Algebraic Logic Research Articles

Structural Derivations in Hilbert Algebras by Endomorphisms

This work introduces the concept of f-derivations in Hilbert algebras, exploring its theoretical foundation alongside a series of illustrative examples. We examine fundamental properties associated with f-derivations through rigorous analysis, shedding light on their algebraic structure and behaviour. In particular, we demonstrate that the kernel Kerdf(A) constitutes a near filter (subalgebra), while the fixed set Fixdf(f) forms a subalgebra within the Hilbert algebra A. These results provide new insights into the interaction between derivations and substructures in Hilbert algebras, offering potential avenues for further exploration in algebraic logic and related fields.

Adding an implication to logics of perfect paradefinite algebras

Abstract Perfect paradefinite algebras are De Morgan algebras expanded with an operation that allows for the full behavior of classical negation to be restored. They form a variety that is term-equivalent to the variety of involutive Stone algebras. Their associated multiple-conclusion (Set-Set) and single-conclusion () order-preserving logics are non-algebraizable self-extensional logics of formal inconsistency and undeterminedness determined by a six-valued matrix. We studied these logics extensively in Gomes et al. ((2022). Electronic Proceedings in Theoretical Computer Science357 56–76.) from both the algebraic and the proof-theoretical perspectives. In the present paper, we continue that study by investigating directions for conservatively expanding these logics with an implication connective (essentially, one that admits the deduction-detachment theorem). We first consider logics given by very simple and manageable non-deterministic semantics whose implication (in isolation) is classical. These, nevertheless, fail to be self-extensional. We then consider the implication realized by the relative pseudo-complement over the six-valued perfect paradefinite algebra. Our strategy is to expand the language of the latter algebra with this connective and study the (self-extensional) Set-Set and order-preserving and $\top$ -assertional logics of the variety induced by the resulting algebra. We provide axiomatizations for such new variety and for such logics, drawing parallels with the class of symmetric Heyting algebras and with Moisil’s “symmetric modal logic.” For the order-preserving Set-Set logic, in particular, we obtain a Set-Set axiomatization that is analytic. We close by studying interpolation properties for these logics and concluding that the new variety has the Maehara amalgamation property.

A 4-valued logic for double Stone algebras

This paper investigates the logical structure of the 4-element chain considered as a double Stone algebra. It has been shown that any element of a double Stone algebra can be identified as monotone ordered triplet of sets. As a consequence, we obtain the 4-valued semantics for the logic LD of double Stone algebras. Furthermore, the rough set semantics of the logic LD is provided by dividing the boundary region (uncertainty) into two disjoint subregions.

Logic Families

AbstractA logic family is a bunch of logics that belong together in some way. First-order logic is one of the examples. Logics organized into a structure occur in abstract model theory, institution theory and in algebraic logic. Logic families play a role in adopting methods for investigating sentential logics to first-order like logics. We thoroughly discuss the notion of logic families as defined in the recent Universal Algebraic Logic book.

AN OUTSPREAD ON VALUED LOGIC SUPERHYPERALGEBRAS

In all classical logical algebras, only two elements can be equated to one element, and this is not possible when we want to equate more than two elements to one or more than one element. In this study, we are looking for a new idea to cover this defect. This paper considers the logic algebra structures and generalizes them to superhyper logic algebra. Indeed, we extended the axioms of logic algebra to neutrosophic superhyper logic algebras.

On Neutrosophic Transitivity and Absorbent Filters of Basic Logic Algebras

On Neutrosophic Transitivity and Absorbent Filters of Basic Logic Algebras

Construction of a model for matching user's linguistic structures to a chat-bot language model

The research object of this work is the linguistic structures of the user when constructing a request to a chatbot with generative artificial intelligence. The study solved the task of improving the communication mediation algorithms of chatbots through the comparison models of linguistic structures by users. Sometimes the user intentionally or due to lack of information forms an inaccurate request. Formally, this is described by logical operations "And" and "And or Not". As a result of the research, a model was built comparing linguistic structures at the input with the information model of the response at the output. The model was based on an approach with recursive creation of an answer. That has made it possible to determine the basic characteristics of the object of the request and form an answer on this basis. Using this approach improved the accuracy of the chatbot response. It also made it possible to consider the linguistic structure of the user through its formalization. The use of logic algebra made it possible to find typical errors of users during dialogs with generative artificial intelligence. A feature of the reported advancement is that the comparison of models of linguistic structures of query formation is carried out through a recurrent algorithm. As a result, it makes it possible to compare the query in such a way as to reduce the absolute error of the primary data by 0.02 % and simplify the process of mathematical calculations. At the same time, the received information becomes more accurate – the number of references increases from 2 to 6 sources. The proposal could be used in practical activities to improve the natural language recognition technologies of users in chatbots with generative artificial intelligence. On this basis, it is possible to devise various applications and services for training and practical activities

Abduction in the Assessment of Special Educational Needs - Learning Disability

The diagnostic categories used to define learning disability are not standardized, and categorization systems are vague. This study aimed to explore the diagnostic methodology and strategies used to identify learning disabilities. The aim is to identify abductions in diagnostics in the field of special education. Interpreting diagnostics in remedial education using abduction can help identify learning disabilities more accurately. In the previous research phase, we conducted a meta-analysis of 11 expert reviews to identify abduction using fuzzy logic, fsQCA, and Boolean algebra. This study allowed for the creation of a new abductive diagnostic model. Based on these results, the reliability of the diagnostic process can be increased, and the diagnostic model can be used to detect learning disabilities or other types of problems and to identify sufficient conditions underlying a given phenomenon. Neither qualitative content analysis nor fsQCA revealed a relationship between all variables at a sufficient depth. Thus, in the present study, we moved on to Bayesian meshes, which shift and attempt to reorder previously identified variables based on conditional probability. We hypothesized that the Bayesian mesh and abduction application together may already be an efficient tool, which also anticipates the possibility of automation.

Laboratory work “Digital circuit design and algebra of logic: creation of digital devices in an online simulator”: competence-based approach for the purposes of continuity of education levels

Laboratory work “Digital circuit design and algebra of logic: creation of digital devices in an online simulator”: competence-based approach for the purposes of continuity of education levels

On Pre-Hilbert and Positive Implicative Pre-Hilbert Algebras

In the paper, pre-Hilbert algebras are defined as a generalization of Hilbert algebras (namely, a Hilbert algebra is just a pre-Hilbert algebra satisfying the property of antisymmetry). Pre-Hilbert algebras have been inspired by Henkin’s Positive Implicative Logic. Their properties and characterizations are investigated. Some important results and examples are given. Moreover, positive implicative pre-Hilbert algebras are introduced and studied, their connections with some algebras of logic are presented. The hierarchies existing between the classes of algebras considered here are shown.

Nearness Cosets of the Nearness Groups

In 2019, Öztürk et al. [Gamma semigroups on weak nearness approximation spaces, Journal of the International Mathematical Virtual Institute 9 (2019) 53–72] defined the first algebraic structure on weak nearness approximation spaces which is the gamma semigroup. After this study, the view on the nearness of algebraic structures has completely changed. This view was first expressed in a previous study in 2021 [M. A. Öztürk, Nearness [Formula: see text]-algebras, Journal of Algebraic Hyperstructures and Logical Algebras 2 (2021) 73–84]. Our aim is to give the concept of modulo in the nearness group [Formula: see text] as in the nearness group [Formula: see text], where [Formula: see text] is the set of integers. Afterward, based on the concept of modulo in the nearness group [Formula: see text], the nearness cosets of the nearness group [Formula: see text] are constructed and its basic properties are examined. Lastly, it is seen that Lagrange theorem is not valid for the nearness subgroups as compared to the usual subgroups.

Semiconic idempotent logic I: Structure and local deduction theorems

Semiconic idempotent logic is a common generalization of intuitionistic logic, relevance logic with mingle, and semilinear idempotent logic. It is an algebraizable logic and it admits a cut-free hypersequent calculus. We give a structural decomposition of its characteristic algebraic semantics, conic idempotent residuated lattices, showing that each of these is an ordinal sum of simpler partially ordered structures. This ordinal sum is indexed by a totally ordered residuated lattice, which serves as its skeleton and is both a subalgebra and nuclear image. We equationally characterize the totally ordered residuated lattices appearing as such skeletons. Further, we describe both congruence and subalgebra generation in conic idempotent residuated lattices, proving that every variety generated by these enjoys the congruence extension property. In particular, this holds for semilinear idempotent residuated lattices. Based on this analysis, we obtain a local deduction theorem for semiconic idempotent logic, which also specializes to semilinear idempotent logic.

A set-theoretic approach to algebraic L-domains

AbstractIn this paper, the notion of locally algebraic intersection structure is introduced for algebraic L-domains. Essentially, every locally algebraic intersection structure is a family of sets, which forms an algebraic L-domain ordered by inclusion. It is shown that there is a locally algebraic intersection structure which is order-isomorphic to a given algebraic L-domain. This result extends the classic Stone’s representation theorem for Boolean algebras to the case of algebraic L-domains. In addition, it can be seen that many well-known representations of algebraic L-domains, such as logical algebras, information systems, closure spaces, and formal concept analysis, can be analyzed in the framework of locally algebraic intersection structures. Then, a set-theoretic uniformity across different representations of algebraic L-domains is established.

Hyper L-algebras

In this paper, we introduced the notion of hyper [Formula: see text]-algebras as a generalization of [Formula: see text]-algebras. We studied basic properties of them and investigated the relationship among hyper [Formula: see text]-algebras and some other hyper logical algebras such as hyper ([Formula: see text],BCI) BCK-algebras and hyper hoops. In the following, we defined the notions of (weak) hyper ideals in hyper [Formula: see text]-algebras and then by considering the congruence relation (relative to a hyper ideal) on the hyper [Formula: see text]-algebra, we constructed the quotient hyper [Formula: see text]-algebras and investigated some properties of them.

Exploration and Practice of Project-based Classroom Teaching Reform with the Support of Knowledge Graph: A Case Study of the “Logic Algebra and Digital Circuits” Course

In response to the prominent issues in teaching the Logic Algebra and Digital Circuits course for undergraduate computer science students, such as insufficient student motivation, inadequate support for instructional design and organization, and ineffective project-based teaching, this study explores and implements innovative reforms in project-based classroom teaching supported by a knowledge graph. The reform is carried out by adopting the Outcome-Based Education (OBE) concept to design the course objectives, integrating and optimizing course resources, constructing a comprehensive and rich knowledge graph centered around key concepts, and developing a problem system that covers these key concepts. It establishes an individualized learning path to provide tailored and personalized services for students, creates a holistic learning environment that promotes intrinsic motivation with external support, and implements a 6-3-3 teaching organization strategy based on BOPPPS (Bridge-In, Outcomes, Pre-Assessment, Participatory Learning, Post-Assessment, and Summary). This approach stimulates a student-centered, eco-friendly teaching model through various methods. Through four rounds of teaching reform practice in the Logic Algebra and Digital Circuits course, significant improvements have been observed in student learning enthusiasm and engineering application abilities, and the outcomes have received broad recognition.

Semirings for probabilistic and neuro-symbolic logic programming

The field of probabilistic logic programming (PLP) focuses on integrating probabilistic models into programming languages based on logic. Over the past 30 years, numerous languages and frameworks have been developed for modeling, inference and learning in probabilistic logic programs. While originally PLP focused on discrete probability, more recent approaches have incorporated continuous distributions as well as neural networks, effectively yielding neuro-symbolic methods. We provide an overview and synthesis of this domain, thereby contributing a unified algebraic perspective on the different flavors of PLP, showing that many if not most of the extensions of PLP can be cast within a common algebraic logic programming framework, in which facts are labeled with elements of a semiring and disjunction and conjunction are replaced by addition and multiplication. This does not only hold for the PLP variations itself but also for the underlying execution mechanism that is based on (algebraic) model counting. In order to showcase and explain this unified perspective, we focus on the ProbLog language and its extensions.

On type-2 cyclic associative groupoids and inflationary pseudo general residuated lattices

This paper explores the relationship between fuzzy logic algebra and non associative groupoid. As a groupoid which can satisfy type-2 cyclic associative (T2CA) law, T2CA-groupoid is characterized by generalized symmetry. Fuzzy logic algebra is a major direction in the study of fuzzy logic. Residuated lattices are a class of fuzzy logic algebras with widespread applications. The inflationary pseudo general residuated lattice (IPGRL), a generalization of the residuated lattice, does not need to satisfy the associative law and commutative law. Moreover, the greatest element of IPGRL is no longer the identity element. In this paper, the notion of T2CA-IPGRL (IPGRL in T2CA-groupoid) is proposed and its properties are investigated in combination with the study of IPGRL and T2CA-groupoid. In addition, the generalized symmetry and regularity of T2CA-groupoid are investigated based on the characteristics of commutative elements. Meanwhile, the decomposition of T2CA-root of band with T2CA-unipotent radical is studied as well. The result shows that every T2CA-root of band is the disjoint union of T2CA-unipotent radicals.

Classification of L-algebras up to order 4

L-algebras, connected to algebraic logic and quantum structures, were first introduced by Rump [13]. In [5], the number of some finite L-algebras is counted. But classifying and finding examples of L-algebras is not always an easy task and is very important. So, in this paper, we classify the set of all non-isomorphic L-algebras up to order 4. For this, we define the notion of partial and total conditions and by using them, we characterize all of the L-algebras of orders 2, 3 and 4. We prove that that there are 5 L-algebras of order 3 and 44 L-algebras of order 4, up to isomorphism.

Mathematical Model of an Automated System for Learning to Play the Piano

Purpose of research. Nowadays, the automation of the distance learning process or self-study is relevant. However, to increase its effectiveness, there is a need for a personality-oriented approach to the student. The creation of adaptive learning technologies is becoming a priority. Development of a mathematical model of an automated system for learning to play the piano was the purpose of this work.Methods. The article discusses one of the approaches to modeling educational processes and systems - the cybernetic approach. The use of this approach allows us to consider the model of the educational system from the point of view of internal blocks and their functions, it allows us to identify sets of input and output data and internal relationships of various components of the system. The mathematical model is based on the operations of the algebra of logic and set theory.Resalts. Based on the cybernetic approach, the article provides a functional diagram of the system, which clearly demonstrates the sequence of actions performed, as well as the flow of various sets of input and output data. The functionality of the blocks is described in the terminology of operators, which determine the sequence and the required number of questionnaire questions to determine the initial knowledge, skills, and abilities of the trainee and his goals, select the most optimal initial training plan, assess progress and adjust the initial training plan depending on the results of the student. For each operator, sets of inputs and outputs are defined. The description of the data shows what information the system must operate in order for it to work correctly and efficiently. Another significant result of the work was the mathematical model of the system obtained as a result of research, which determines the relationship between input and output data.Conclusion. The cybernetic approach to modeling educational activities, including in the system of additional education, allows you to structure complex logical connections in the educational system, to determine the sequence and options for the system. The model will form the basis of the information system for piano playing teaching.

Applying APOS Theory to Enhance Algebraic Logic Skills: Comparing Traditional Teaching and Computer-Assisted Instruction

This study aims to assess how effectively applied science students use equations to express connectivity in algebraic logic and understand their cognitive framework. This research measures students' understanding of mathematical logic connectors in line with the polytechnic curriculum and uses APOS theory as a basis. It compares the effectiveness of traditional teaching methods with the computer-assisted instruction (CAI) approach. The results show that students learning with CAI perform better in applying connectivity equations. These findings underscore the importance of students' cognitive schemes in understanding the relationship between algebra and logic. Students with a more mature conceptual understanding tend to overcome conceptual application difficulties better. For future research, it is suggested to explore the application of CAI methods in various learning contexts and other disciplines, as well as to assess its long-term impact on students' cognitive skills and mathematical understanding. Additionally, it is essential to compare the effectiveness of CAI with other innovative learning methods to determine the best approach to developing students' logic and algebra skills.