Implication Operators Research Articles

An Approach to a Priori Assessment of Fuzzy Classification Models in Monitoring Tasks

The article addresses the problems of using automation tools to perform monitoring and management tasks applicable to assessing the quality of fuzzy classification models, where the classification procedure is implemented on the basis of knowledge (rules) in the absence of the training set. An approach is proposed to obtain a priori assessments of the classification quality based on the study of the used model sensitivity to changes in the values of internal parameters during the corresponding modeling. The interpretation of the modeling results in the form of risk assessment caused by the self-imperfection of the classification models is obtained. The article provides an example of a fuzzy classification model based on a comparison of the current state of a monitoring object described using fuzzy features with a set of predefined typical states, which form corresponding fuzzy equal (close) states (monitoring situations). The comparison is carried out using the fuzzy implication operation provided that the required reliability is met. The example of this model demonstrates how the type of implication operation, as well as the internal features of the model, affect the results of classification, and appropriate indicators are proposed, which are both an interpretation of generally accepted indicators for assessing the classification quality, and unique, inherent in the considered model. Computational experiments were carried out, which made it possible to obtain graphs of changes in classification quality assessment indicators for the considered model and its modification and visualize the influence of internal parameters of the model on the results of its application. A number of indicators are proposed that allow for an a priori assessment of the risks arising from the application of the model before its actual application.

Application of Fuzzy Reasoning with Fusion Rules in Intelligent Information Processing

The core of artificial intelligence (AI) research is intelligent information processing, and fuzzy reasoning and fuzzy neural networks are significant research areas in models of intelligent information processing. The study examines the robustness of fuzzy reasoning by using the property characteristics of triangular modal operators and implication operators, which are used to analyse the Lipschitz property of fuzzy operators and its corresponding property characteristics of triangular modal operators and implication operators. Inference operators are used to investigate the robustness and error-control capabilities of fuzzy operators for fuzzy reasoning. The results showed that the robustness of quasi-copula fuzzy reasoning with fusion rules is better, and the corresponding maximum output perturbation is lowest at 0.36 when the fuzzy operator is T[Formula: see text]&R[Formula: see text] and the corresponding maximum output error is lowest at 0.41 when the fuzzy operators are T[Formula: see text]&R[Formula: see text] and T[Formula: see text]&R[Formula: see text]. This shows the optimal performance of the fuzzy associative memory model with fuzzy rule for quasi-copula fuzzy operators and can improve the theoretical technology. The best-performance fuzzy copula operator of the fuzzy associative memory model can enhance the theoretical foundation and technical support for the processing of intelligent information.

The L-Fuzzy Prime Filter Degrees on Lattices and Its Induced L-Fuzzy Convex Structure

The aim of this paper is to examine the L-fuzzy prime filter degrees on lattices and their induced L-fuzzy convex structure. Firstly, the notion of L-fuzzy prime filter degrees on lattices is established using the implication operator when L is a completely distributive lattice. Secondly, an equivalent characterization of L-fuzzy prime filter degrees on lattices is provided. The equivalence relation, through the definitions of reflexivity, symmetry, and transitivity, provides a method for partitioning subsets within a lattice that possesses the prime filter property. Finally, the L-fuzzy convex structure induced by the L-fuzzy prime filter degrees on lattices is examined. Simultaneously, the properties of L-fuzzy prime filter degrees on lattices in relation to images and preimages under homomorphic mappings are discussed.

Some implication operators on bounded lattices

In this paper, first we give a construction method for implications by an arbitrary element on bounded lattice. We present some implications obtained from binary operations on the bounded lattices, that is, from basic logic connectives, viz., t-norms, t-conorms and negations. Finally, we examine the main properties of the constructed implications.

A study of rough inclusion on algebras with quasi-Boolean base

Pre-rough algebra has emerged from the rough set theory, which is a quasi-Boolean algebra with a few additional axioms. Four types of abstract rough inclusion are defined in pre-rough algebra which give rise to four different implication operators within this algebra. Properties of these four implications are studied from the angle of residuation. Logics of pre-rough algebra with respect to different implications are presented.

Regular Double p-Algebras: A Converse to a Katriňák Theorem and Applications

ABSTRACT In 1973, Katriňák proved that regular double p-algebras can be regarded as (regular) double Heyting algebras by ingeniously constructing binary terms for the Heyting implication and its dual in terms of pseudocomplement and its dual. In this paper, we prove a converse to Katriňák’s theorem, in the sense that in the variety ℝ D ℙ ℂ ℍ \[\mathbb{R}\mathbb{D}\mathbb{P}\mathbb{C}\mathbb{H}\] of regular dually pseudocomplemented Heyting algebras, the implication operation → satisfies Katriňák’s formula. As applications of this result together with the above-mentioned Katriňák’s theorem, we show that the varieties ℝ D B L ℙ \[\mathbb{R}\mathbb{D}\mathbb{B}\mathbb{L}\mathbb{P}\] , ℝ D ℙ ℂ ℍ \[\mathbb{R}\mathbb{D}\mathbb{P}\mathbb{C}\mathbb{H}\] , ℝ ℙ ℂ ℍ d \[\mathbb{R}\mathbb{P}\mathbb{C}{{\mathbb{H}}^{d}}\] and ℝ D B L ℍ \[\mathbb{R}\mathbb{D}\mathbb{B}\mathbb{L}\mathbb{H}\] of regular double p-algebras, regular dually pseudocomplemented Heyting algebras, regular pseudocomplemented dual Heyting algebras, and regular double Heyting algebras, respectively, are term-equivalent to each other and also that the varieties ℝ D M ℙ \[\mathbb{R}\mathbb{D}\mathbb{M}\mathbb{P}\] , ℝ D M ℍ \[\mathbb{R}\mathbb{D}\mathbb{M}\mathbb{H}\] , ℝ D M D B L ℍ \[\mathbb{R}\mathbb{D}\mathbb{M}\mathbb{D}\mathbb{B}\mathbb{L}\mathbb{H}\] , ℝ D M D B L ℙ \[\mathbb{R}\mathbb{D}\mathbb{M}\mathbb{D}\mathbb{B}\mathbb{L}\mathbb{P}\] of regular De Morgan p-algebras, regular De Morgan Heyting algebras, regular De Morgan double Heyting algebras, and regular De Morgan double p-algebras, respectively, are also term-equivalent to each other. From these results and recent results of Adams, Sankappanavar and Vaz de Carvalho on varieties of regular double p-algebras and regular pseudocomplemented De Morgan algebras, we deduce that the lattices of subvarieties of all these varieties have cardinality 2 ℵ 0 \[{{2}^{{{\aleph }_{0}}}}\] . We then define new logics, ℛ D P C ℋ \[\mathcal{R}\mathcal{D}\mathcal{P}\mathcal{C}\mathcal{H}\] , ℛ P C ℋ d \[\mathcal{R}\mathcal{P}\mathcal{C}{{\mathcal{H}}^{d}}\] and ℛ D ℳ ℋ \[\mathcal{R}\mathcal{D}\mathcal{M}\mathcal{H}\] , and show that they are algebraizable with ℝ D ℙ ℂ ℍ \[\mathbb{R}\mathbb{D}\mathbb{P}\mathbb{C}\mathbb{H}\] , ℝ ℙ ℂ ℍ d \[\mathbb{R}\mathbb{P}\mathbb{C}{{\mathbb{H}}^{d}}\] and ℝ D M ℍ \[\mathbb{R}\mathbb{D}\mathbb{M}\mathbb{H}\] , respectively, as their equivalent algebraic semantics. It is also deduced that the lattices of extensions of all of the above mentioned logics have cardinality 2 ℵ 0 \[{{2}^{{{\aleph }_{0}}}}\] .

Picture fuzzy reasoning full implication method and its application

In this paper, we focus on studying picture fuzzy reasoning full implication methods. First, picture associated t-norm and its residuated implication operator are given. Then, picture fuzzy reasoning full implication principles and the solutions of picture fuzzy reasoning full implication methods based on picture associated t-norm are given. Moreover, the reducibility and robustness of picture fuzzy reasoning full implication methods based on picture associated t-norm are proved. Finally, pattern recognition problems can be successfully solved by the proposed method.

Ternary Łukasiewicz logic using memristive devices

Memristive devices based on the Valence Change Mechanism (VCM) are promising devices for storage class memory, neuromorphic computing and logic-in-memory (LIM) applications. They are suited for such a wide range of applications, due to their possibility for extreme dense integration, low power consumption and multilevel capabilities. Through LIM concepts, Boolean logic operations can be performed directly in memory. In many of these concepts, the resistance state of the device is interpreted as the logical input and output of the logic function, which is why these concepts are called ‘stateful’ logic. Most of the proposed ideas, however, are defined based on only binary switching VCM devices and neglect their multi-level capabilities. Extending LIM concepts towards multinary logic, e.g. a ternary logic, would increase the data density inside the memory array and reduce the number of devices required to perform a certain operation. In this work, we discuss two possibilities of realizing a ternary logic based on the analog switching in the RESET or in the SET direction. For both directions we verify the logic functionality by showing the basic operations of implication, negation and false operation, which together form a functionally complete logic. Additionally, for both switching directions, we discuss a 41-Trit ( 64-Bit) addition. For all investigations the physics-based compact model JART VCM v1b is used, which has been verified on the RESET direction multilevel properties of the TaO devices.

Retracted: Interval Information Content of Fuzzy Relation and the Application in the Fuzzy Implication Operators

Retracted: Interval Information Content of Fuzzy Relation and the Application in the Fuzzy Implication Operators

Intuitionistic Modal Algebras

Recent research on algebraic models of quasi-Nelson logic has brought new attention to a number of classes of algebras which result from enriching (subreducts of) Heyting algebras with a special modal operator, known in the literature as a nucleus. Among these various algebraic structures, for which we employ the umbrella term intuitionistic modal algebras, some have been studied since at least the 1970s, usually within the framework of topology and sheaf theory. Others may seem more exotic, for their primitive operations arise from algebraic terms of the intuitionistic modal language which have not been previously considered. We shall for instance investigate the variety of weak implicative semilattices, whose members are (non-necessarily distributive) meet semilattices endowed with a nucleus and an implication operation which is not a relative pseudo-complement but satisfies the postulates of Celani and Jansana’s strict implication. For each of these new classes of algebras we establish a representation and a topological duality which generalize the known ones for Heyting algebras enriched with a nucleus.

Research on the problem of aggregation of multiple rules in fuzzy inference systems

In this paper, we establish the matching relation between implication operator and aggregation operator, which provides a new solution for the design and construction of multi-rule fuzzy inference system. Firstly, according to the definition and monotonicity of implication operator, a new classification method of implication operator is proposed, and then the fuzzy inference process using different implication operators is classified. Then, dynamic maximum aggregation operator and dynamic minimum aggregation operator are proposed. Based on the compositional rule of inference (CRI) method, a matching method and basis of implication operator and aggregation operator for fuzzy inference systems is given and illustrated with examples. Finally, the applicability of the proposed method in this paper is further illustrated by comparing the method with existing methods in the literature and using the nearness degree as an evaluation index.

Combination of linguistic Z-numbers and IF-THEN rules based on rectangular coordinate system

There is a growing body of literature that recognises the importance of Z-numbers proposed by Prof Zadeh. However, due to the complicated structure and short presentation time, many unknowns about Z-numbers still exist. To fill these gaps, this study aims to make use of rectangular coordinate system to express linguistic Z-numbers. Simultaneously, this study sets out to design a score function to quantify the information contained in different Z-numbers. Subsequently, distance measure and similarity measure are also presented from the perspective of coordinate system. Besides, linguistic discrete Z-numbers and belief rule base (BRB) model are combined to construct a novel reasoning model on the basis of implication operators. To verify the validity of the proposed method, three representative examples of epidemic level assessment, multicriteria group decision-making (MCGDM), and network security assessment are employed. The comparison with other widely used methods are performed to further demonstrate the superiority of the proposed method.

A Fundamental Non-Classical Logic

We give a proof-theoretic as well as a semantic characterization of a logic in the signature with conjunction, disjunction, negation, and the universal and existential quantifiers that we suggest has a certain fundamental status. We present a Fitch-style natural deduction system for the logic that contains only the introduction and elimination rules for the logical constants. From this starting point, if one adds the rule that Fitch called Reiteration, one obtains a proof system for intuitionistic logic in the given signature; if instead of adding Reiteration, one adds the rule of Reductio ad Absurdum, one obtains a proof system for orthologic; by adding both Reiteration and Reductio, one obtains a proof system for classical logic. Arguably neither Reiteration nor Reductio is as intimately related to the meaning of the connectives as the introduction and elimination rules are, so the base logic we identify serves as a more fundamental starting point and common ground between proponents of intuitionistic logic, orthologic, and classical logic. The algebraic semantics for the logic we motivate proof-theoretically is based on bounded lattices equipped with what has been called a weak pseudocomplementation. We show that such lattice expansions are representable using a set together with a reflexive binary relation satisfying a simple first-order condition, which yields an elegant relational semantics for the logic. This builds on our previous study of representations of lattices with negations, which we extend and specialize for several types of negation in addition to weak pseudocomplementation. Finally, we discuss ways of extending these representations to lattices with a conditional or implication operation.

Pseudo-Quasi Overlap Functions and Related Fuzzy Inference Methods

The overlap function, a particular kind of binary aggregate function, has been extensively utilized in decision-making, image manipulation, classification, and other fields. With regard to overlap function theory, many scholars have also obtained many achievements, such as pseudo-overlap function, quasi-overlap function, semi-overlap function, etc. The above generalized overlap functions contain commutativity and continuity, which makes them have some limitations in practical applications. In this essay, we give the definition of pseudo-quasi overlap functions by removing the commutativity and continuity of overlap functions, and analyze the relationship of pseudo-t-norms and pseudo-quasi overlap functions. Moreover, we present a structure method for pseudo-quasi overlap functions. Then, we extend additive generators to pseudo-quasi overlap functions, and we discuss additive generators of pseudo-quasi overlap functions. The results show that, compared with the additive generators generated by overlap functions, the additive generators generated by pseudo-quasi overlap functions have fewer restraint conditions. In addition, we also provide a method for creating quasi-overlap functions by utilizing pseudo-t-norms and pseudo automorphisms. Finally, we introduce the idea of left-continuous pseudo-quasi overlap functions, and we study fuzzy inference triple I methods of residual implication operators induced by left-continuous pseudo-quasi overlap functions. On the basis of the above, we give solutions of pseudo-quasi overlap function fuzzy inference triple I methods based on FMP (fuzzy modus ponens) and FMT (fuzzy modus tollens) problems.

Novel variable precision fuzzy rough sets and three-way decision model with three strategies

Variable precision (fuzzy) rough sets are interesting generalizations of Pawlak rough sets and can handle uncertain and imprecise information well due to their error tolerance capability. The comparable property (CP for short), i.e., the lower approximation is included in the upper approximation, is fundamental in Pawlak rough sets since many theories and applications rely on this property. However, the CP is lost in many existing variable precision (fuzzy) rough sets. Therefore, a novel variable precision fuzzy rough set model with CP is proposed and an associated three-way decision model is developed. Firstly, a pair of variable precision fuzzy upper and lower approximation operators are defined and studied through fuzzy implication and co-implication operators. We show that this pair of approximation operators satisfy various properties (including CP) that the existing variable precision fuzzy approximation operators do not possess. Secondly, new methods for constructing loss functions and conditional probabilities are given by using CP, and then a new three-way decision model with three strategies (optimistic, pessimistic, and compromise strategies) is established. Finally, an explanatory example is provided to examine validity, stability and sensitivity of the proposed three-way decision model.

Three-Way Fuzzy Sets and Their Applications (III)

Three-way fuzzy inference is the theoretical basis of three-way fuzzy control. The proposed TCRI method is based on a Mamdani three-way fuzzy implication operator and uses one inference and simple composition operation. In order to effectively improve the TCRI method, this paper proposes a full implication triple I algorithm for three-way fuzzy inference and gives the triple I solution to the TFMP problem. The emphasis of our research is R0 and Go¨del triple I solution, which is related to three-way residual implication, as well as Zadeh’s and Mamdani’s triple I solution, which is based on three-way fuzzy implication operator. Then the three-way fuzzy controller is constructed by the proposed Zadeh’s and R0 triple I algorithm. Finally, the proposed triple I algorithm is applied to the three-way fuzzy control system, and its advantage is illustrated by the three-dimensional surface diagram of the control variable.

Pseudo Overlap Functions, Fuzzy Implications and Pseudo Grouping Functions with Applications

Overlap and grouping functions are important aggregation operators, especially in information fusion, classification and decision-making problems. However, when we do more in-depth application research (for example, non-commutative fuzzy reasoning, complex multi-attribute decision making and image processing), we find overlap functions as well as grouping functions are required to be commutative (or symmetric), which limit their wide applications. For the above reasons, this paper expands the original notions of overlap functions and grouping functions, and the new concepts of pseudo overlap functions and pseudo grouping functions are proposed on the basis of removing the commutativity of the original functions. Some examples and construction methods of pseudo overlap functions and pseudo grouping functions are presented, and the residuated implication (co-implication) operators derived from them are investigated. Not only that, some applications of pseudo overlap (grouping) functions in multi-attribute (group) decision-making, fuzzy mathematical morphology and image processing are discussed. Experimental results show that, in many application fields, pseudo overlap functions and pseudo grouping functions have greater flexibility and practicability.

An embedded deep fuzzy association model for learning and explanation

This paper explores the complementary benefits of embedding a deep learning model as a fully data-driven fuzzy implication operator of a five-layer neuro-fuzzy system for learning and explanations for the predictions of both steady-state and dynamically changing data. In traditional Mandani-type neuro-fuzzy systems, the entailment performed by the implication is realized using the fuzzy implication operator based on the fuzzy rules formed in the rule base during encoding and recall. Given the presence of a group of test data that are significantly different from the training data, the realization of entailments through the use of the implication operator based on the fuzzy rules formed in traditional neuro-fuzzy systems may not be adequate. This paper attempts to adopt a more direct approach by embedding a deep learning model in the neuro-fuzzy system to serve as a fuzzy implication operator, thereby allowing the data-driven learning of fuzzy implication using the deep structure to provide a close correspondence to the real-world entailment of data. In addition, embedding the neuro-fuzzy architecture within the deep learning model allows the comprehension of the learning and explanation of the reasoning of the deep network. The induced fuzzy association rules impart transparency to the deep learning based implication using a common set of semantic meanings, which are amenable to human interpretability. The effectiveness of the proposed model is evaluated on a continuously stirred tank reactor dataset and three financial stock prediction datasets. Experimental results showed that the proposed model outperformed other state-of-the-art techniques based on the four datasets, which contain high levels of uncertainties.

Monoidal closedness of the category of $\top$-semiuniform convergence spaces

Lattice-valued semiuniform convergence structures are important mathematical structures in the theory of lattice-valued topology. Choosing a complete residuated lattice $L$ as the lattice background, we introduce a new type of lattice-valued filters using the tensor and implication operations on $L$, which is called $\top$-filters. By means of $\top$-filters, we propose the concept of $\top$-semiuniform convergence structures as a new lattice-valued counterpart of semiuniform convergence structures. Different from the usual discussions on lattice-valued semiuniform convergence structures, we show that the category of $\top$-semiuniform convergence spaces is a topological and monoidal closed category when $L$ is a complete residuated lattice without any other requirements.

Host-guest interaction aided Zinc carry and delivery by ESIPT active 2-(2′-hydroxyphenyl)benzoxazole

The effect of solvents and supramolecular hosts on the binding of metal ion with an excited state intramolecular proton transfer (ESIPT) active fluorophore 2-(2′-hydroxyphenyl)benzoxazole (HPBO) are investigated to scrutinize a possible metal ion carry and delivery system. The fluorophore forms strong fluorescent complex with Zn2+ ion. In aqueous medium, β-cyclodextrin (β-CD) breaks the HPBO–Zn2+ complex and encapsulate the freed fluorophore. Hence, the initially blocked ESIPT process is restored by forming an inclusion complex with the host molecules. However, in dimethyl sulphoxide (DMSO), β-CD does not break the complex. But cucurbit[7]uril (CB-7) breaks the complex in both DMSO and water. The tuned emission characteristics are considered for constructing different molecular logic gates. BUFFER, NOT, PASS, IMPLICATION and INHIBIT logic operations are substantiated based on Zn2+, CB-7 and β-CD response.