Implication Operators Research Articles

Reverse triple I method based on the Pythagorean fuzzy inference model and its application

This paper is devoted to discussing the reverse triple I method based on the Pythagorean fuzzy set (PFS). We first propose the concepts of Pythagorean t-norm, Pythagorean t-conorm, residual Pythagorean fuzzy implication operator (RPFIO) and Pythagorean fuzzy biresiduum. The reverse triple I methods for Pythagorean fuzzy modus ponens (PFMP) and Pythagorean fuzzy modus tollens (PFMT) are also established. In addition, some interesting properties of the reverse triple I method of PFMP and PFMT inference models are analysed, including the robustness, continuity and reversibility. Finally, a practical problem is provided to illustrate the effectiveness of the reverse triple I method for PFMP in decision-making problems. The advantages of the new method over existing methods are also expounded. Overall, compared with the existing methods, the proposed methods are based on logical reasoning rather than using aggregation operators, so the novel methods are more logical, can better deal with the uncertain problems in complex decision-making environments and can completely reflect the decision-making opinions of decision-makers.

Attribute Reduction Based on the Fuzzy Rough Set with Lukasiewicz Implication Operator

Attribute reduction is a common technique and has made breakthroughs in many aspects. One of the major development directions of attribute reduction is the extended fuzzy rough set models, which is embodied in the selection of fuzzy similarity relations and operators, eventually the derived membership functions. In view of the relatively simple selection of implication operators in related research, this article discusses the impact of different implication operators in the fuzzy rough sets. Secondly, the rough set model that relies on Lukasiewicz implication operator is further improved, and the proof of the closure of the new operator on the positive field is given. Finally, a new algorithm is given, and experiments are designed to prove the feasibility of the new algorithm based on eight public data sets.

PYTHAGOREAN PICTURE FUZZY SETS(PPFS), PART 2- SOME MAIN PICTURE LOGIC OPERATORS ON PPFS AND SOME PICTURE INFERENCE PROCESSES IN PPF SYSTEMS

Pythagorean picture fuzzy set (PPFS) - is a combination of Picture fuzzy set with the Yager’s Pythagorean fuzzy set [12-14]. In the first part of the paper [17] we considered basic notions on PPFS as set operators of PPFS. Unfortunately, we have not papers [18,19, 20] about spherical fuzzy sets with the same definition with some operators and applications to multi attribute group decision making problems. Now in the second part, we will present some main operators in picture fuzzy logic on PPFS: picture negation operator, picture t-norm, picture t-conorm, picture implication operators on PPFS. Last, the compositional rule of inference in PPFS setting should be presented and an numerical example was given.

Pythagorean Fuzzy Full Implication Triple I Method and Its Application in Medical Diagnosis

This paper is devoted to the research of full implication triple I method under Pythagorean fuzzy environment. We first propose the concepts of Pythagorean t-norm, Pythagorean t-conorm, residual Pythagorean fuzzy implication operator (RPFIO) and Pythagorean fuzzy biresiduum. The full implication triple I method for Pythagorean fuzzy modus ponens (PFMP) and Pythagorean fuzzy modus tollens (PFMT) are also established. In addition, the properties of full implication triple I method of PFMP and PFMT models including the robustness, continuity and reversibility are analyzed. Finally, a practical problem is discussed to demonstrate the effectiveness of the Pythagorean fuzzy full implication multiple I method in medical diagnosis. The advantages of the new method are also explained. Overall, compared with the existing methods, the proposed methods are based on logical reasoning rather than using aggregation operators, so they can more accurately and completely express decision information.

Robustness of Interval-Valued Intuitionistic Fuzzy Reasoning Quintuple Implication Method

The interval-valued intuitionistic fuzzy quintuple implication algorithm, as an extension of the fuzzy reasoning algorithm, may better characterize and deal with uncertainty in the reasoning, but how to select distance measure and analyze the algorithm’s robustness is an important and unsolved topic. In this paper, a novel distance measure of interval-valued intuitionistic fuzzy sets is constructed based on interval-valued intuitionistic fuzzy biresiduum similarity. The unified form of the conclusion about the robustness of interval-valued intuitionistic fuzzy reasoning quintuple implication algorithm for interval-valued intuitionistic fuzzy modus ponens(IVIFMP) and interval-valued intuitionistic fuzzy modus tollens(IVIFMT) is obtained. Especially, the robustness of the interval-valued intuitionistic fuzzy reasoning quintuple implication algorithm based on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$G\ddot {o}del$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Lukasiewicz$ </tex-math></inline-formula> , and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Goguen$ </tex-math></inline-formula> implication operators is presented. An application example and experiment are offered to demonstrate the validity of the obtained conclusion. Furthermore, the new distance metric is compared to traditional distances, and its benefits and limits are discussed. The results show that our approach to research the robustness is simpler and more representative, and the robustness of the algorithm based on other implication operators can be obtained by simple substitution.

Compositional Rule of Inference with a Complex Rule Using Lukasiewicz t-Norm

Inference systems are intelligent software performed generally to help people take appropriate decisions and solve problems in specific domains. Fuzzy inference systems are a kind of these systems that are based on fuzzy knowledge. To handle the fuzziness in the inference, the compositional rule of inference is used, which has two parameters: a t-norm and an implication operator. However, most of the combinations of t-norm/implication do not give an adequate inference result that coincides with human intuitions. This was the motivation for several works to study these combinations and to identify those that are compatible, in order to guarantee a performance close to that of humans. We are interested in this paper to a more general form of rules, which is complex rules, whose premise is a conjunction of propositions. To obtain the consequence in a fuzzy inference system using the compositional rule of inference with a complex rule, we study, in this work, Lukasiewicz t-norm which was not investigated before in this context. We combine it with known implications, and we verify the satisfaction of some criteria that model human intuitions.

Lukasiewicz Fuzzy Implication Operator on Pythagorean Fuzzy Tautological Matrices

In this paper, introduced Pythagorean fuzzy tautologial matrices and Pythagorean fuzzy cotautological matrices and some properties of Lukasiwicz implication operator over Pythagorean fuzzy tautologial matrices and Pythagorean fuzzy cotautological matrices are discussed. Also discussed the relation between implication with Lukasiewicz disjunction and conjunction operations of PFCMs and PFCTMs.

Pythagorean fuzzy full implication multiple I method and corresponding applications

The overwhelming majority of existing decision-making methods combined with the Pythagorean fuzzy set (PFS) are based on aggregation operators, and their logical foundation is imperfect. Therefore, we attempt to establish two decision-making methods based on the Pythagorean fuzzy multiple I method. This paper is devoted to the discussion of the full implication multiple I method based on the PFS. We first propose the concepts of Pythagorean t-norm, Pythagorean t-conorm, residual Pythagorean fuzzy implication operator (RPFIO), Pythagorean fuzzy biresiduum, and the degree of similarity between PFSs based on the Pythagorean fuzzy biresiduum. In addition, the full implication multiple I method for Pythagorean fuzzy modus ponens (PFMP) is established, and the reversibility and continuity properties of the full implication multiple I method of PFMP are analyzed. Finally, a practical problem is discussed to demonstrate the effectiveness of the Pythagorean fuzzy full implication multiple I method in a decision-making problem. The advantages of the new method over existing methods are also explained. Overall, the proposed methods are based on logical reasoning, so they can more accurately and completely express decision information.

Similarity of intuitionistic fuzzy sets and its applications

This paper mainly discusses three points. First of all, four kinds of intuitionistic similarities based on implication operator and the corresponding logical metric spaces are constructed. Secondly, from the perspective of perturbation parameters, different intuitionistic similarities are compared and analyzed by means of topological method. Finally, the applications of similarities are shown. On the one hand, the constructed similarities are used to solve a pattern recognition problem. On the other hand, the robustness of the Triple I method of intuitionistic reasoning is analyzed by adopting the constructed similarity as the perturbation parameter.

Preference implication-based approach to ranking fuzzy numbers

Dombi and Baczy'{n}ski presented a new approach to the problem of implication operation by introducing the preference implication, which has very advantageous properties. In this paper, it is presented how the preference implication is connected with soft inequalities and with sigmoid functions. Utilizing this connection the preference implication-based preference measure for two fuzzy numbers is introduced and its key properties, including the reciprocity, are described. Then, the exact expression for computing the new preference measure for trapezoidal fuzzy numbers is presented. Here, using the new preference measure, two crisp relations over trapezoidal fuzzy numbers are introduced. It is shown that one of them is a strict (but not a total) order relation, and the other one is an equivalence relation. The strict order relation can be used to rank comparable fuzzy numbers, while the equivalence relation, which we call the indifference relation, expresses that the order of some fuzzy numbers is indifferent. These two crisp relations can be used to rank a collection of trapezoidal fuzzy numbers. Lastly, the proposed ranking method is compared with some well-known existing fuzzy number ranking methods.

IMPLICATION - BASED FUZZY TERNARY SEMIGROUPS FOR ISOLATED SYSTOLIC HYPERTENSION USING PENTAGONAL FUZZY NUMBER MATRIX

In this paper, medical diagnosis concerning isolated systolic hypertension and a study on implication operators is introduced, but fuzzy pentagonal number matrix concepts developed to give a numerical example and graphical representation are also established.

Logic and in-memory computing achieved in a single ferroelectric semiconductor transistor

Exploring materials with multiple properties who can endow a simple device with integrated functionalities has attracted enormous attention in the microelectronic field. One reason is the imperious demand for processors with continuously higher performance and totally new architecture. Combining ferroelectric with semiconducting properties is a promising solution. Here, we show that logic, in-memory computing, and optoelectrical logic and non-volatile computing functionalities can be integrated into a single transistor with ferroelectric semiconducting α-In2Se3 as the channel. Two-input AND, OR, and non-volatile NOR and NAND logic operations with current on/off ratios reaching up to five orders, good endurance (1000 operation cycles), and fast operating speed (10 μs) are realized. In addition, optoelectrical OR logic and non-volatile implication (IMP) operations, as well as ternary-input optoelectrical logic and in-memory computing functions are achieved by introducing light as an additional input signal. Our work highlights the potential of integrating complex logic functions and new-type computing into a simple device based on emerging ferroelectric semiconductors.

Residuated Algebraic Structures in the Vicinity of Pre-rough Algebra and Decidability

Varieties of topological quasi-Boolean algebras in the vicinity of pre-rough algebras [28, 29] are expanded to residuated algebraic structures by introducing a new implication operation and its residual in these structures. Sequent calculi for some classes of residuated algebraic structures are established. These sequent calculi have the strong finite model property which yields the decidability of the word problem for corresponding classes of algebraic structures.

An approach for solving fuzzy multi-criteria decision problem under linguistic information

Linguistic information processing exists in multi-criteria decision making, and linguistic truth-valued lattice implication algebra (LTV-LIA) has definite advantages in handling comparable and incomparable linguistic values. To deal with the preference relations with linguistic evaluation information, we establish a novel approach for solving fuzzy multi-criteria decision problem under linguistic information based on LTV-LIA. In this paper, we propose linguistic lattice-valued preference relation (LLVPR). LLVPR positive and negative matrixes are introduced to evaluate the advantages and disadvantages of alternatives respectively. In order to get a reasonable result, we introduce a new algorithm to check and repair the consistency of a LLVPR. A linguistic lattice-valued 2-tuple representation model (LLV2-tuple) and some new aggregation operations are presented to get the comprehensive linguistic information without information loss. Considering different decision makers have different preferences, a multiple preferences implication operation of LLV2-tuple is introduced. Finally, we propose a novel linguistic analytic hierarchy process embedded in aggregation layer and implication layer, introducing algorithm and numerical examples. A comparative analysis is adopted to illustrate the rationality.

Measurement of Countable Compactness and Lindelöf Property in RL -Fuzzy Topological Spaces

Based on the concepts of pseudocomplement of L -subsets and the implication operator where L is a completely distributive lattice with order-reversing involution, the definition of countable RL -fuzzy compactness degree and the Lindelöf property degree of an L -subset in RL -fuzzy topology are introduced and characterized. Since L -fuzzy topology in the sense of Kubiak and Šostak is a special case of RL -fuzzy topology, the degrees of RL -fuzzy compactness and the Lindelöf property are generalizations of the corresponding degrees in L -fuzzy topology.

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

In rule optimization, some rule characteristics were extracted to describe the uncertainty correlations of fuzzy relations, but the concrete numbers cannot express correlations with uncertainty, such as “at least 0.1 and up to 0.5.” To solve this problem, a novel definition concerning interval information content of fuzzy relation has been proposed in this manuscript to realize the fuzziness measurement of the fuzzy relation. Also, its definition and expressions have also been constructed. Meanwhile based on the interval information content, the issues of fuzzy implication ranking and clustering were analyzed. Finally, utilizing the combination of possibility’s interval comparison equations and interval value’s similarity measure, the classifications of implication operators were proved to be realizable. The achievements in the presented work will provide a reasonable index to measure the fuzzy implication operators and lay a solid foundation for further research.

Willmott Fuzzy Implication in Fuzzy Databases

The object of the research are fuzzy functional dependencies on given relation scheme, and the question of their obtaining using the classical and innovated techniques. The attributes of the universal set are associated to the elements of the unit interval, and are turned into fuzzy formulas in this way. We prove that the dependency (which is treated as a fuzzy formula with respect to appropriately chosen valuation) is valid whenever it agrees with the attached two-elements fuzzy relation instance. The opposite direction of the claim is proven to be incorrect in this setting. Generalizing things to sets of attributes, we prove that particular fuzzy functional dependency follows form a set of fuzzy dependencies (in both, the world of two-element and the world of arbitrary fuzzy relation instances) if and only if the dependency is valid with respect to valuation anytime the set of fuzzy formulas agrees with the valuation. The results derived in paper show that the classical techniques in the procedure for generating new fuzzy dependencies may be replaced by the resolution ones, and hence automated. The research is conducted with respect to Willmott fuzzy implication operator

Fuzzy Functional and Multivalued Dependencies for Frank’s Class of Additive Generators

In this paper we consider all possible dependencies that can be built upon similarity-based fuzzy relations, that is, fuzzy functional and fuzzy multivalued dependencies. Motivated by the fact that the classical obtaining of new dependencies via inference rules may be tedious and uncertain, we replace it by the automated one, where the key role is played by the resolution principle techniques and the fuzzy formulas in place of fuzzy dependencies. We prove that some fuzzy multivalued dependency is actively correct with respect to given fuzzy relation instance if and only if the corresponding fuzzy formula is in line with the attached interpretation. Additionally, we require the tuples of the instance to be conformant (up to some extent) on the leading set of attributes. The equivalence as well as the conclusion are generalized to sets of attributes. The research is conducted by representing the attributes and fuzzy dependencies in the form of fuzzy formulas, and the application of fuzzy implication operators derived from carefully selected Frank’s classes of additive generators

Computational characterization of thermal and mechanical properties of single and bilayer germanene nanoribbon

Two-dimensional germanene has provided a cornucopia of new functionalities in the field of nanotechnology owing to its remarkable electronic and thermoelectric attributes. The robust spin–orbit coupling and high carrier mobility give rise to many salient features including non-trivial topological properties, quantum spin-Hall state near room temperature, and topological superconductivity, rendering it an excellent contender for valleytronics, spintronics, and quantum computation. As such, an in-depth characterization of thermal and mechanical properties of germanene is crucial for its practical implication and efficient operation, which remains elusive. Here, we employed equilibrium molecular dynamics simulations utilizing Stillinger Weber potential to reveal the mechanical strength, melting temperature, and phonon thermal conductivity (PTC) of single-layer germanene nanoribbon (SLGeNR) and bilayer germanene nanoribbon (BiLGeNR). Effects of temperature, biaxial tensile and compressive strain, monovacancy defects, length and width of the nanoribbon on the PTC have been rigorously investigated. It has been found that PTC of SLGeNR could be substantially reduced by BiLGeNR. Our simulation results suggest that PTC of SLGeNR demonstrates an inverse relation with temperature, biaxial compressive strain, and monovacancy defects while biaxial tensile strain, length and width of the nanoribbon increases the PTC of SLGeNR significantly. To understand the PTC more profoundly, phonon density of state (PDOS) profiles have been studied. The BiLGeNR demonstrates more tensile strength as well as melting temperature compared to SLGeNR. This study offers a comprehensive guideline for engineering the TC as well as discloses important mechanical and melting characteristics of the SLGeNR and BiLGeNR for a wide range of applications in flexible nano-electronics and thermoelectric nanodevices.

Generalizations of implication-based fuzzy subalgebras in BCK/BCI-algebras

In [2], Jun discussed implication-based subalgebras in BCK/BCI-algebras. In this article, more general forms than Jun’s results are discussed. We provide an example to show that a fuzzy subalgebra with thresholds 0 and 0.5 is not an implication-based subalgebra under the Łuckasiewicz implication operator, and then conditions for a fuzzy subalgebra with thresholds 0 and 0.5 to be an implication-based subalgebra under the Łuckasiewicz implication operator are provided.