Set Of Algebraic Operations Research Articles

Algebraic Operations on Pythagorean neutrosophic sets (PNS): Extending Applicability and Decision-Making Capabilities

Pythagorean neutrosophic sets (PNS) have been recognized as a highly effective mechanism for managing situations characterized by indeterminacy and inconsistency within decision-making procedures. This paper delves into the examination of algebraic operations performed on PNS, thereby expanding their scope of application, and enhancing their utility. In this study, we put forth a set of algebraic operations that can be applied to PNS. These operations encompass addition, multiplication, scalar multiplication, and power. These operations facilitate the efficient manipulation and combination of PNS, thereby enhancing decision-making in scenarios characterized by uncertainty and vagueness. To demonstrate the efficacy of these operations, we will present several illustrative examples accompanied by corroborating proofs. The introduction of algebraic operations enhances the capabilities of PNS, thereby creating opportunities for their practical application.

Classic Probability Revisited (I): Mathematical Models of an Extended Probability Theory

Part I of this paper presents a set of extended mathematical models of probability theory in order to explain the nature, properties, and rules of general probability. It is found that probability is a hyperstructure beyond those of the traditional monotonic and one-dimensional discrete structures. The sample space of probability is not invariant in general cases. Types of vents in the sample space may be refined as joint or disjoint and dependent, independent, or mutuallyexclusive. These newly identified properties lead to a three-dimensional dynamic model of probability structures constrained by the type of sample spaces, the relation of events, and the dependency of events. A set of algebraic operators on the mathematical structures of the general probability theory is derived based on the extended mathematical models of probability. It is revealed that the Bayes’ law needs to be extended in order to fit more general contexts on variant sample spaces and complex event properties in fundamental probability theories. The revisited probability theory enables a rigorous treatment of uncertainty events and causations in formal inference, qualification, quantification, and semantic analysis in contemporary fields such as cognitive informatics, computational intelligence, cognitive robots, complex systems, soft computing, and brain informatics.

Fuzzy data modeling and algebraic operations in RDF

The Resource Description Framework (RDF), a language proposed by the World Wide Web Consortium (W3C), is a flexible model for representing information about resources on the Web and has been applied in many application areas. At the same time, information is often imprecise and uncertain in many real-world applications. The existing works of RDF fall short in the ability of modeling imprecise and uncertain data in RDF. In this paper, we introduce some basic notions of fuzzy RDF graph model based on fuzzy sets. Along with the fuzzy RDF model, a fuzzy RDF algebra is formally proposed and a set of algebraic operations are developed. We also present some algebraic equivalences based on data graph isomorphism which can be used to rewrite algebra expressions in a form that satisfies certain needs. Finally, we show how to apply the proposed algebraic operations to capture the queries expressed in SPARQL query language. Based on the proposed data model and algebraic operations, we discuss the implementation issues.

Towards the abstract system theory of system science for cognitive and intelligent systems

Basic studies in system science explore the theories, principles, and properties of abstract and concrete systems as well as their applications in system engineering. Systems are the most complicated entities and phenomena in abstract, physical, information, cognitive, brain, and social worlds across a wide range of science and engineering disciplines. The mathematical model of a general system is embodied as a hyperstructure of the abstract system. The theoretical framework of system science is formally described by a set of algebraic operations on abstract systems known as system algebra. A set of abstract structures, properties, behaviors, and principles is rigorously formalized in contemporary system theories. Applications of the formal theories of system science in system engineering, intelligent engineering, cognitive informatics, cognitive robotics, software engineering, cognitive linguistics, and cognitive computing are demonstrated, which reveals how system structural and behavioral complexities may be efficiently reduced in system representation, modeling, analysis, synthesis, inference, and implementation.

Analytical Standard Uncertainty Evaluation Using Mellin Transform

Uncertainty evaluation plays an important role in ensuring that a designed system can indeed achieve its desired performance. There are three standard methods to evaluate the propagation of uncertainty: 1) analytic linear approximation; 2) Monte Carlo (MC) simulation; and 3) analytical methods using mathematical representation of the probability density function (pdf). The analytic linear approximation method is inaccurate for highly nonlinear systems, which limits its application. The MC simulation approach is the most widely used technique, as it is accurate, versatile, and applicable to highly nonlinear systems. However, it does not define the uncertainty of the output in terms of those of its inputs. Therefore, designers who use this method need to resimulate their systems repeatedly for different combinations of input parameters. The most accurate solution can be attained using the analytical method based on pdf. However, it is unfortunately too complex to employ. This paper introduces the use of an analytical standard uncertainty evaluation (ASUE) toolbox that automatically performs the analytical method for multivariate polynomial systems. The backbone of the toolbox is a proposed ASUE framework. This framework enables the analytical process to be automated by replacing the complex mathematical steps in the analytical method with a Mellin transform lookup table and a set of algebraic operations. The ASUE toolbox was specifically designed for engineers and designers and is, therefore, simple to use. It provides the exact solution obtainable using the MC simulation, but with an additional output uncertainty expression as a function of its input parameters. This paper goes on to show how this expression can be used to prevent overdesign and/or suboptimal design solutions. The ASUE framework and toolbox substantially extend current analytical techniques to a much wider range of applications.

On Granular Algebra: A Denotational Mathematics for Modeling Granular Systems in Granular Computing

Granular computing is an emerging technology for modeling and implementing complex computing architectures, behaviors, and big data manipulations. This paper presents a recent development in denotational mathematics known as granular algebra, which enables a rigorous treatment of computing granules as a generic abstract mathematical structure and granular behaviors as a set of algebraic operations. A formal granule is introduced as a mathematical model that elicits a set of basic properties of computing granules. A set of algebraic operations on formal granules is rigorously defined for the relational, reproductive, and compositional operations. A real-world case study is presented that demonstrates how concrete granules and their algebraic operations are manipulated by granular algebra. This work demonstrates that granular algebra is not only a powerful conceptual modeling methodology for granular systems, but also a practical functional specification methodology for granular computing.

On Semantic Algebra: A Denotational Mathematics for Natural Language Comprehension and Cognitive Computing

Semantics is composed meaning of language expressions and perceptions at the levels of words, phrase, sentence, paragraph, and essay. The basic unit of formal semantics is a concept as a formal model of words in a natural language. Cognitive linguistics focuses on cognitive semantics and its interaction with underlying syntactic structures. A denotational mathematics known as semantic algebra is presented for rigorous semantics manipulations in cognitive linguistics. Semantic algebra deals with formal semantics of linguistics by a general mathematical model of semantics and a set of algebraic operators. On the basis of semantic algebra, language semantics may not only be deductively analyzed based on syntactic structures from the top down, but also be synthetically composed by the algebraic semantic operators from the bottom up at different levels of language units. Semantic algebra enables a wide range of applications in cognitive informatics, cognitive linguistics, computational linguistics, semantic computing, cognitive computing, machine learning, computing with words, as well as natural language analysis, synthesis, and comprehension.

Inference Algebra (IA)

Inference as the basic mechanism of thought is abilities gifted to human beings, which is a cognitive process that creates rational causations between a pair of cause and effect based on empirical arguments, formal reasoning, and/or statistical norms. It’s recognized that a coherent theory and mathematical means are needed for dealing with formal causal inferences. Presented is a novel denotational mathematical means for formal inferences known as Inference Algebra (IA) and structured as a set of algebraic operators on a set of formal causations. The taxonomy and framework of formal causal inferences of IA are explored in three categories: a) Logical inferences; b) Analytic inferences; and c) Hybrid inferences. IA introduces the calculus of discrete causal differential and formal models of causations. IA enables artificial intelligence and computational intelligent systems to mimic human inference abilities by cognitive computing. A wide range of applications of IA are identified and demonstrated in cognitive informatics and computational intelligence towards novel theories and technologies for machine-enabled inferences and reasoning. This work is presented in two parts. The inference operators of IA as well as their extensions and applications will be presented in this paper; while the structure of formal inference, the framework of IA, and the mathematical models of formal causations has been published in the first part of the paper in IJCINI 5(4).

Inference Algebra (IA)

Inference as the basic mechanism of thought is one of the gifted abilities of human beings. It is recognized that a coherent theory and mathematical means are needed for dealing with formal causal inferences. This paper presents a novel denotational mathematical means for formal inferences known as Inference Algebra (IA). IA is structured as a set of algebraic operators on a set of formal causations. The taxonomy and framework of formal causal inferences of IA are explored in three categories: a) Logical inferences on Boolean, fuzzy, and general logic causations; b) Analytic inferences on general functional, correlative, linear regression, and nonlinear regression causations; and c) Hybrid inferences on qualification and quantification causations. IA introduces a calculus of discrete causal differential and formal models of causations; based on them nine algebraic inference operators of IA are created for manipulating the formal causations. IA is one of the basic studies towards the next generation of intelligent computers known as cognitive computers. A wide range of applications of IA are identified and demonstrated in cognitive informatics and computational intelligence towards novel theories and technologies for machine-enabled inferences and reasoning.

The complexity of soft constraint satisfaction

Over the past few years there has been considerable progress in methods to systematically analyse the complexity of constraint satisfaction problems with specified constraint types. One very powerful theoretical development in this area links the complexity of a set of constraints to a corresponding set of algebraic operations, known as polymorphisms. In this paper we extend the analysis of complexity to the more general framework of combinatorial optimisation problems expressed using various forms of soft constraints. We launch a systematic investigation of the complexity of these problems by extending the notion of a polymorphism to a more general algebraic operation, which we call a multimorphism. We show that many tractable sets of soft constraints, both established and novel, can be characterised by the presence of particular multimorphisms. We also show that a simple set of NP-hard constraints has very restricted multimorphisms. Finally, we use the notion of multimorphism to give a complete classification of complexity for the Boolean case which extends several earlier classification results for particular special cases.

Algebraic retrieval of fragmentarily indexed video

When dealing with long video data, the task of identifying and indexing all meaningful subintervals that become answers to some queries is infeasible. It is infeasible not only when done by hand but even when done by using latest automatic video indexing techniques. Whether manually or automatically, it is only fragmentary video intervals that we can identify in advance of any database usage. Our goal is to develop a framework for retrieving meaningful intervals from such fragmentarily indexed video data. We propose a set of algebraic operations that includes ourglue join operations, with which we can dynamically synthesize all the intervals that are conceivably relevant to a given query. In most cases, since these operations also produce irrelevant intervals, we also define variousselection operations that are useful in excluding them from the answer set. We also show the algebraic properties possessed by those operations, which establish the basis of an algebraic query optimization.

A framework for design of multivalued logic functions and its application using CMOS ternary switches

Two important problems in implementing multi-valued logic (MVL) as compared to binary logic (BL) are the lack of an efficient logic minimization technique and larger chip area and power consumption for realizing an MVL function. In this paper, a new theory for the implementation of MVL is proposed in an attempt to address these problems. In the proposed theory, an MVL function is decomposed into a set of subfunctions that can be efficiently realized using switches that are multivalued-in-nature. Each switch consists of a group of more elementary switches, called subswitches. A complete set of algebraic operators and relations are presented to facilitate the construction of the switches from a set of subswitches. The proposed algebra can be used to minimize the number of subswitches required for each switch in a function realization. Application of the proposed theory to implementing a ternary logic (TL) truth table is illustrated which is expected to encourage further investigation into exploring the possibility of using TL as a competitor to BL. The realization of subswitches is done using CMOS transistors that has potential for a VLSI implementation which can have a small chip area and consume low power.

Graphical query formulation for an entity-relationship model

A query language is defined in which users graphically manipulate entity-relationship (ER) diagrams to formulate queries. The theoretical foundation for the graphical language is algebraic. Each algebraic operator maps an ER diagram to an ER diagram, and every ER diagram represents a possible user query. A set of operators is formally defined and a suggested interactive implementation is given. The set of algebraic operators is both entity-set conservative, which guarantees that a user can interpret any transformed diagram in terms of known entity sets, and relationally complete, which guarantees an acceptable level of retrieval power.

Multidimensional Polynomial Algebra for Bubble Circuits

The paper expands the basic concepts of the coding theorists in the representation of data strings by algebraic polynomials, and develops the representation of both time and location of individual binary positions in the same polynomial. Further, it advances a set of algebraic operations on such polynomials to correspond to the various subfunctions that are accomplished in the actual domain circuits. The specific applications of the techniques proposed in this paper for the design and synthesis of such circuits is presented in a companion paper.