials. Based on the Scarpi’s variable-order fractional derivative, the fractional Zener model and fractional viscoelastic-visoplastic models are established, and the parameters sensitivity on stress-strain relation is discussed. Then, the proposed models are then applied to depict different rheological behaviors of polymers at various temperatures and strain rates, and the results show that they have excellent agreement for experimental data compared to the reference model. It is observed that the order increases linearly with strain of the viscoelastic model and decreases exponentially with strain of the viscoplastic model. Furthermore, the physical significance of the corresponding fractional order parameters is investigated. It is found that they have the variation relation with temperatures and strain rates. It reveals that the variable fractional order can be regarded as an index to characterize the evolution of mechanical behaviors of polymers.

Meningkatkan Hasil Belajar Siswa Melalui Pembelajaran Kooperatif Student Teams Achievement Division (STAD)

The aim of this research is to determine whether the application of Student Teams Achievement Divisions (STAD) cooperative learning can improve learning outcomes. This research was carried out at SMA Martia Bhakti Bekasi Jl. Jendral Sudirman Km. 32 Kayuringin Jaya Village, South Bekasi District, Bekasi City. In this research, the samples to be taken are class X.2 as an experiment with a total of 35 students, and class X.3 as a control class with a total of 35 students. So the total sample used was 70 students. The cooperative learning model requires students' skills and cooperation in their groups, training students in critical thinking so that students' ability to understand the subject matter presented can increase. Meanwhile, researchers chose STAD cooperative learning because STAD cooperative learning is the simplest cooperative learning and easiest to apply in classes that have never implemented cooperative learning. Thus, it can be concluded that providing the Student Team Achievement Division (STAD) type cooperative learning model has a positive effect on mathematics learning outcomes on the subject of Mathematical Logic. From the research results, it can be concluded that providing the Student Team Achievement Division (STAD) type cooperative learning model has a positive effect on mathematics learning outcomes on the subject of Mathematical Logic.

Electronic Proves as One of the Modern Independent Means of Evidence

Objective: the purpose of this study is a determination of the procedural status of electronic documents as electronic evidence and identification of gaps in legislation regarding their legal regulation.
 
 Theoretical framework: theoretical materials were based on international scientific publications, reports, and scientific papers. And also for a more complete and objective presentation of the problem being studied, also practical materials from criminal cases were used.
 
 Method: is a dialectical method of understanding general patterns and particular manifestations of the essence of phenomena of objective reality. The comparative legal method made it possible to qualitatively study foreign legislation from the point of view of the legal regulation of electronic evidence and their application in practice. The method of mathematical analysis and logic made it possible to analyze everything that could relate to electronic evidence, identifying their characteristics.
 
 Results and conclusion: electronic evidence is information in electronic digital form, suitable for communication, interpretation or processing. In the electronic digital environment, the main interacting objects are various kinds of information objects (data sets, files, programs, etc.) with a complex hierarchical structure. Data in electronic digital form are intended to transmit information through media, information systems and telecommunication networks, and in the event of a crime being committed against them, their protection is of primary interest, and the nature and content of this data fades into the background.
 
 Originality/value: the value of the study lies in the fact that, based on a comprehensive analysis, electronic evidence has a unique structure and ensuring their safety is one of the criteria for their use, as well as the possibility of introducing a copying function will expand the process of processing evidence. The concept of “computer information” is the author’s definition, which will allow us to consolidate electronic evidence as an independent type.

Zhao Xiaosheng’s Tai Chi:


 The implementation of China Reform and Opening-up policy in 1978 gave birth to the Chinese new wave music, with an idiosyncratic manner of rebelliousness, individuality and innovation that brought a break-through in Chinese new music of the past. In this paper, we focused on a representative piano solo work Tai Chi (1987) by composer Zhao Xiaosheng who explored the ancient Chinese philosophy I Ching and the Tang Dynasty musical form Tang Daqu using Allen Forte’s pitch class set theory. We analyzed how Zhao employed the Western sonata and Chinese Tang Daqu to reflect the yin-yang dualism, based on the composer’s statement in the interview, researchers’ analyses, and authors’ suggestion. By decoding the pitch-class permutations that correspond to the 64 hexagrams of I Ching, the mathematical logic becomes the main way to combine the I Ching with serial music. Nevertheless, the compatibility between musical expression and commentaries of hexagrams is not reflected in the piano solo Tai Chi. Therefore, we discussed the entanglement of a compositional identity formation based on contradicting elements from that of the traditional Chinese musical culture and Western art music, with questions about the accretion and authenticity in the contemporaneity and musicalization of the ancient I Ching. 

The process of mathematics students’ reflective abstraction in solving continuous random variable problems in the probability course

Reflective abstraction is a mental mechanism in mental constructs in which all the structures of mathematical logic are developed in the mind of the individual. The mental mechanisms in question are interiorization, coordination, reversal, encapsulation, and generalization. This research is classified as exploratory research with a qualitative descriptive approach. This research data was taken from three fifth-semester mathematics students. The instruments used in this study were researchers as key instruments, essay tests, and interview guidelines. Data analysis included: 1) preparing the data for analysis, 2) reading the entire data, 3) encoding the data, 4) describing the data and presenting it in the narrative, and 5) interpreting the data and testing the validity of the data using data triangulation. The research found that (1) The first participant went through four components of reflective abstraction: interiorization – coordination – encapsulation – generalization. (2) The second participant went through five components of reflective abstraction thinking namely interiorization – coordination – reversal – encapsulation – generalization. (3) The third participant failed to solve the problem because the thought process in his mental mechanism had not yet manifested coordination, encapsulation, and generalization. Researchers found that the mental construction that the first participant had gone through could illustrate that the minimum mental construction that must be passed in solving the problem of continuous random variables required components- interiorization, coordination, encapsulation, and generalization.

A Proof That the Set of NP-problems is Bigger Than the Set of P-problems by Using a Logical Consideration

The field of informatics is the domain that emerged by applying the mathematical logic on electronic devices called computers in order to simplify many tasks for humans. The application of informatics in all economic and scientific areas is the most important factor that made our civilization reach our current phase of development. Nowadays, the experts and even the beginners of informatics are eager to use quantum computers. However, there is still an unsolved problem of classical theoretical informatics in ordinary electronic computers. It is the famous philosophical problem of the “Millenium Prize” of the Clay Mathematics Institute concerning the complexity of problems that has been treated by many other researchers but without acceptable sufficient answers. A solution to this problem can make all the fields based on informatics make huge progress. And thus, thanks to my short studies about informatics, I present to you this mathematical proof that deals with the sets of P problems, NP-Complete problems and NP-Hard problems in the field of classical electronic computers in order to prove new formulas about the cardinals of each group of complexity problems and about the intersections of each one of these sets. The aim is to contribute to an acceptable solution for this Millenium Problem and the methodology is purely logical and mathematical. The readers won’t need any complicated notions from the background of previous informatics or mathematics research in order to understand the demonstrations of this article since the proof is based only on notions of sets by starting with easy logical considerations. Furthermore, you will find in this work a proof of an interesting theorem about the complexity of problems that allows us to identify NP-problems even if their algorithms have infinite time of execution. This paper ends by proving that the set of NP-Problems is definitely bigger than the set of PProblems. Hence, all the readers are invited to understand and develop this work by inspecting the applied logical considerations in order to succeed in finding a sufficient solution to the interesting Millenium Problem of complexity.

ASESMEN KOMPETENSI MINIMUM UPAYA DALAM MENINGKATKAN LITERASA BAHASA DAN NUMERASI DI SEKOLAH DASAR

Language and numeracy literacy are the basic skills required for students to analyze reading or numbers in different contexts. Literacy and numeracy skills must be mastered and possessed in order to increase competitiveness and improve the quality of human resources. However, what happens is the low reading ability of Indonesian school children known from the results of the PISA test. In 2018, Indonesia ranked 70th out of 78 countries. Students' difficulties in critical thinking are known due to the lack of students who can solve Higher Thinking Skills (HOTS) questions and teachers who have not been able to create HOTS-based questions. Therefore, the Ministry of Education and Culture decided to implement competency assessment for all schools. The purpose of this study is to provide insight into the concept of Minimum Competency Assessment (MCA) to improve reading, writing and arithmetic skills in primary schools. The research method used is literature research. Based on the review of the literature approach, it was found that AKM focuses more on the ability to reason using language or literacy and using mathematical logic or numeracy.

Analysis of the impact of public procurement indicators on the achievement of public policy goals

The study is devoted to the study of the dynamics of the achievement of indicators of the state macroeconomic policy of Russia during 2013—2022 and the assessment of the impact of public procurement spheres on this process. The article presents statistics of absolute and relative indicators of public policy, as well as gross indicators and effective indicators of public procurement. The article contains the results of a correlation analysis of the impact of public procurement indicators implemented under Federal Law No. 44-FZ of April 5, 2013 "On the Contract System in the Field of procurement of Goods, Works, Services for State and Municipal Needs" on the achievement of public policy targets, including the dynamics of GDP and changes in the level of budget savings in the implementation of public procurement activities.The scientific novelty of the study consists in the application of the mathematical and statistical method of correlation with respect to macroeconomic indicators of the functioning of the national economy of the Russian Federation to identify the direction and strength of the relationship between the studied areas of economic activity. An element of scientific novelty is also the justification of the time shift of the interdependent studied indicators of public policy and public procurement based on the logic of the process impact of public procurement activities on the results of public policy and information restrictions imposed by the current mechanism for collecting, summarizing and publishing official state statistics. The author's contribution to the development of scientific research devoted to the study and attempts to optimize public procurement activities as a tool for achieving the goals of public policy consists in the use of mathematical logic tools to substantiate the performance indicators of public procurement, providing the best economic results of public policy in the time period under study. Based on the results of the study, it is concluded that it is possible to establish and quantify the relationship between the gross indicators of the functioning of public procurement and the achievement of public policy indicators, including ensuring economic growth in terms of GDP dynamics and managing the level of budget savings generated as a result of public procurement activities.

Combining Building Block Process With Computational Thinking Improves Learning Outcomes of Python Programming With Peer Assessment

The capability of computer programming language logic is one of the basics of technical education. How to improve students “interest in program logic design and help overcome students” fears of coding has become vital for educators. Cultivating practical talents with information technology application and basic programming development will become one of the important topics in the department of information related science. The objective of this research is to improve the ability of learning basic programming courses by using Zuvio interactive software. Zuvio employs the mathematical logic of computational thinking to analyze problems and enhance learners’ interest in learning programming skills through a graphical interface tool with building blocks. It uses innovative interactive teaching to use peer and self-assessment to study the content of the course. Zuvio improves the design ability of different groups of class learning Python programming. In line with the innovative teaching policy of the schools and the current stage of the learner’s learning model, learning effectiveness can be achieved. The research results were analyzed by midterm and final experimental group scores, and the progress of the experimental group’s scores was examined through descriptive statistics. The average and standard deviation of the assessment were used to analyze the progress of the experimental group students in the programming course. In the classroom, assessment criteria were set up as the basis for peer assessment scoring. After the midterm and final exams, the teacher assessment and peer assessment scores were analyzed for cognitive differences, and possible learning differences were analyzed. The students’ professional ability was examined to see if it met the professional standards required by the course, and whether innovative teaching methods could improve the learning outcomes of learners with different professional backgrounds in Python programming.

Комплексная оценка трудноформализуемых вентиляционно-технологических процессов на угольных шахтах

Introduction. Automated process control systems, as a set of characteristics related to the ability of an object to satisfy certain needs, are one of the most important indicators of the quality of complex technical systems consisting of a large number of interacting mechanisms, apparatus and instruments. Their failure can lead to accidents and emergencies, resulting in large economic losses, destruction of industrial facilities, and sometimes human casualties. Methodology and research methods. The solution to the assigned problems is based on modern methods of artificial intelligence, automatic control theory, mathematical modeling of technological processes, and mathematical statistics; graph theory; decision theories; apparatus of mathematical logic; set theory and systems analysis. Discussion of research results. The article presents the results of studies of the patterns of ventilation and technological processes in coal mines. It has been established that studies of patterns in coal mines are becoming more relevant in connection with the formulation of the task of modernizing production and the Russian economy as a whole. It has been revealed that the methodological foundations for the synthesis of control decisions that ensure the achievement of a stable state make it possible to formulate acceptable organizational and managerial decisions and evaluate their effectiveness using metered characteristics of the proximity of vaguely described states. Conclusion. Based on the research carried out, new theoretical and applied problems were posed and solved, system connections and patterns of the ventilation technological process (VTP) were identified, models, methods and algorithms for automated ventilation technological process control systems (ACVTP) were developed, aimed at increasing the efficiency of the technical air supply system coal mines. The theoretical foundations for modeling and controlling ventilation and technological processes have been created, including concepts, models, methods and control algorithms. The research results can be useful in controlling ventilation and technological processes. Conclusions on the article. The article presents the results of studies of ETP conducted in coal mines. It has been established that, using the developed methodology, it is possible to: design high-tech equipment with specified quality indicators for intelligent control systems; choose from a variety of structural diagram options the best in terms of operational efficiency; carry out structural-parametric adjustment and change the types of mathematical models, both for an individual element and for the ETP as a whole; identify the weakest points in the HTP automated control system and adapt the methodology for controlling ventilation and technological processes, evaluate the effectiveness of using structural redundancy. Suggestions for practical applications and directions for future research. The research results can be useful when carrying out mining operations at all enterprises where air supply is necessary.

Учет переменного характера расчетных нормативов плана формирования поездов с использованием технологий вычислительного интеллекта

Purpose: The train formation plan is the most important logistics tool for cargo transportation management. But the use of discrete values of design standards in its development — the quantity of car-hours for the accumulation of trains and savings from passing of cars through technical station without processing — do not always guarantee the optimal solution due to the objectively existing unevenness of operational work. It is precisely this factor that generates uncertainty, which is not inherently stochastic, and necessitates adjustments to the train formation plan throughout its lifecycle. In addition to fluctuations in car traffic, it causes a change in the values of the calculated standards. To manage such uncertainty, there are special methods, one of which is fuzzy logic. The article describes a way to account for changes in all calculation standards of the train formation plan. Methods: The methods of one of the technologies of computational artificial intelligence are used — fuzzy logic, fuzzy sets and fuzzy mathematics. Results: The dependence has been established on how changes in the calculated parameters of the train formation plan are affected by fluctuations in car flows on specific destinations. The formulas obtained on the basis of known analytical expressions allow us to determine the values of the standards of the formation plan without using auxiliary tables and graphs. Practical significance: The use of the obtained dependencies in the development of the formation plan will improve the accuracy of its calculation by taking into account fluctuations not only of car traffic, but also the values of the calculated standards depending on them.

Education-oriented Proof Assistant Based on Calculational Logic: Proof Theory Algorithms and Assessment Experience

This work presents an interactive proof assistant, based on Dijkstra-Scholten logic, aimed at teaching logic and discrete mathematics in higher education. The assistant interface is web and easy to use, since inferences can be made just with the mouse. The educational experience is presented showing a correlation between the grades of the assessments in class and those made with the application web. Additionally, an algorithm proof theory for the Disjktra-Scholten system are made and the following algorithms are shown: 1) a versatile printing algorithm that allows the administrator to configure the symbols of a theory, by assigning the desired presentation with LaTeX; 2) An algorithm, based on Broda and Damas combinators, for generate monotonic or anti monotonic inferences in the Dijkstra-Scholten logic; 3) An algorithm to generate the proofs of dual theorems in Boolean Algebra theory.

Bertrand Russell’s Philosophical Logic and its Logical Forms

From 1901 to 1919, Russell persistently maintained that there were two kinds of logic and distinguished between one and the other as mathematical logic and philosophical logic. In this paper, we discuss the concept of philosophical logic, as used by Russell. This was only a tentative program that Russell did not clarify in detail; therefore, our task will be to make it explicit. We shall show that there are three (-and-a-half) kinds of Russellian philosophical logic: (i) “pure logic”; (ii) philosophical logic investigating the logical forms of propositions; (iii) philosophical logic exploring the logical forms of facts: in epistemology and in the external world. In particular, Russell’s program or philosophical logic of the facts of the external world remained less than sketchily outlined. Keywords: Russell, mathematical logic, philosophical logic, Wittgenstein

Russell Paradox Analysis: Descriptions and Logical Cases

The Russell Paradox is a classic problem in mathematical logic that challenges the underlying basics of set theory and the concept of infinity. This study provides a comprehensive analysis of the paradox, including its history, development, and recent research. The paradox arises from the set whose members are sets not containing themselves, which results in being contradictory. Bertrand Russell and Ernst Zermelo proposed early solutions, while the axiomatic set theory provided a rigorous foundation for mathematics. Recent studies suggest alternative solutions, including distinguishing between object-language and meta-language and introducing a new concept called a "Russell-class." This paper summarizes the history of the paradox, summarizes recent studies, and presents a framework for understanding and evaluating solutions. According to the analysis, it aims to deepen the understanding of the paradoxical situation and why it is of enormous importance for the underlying basics of mathematics. This paradox remains a subject of intense study and continues to inspire the development of various theories and frameworks to resolve it. Overall, these results shed light on guiding further exploration of the paradoxical situation and its enormous importance for the underlying basics of mathematics. They also present a framework for understanding and evaluating solutions, with the goal of deepening our understanding of the paradox and inspiring the development of various theories and frameworks to resolve the issue.

The umbilical cord of finite model theory

Abstract Model theory was born and developed as a part of mathematical logic. It has various application domains but is not beholden to any of them. A priori, the research area known as finite model theory would be just a part of model theory but it didn’t turn out that way. There is one application domain, relational database management, that finite model theory had been beholden to during a substantial early period when databases provided the motivation and were the main application target for finite model theory. Arguably, finite model theory was motivated even more by complexity theory. But the subject of this paper is how relational database theory influenced finite model theory. This is NOT a scholarly history of the subject with proper credits to all participants. My original intent was to cover just the developments that I witnessed or participated in. The need to make the story coherent forced me to cover some additional developments.

What Are Statistics: It's Time to set the Record Straight

The authors present their views on the modern concept of statistics as a science, academic discipline, and field of activity. The urgency of the matter lies in an adequate understanding of the essence of statistics, which determines the direction for its further evolution as a tool for increasing the effectiveness of the national socio-economic processes.According to the authors, by the beginning of the XXI century, de jure statistics are associated with activities, measured by data, that reflect a set of phenomena of diverse nature while ignoring de facto activities, associated with an assessment of the quantitative side of mass phenomena as part of information support of cognitive processes and management decision-making in the socio-economic sphere. In the former case, theoretical, methodological, and practical statistical activities are based on the consideration of the statistical population as such, for which is being developed an appropriate toolkit based primarily on the unity of methods of mathematical logic, mathematical statistics, and big data analysis. In the latter case, theoretical, methodological, and practical activities are interpreted within the framework of the concept implying that the quantitative side of mass social phenomena is the object of cognition and management as an objective reality. It is based on the following categories: a statistical indicator, a system of indicators, and an information model.Analysis of the main features of the two types of activity show that their convergence is impossible. The paper concludes that while maintaining the independent status of the first type of activity, called Statistics, it would be appropriate to acknowledge the independent status of the second type of activity, called Socio-economic statistics, as a type of occupation, a specialty and area of training in higher education, a scientific specialty – a branch of science, incorporating it in the professional standard «Statistician», the educational standard of higher education «Socio-economic statistics», and the standard of the scientific specialty «Socio-economic statistics».

Digital flipbook module in circulatory system to facilitate student different intelligent and learning style

Students acquire and improve various intellectual abilities through learning process facilitated by learning methods. The study provides information on the development process of digital flipbook modules to facilitate student intelligence and learning style for the circulatory system, determine its feasibility, and students’ responses. The method used was an R&D with 4D development models. It consisted of four stages: define, design, develop, and disseminate. However, this research was limited to the development stage. Development of digital flipbook modules based on nine types of intelligence: linguistic, mathematical logic, kinesthetic, visual-spatial, musical, interpersonal, intrapersonal, naturalist, and existential. The instruments used expert feasibility questionnaires (materials and media) and student response questionnaires. The result of the assessment test of experts in the material aspect was 96.4% in the very eligible category, and the media aspect was 92.8% in the very suitable category. The score of students’ responses was 90,14%, in the very eligible category. Based on the analysis results, the digital flipbook module was suitable biology teaching material for the circulatory system