Implication Operators Research Articles

On quantum implication

Logic is an algebraic structure that defines a set of abstract rules which govern an area of interest. The abstraction property of the rules makes them reusable tools to model different problems and to reason with them. The proliferation of quantum theory brought attention to quantum logic which is a lattice of projectors and it is of importance to quantum computing. Unfortunately, basic tools like implication are not sufficiently studied in that logic, which prevents us from exploiting the power of quantum mechanics in reasoning. This note investigates the implication issue in quantum logic and defines a quantum implication operator for compatible events as well as for incompatible events. The suggested operator depends both on the angle between the vector sub-spaces of the involved events and the angles between the system state and the vector sub-spaces. It differentiates between three cases depending on the angle between the events’ sub-spaces. The article further shows through an example that some classical reasoning rules such as Modus Ponens and Modus Tollens hold given the suggested implication.

NEW OPERATIONS FOR PYTHAGOREAN FUZZY MATRICES

Objectives: This article is to extend and present an idea related to intuitionistic fuzzy matrix to Pythagorean fuzzy matrix. Methods/Statistical Analysis: The main feature of the Pythagorean fuzzy matrix is to relax the condition that the sum of the membership degree and the non-membership degree to which an alternative satisfying a criterion provided by an expert may be bigger than one, but they square off is equal to or less than one. Particularly those involving the operation A → B define a standard the Pythagorean fuzzy implication with other operations. Findings: We define some new operations for Pythagorean fuzzy matrices (→, $, # ) and discuss their algebraic properties with some existing operations (∨, ∧, ⊕, @ ) in detail. Also, we prove some new results associated with the standard Pythagorean fuzzy implication (→). Finally, implication operation A → B has been extended for Pythagorean fuzzy matrices. Application: An application of Pythagorean fuzzy decision matrix and its aggregation operators constructed by Yager and which are used to solve multicriteria decision-making problems. Recently, a new model based on Pythagorean fuzzy matrix has been presented to manage the uncertainty in real-world decision-making problems. Keywords: Algebraic Sum And Algebraic Product, Implication Operation, Intuitionistic Fuzzy Matrix, Pythagorean Fuzzy Matrix

An Application of Peircean Triadic Logic: Modelling Vagueness

Development of decision-support and intelligent agent systems necessitates mathematical descriptions of uncertainty and fuzziness in order to model vagueness. This paper seeks to present an outline of Peirce’s triadic logic as a practical new way to model vagueness in the context of artificial intelligence (AI). Charles Sanders Peirce (1839–1914) was an American scientist–philosopher and a great logician whose triadic logic is a culmination of the study of semiotics and the mathematical study of anti-Cantorean model of continuity and infinitesimals. After presenting Peircean semiotics within AI perspective, a mathematical formulation of a Peircean triadic set is given in relationship with classical and fuzzy sets. Using basic logical operators, all possible respective implication operators, bi-equivalence operators, valid rules of inference, and associative, distributive and commutative logical properties are derived and verified through the truth function approach. In order to suggest practical directions, aggregation operators for Peirce’s triadic logic have been formulated. A mathematical formulation of a medical diagnostic problem and ER diagram of a library management system using Peirce’s triadic relation show potential for further applications of the proposed triadic set and triadic logic. Alongside, a classical AI game—The Wumpus World—is implemented to show practical efficacy in comparison with binary implementation. Besides giving some preliminary formulations for trichotomous set theory and definition of finite automaton, development of hybrid architectures for intelligent agents and evolutionary computations are discussed as potential practical avenues for Peirce’s triadic logic.

Similarity measures, penalty functions, and fuzzy entropy from new fuzzy subsethood measures

In this study, we discuss a new class of fuzzy subsethood measures between fuzzy sets. We propose a new definition of fuzzy subsethood measure as an intersection of other axiomatizations and provide two construction methods to obtain them. The advantage of this new approach is that we can construct fuzzy subsethood measures by aggregating fuzzy implication operators which may satisfy some properties widely studied in literature. We also obtain some of the classical measures such as the one defined by Goguen. The relationships with fuzzy distances, penalty functions, and similarity measures are also investigated. Finally, we provide an illustrative example which makes use of a fuzzy entropy defined by means of our fuzzy subsethood measures for choosing the best fuzzy technique for a specific problem.

Fuzzy [formula omitted]-covering based [formula omitted]-fuzzy rough set models and applications to multi-attribute decision-making

Multi-attribute decision-making (MADM) can be regarded as a process of selecting the optimal one from all alternatives. Traditional MADM problems with fuzzy information are mainly focused on a fundamental tool which is a fuzzy binary relation. However some complicated problems cannot be effectively solved by a fuzzy relation. For this reason, in order to solve these issues, we set forth two decision-making methods that are stated in terms of novel and flexible fuzzy rough set models. For the purpose of defining these models we employ a fuzzy implication operator I and a triangular norm T. With these adaptable tools we design four kinds of fuzzy β-coverings based (I,T)-fuzzy rough set models. The elements that make these models different are the combination of fuzzy β-neighborhoods that intervene in the definitions of the lower (upper) approximations. Then, we discuss the relationships among these given four types of models. Finally, we propose two novel methodologies to solve MADM problems with evaluation of fuzzy information, which rely on these models. Through the analysis of the ranking results of these two methods, we observe that the optimal selected alternative is the same, which means that these two decision-making methods are reasonable. In addition, by comparing the ranking results of these two methods and the existing traditional methods (WA operator and TOPSIS), we observe that our proposed methods can solve the ranking problems that traditional methods cannot solve, which means that our proposed methods are superior to traditional methods.

Triple-I FMP algorithm for double hierarchical fuzzy system based on manifold learning

As is well known, Zadeh presented the compositional rule of inference to discuss complex inference modes. After that other researchers also investigated this problem. However, the classic fuzzy control systems in the application often encounter the curse of dimensionality. To overcome these difficulties in the classic fuzzy control systems with high-dimensional input variables, the entire triple-I algorithm for double fuzzy control systems and manifold learning of dimensionality reduction will be discussed in this paper. Specifically, triple-I FMP algorithm is presented for double hierarchical fuzzy control system based on Guojun Wang’s implication operator and its reversibility is proved. Using manifold learning, dimension reduction SNE algorithm is given for double-layer hierarchical fuzzy control systems to keep the distribution of peak possibly point, so as to minimize the control stability impact due to reducing dimension. Since any type of multi-layer fuzzy control system is regarded as multiple fusion of double-layer hierarchical fuzzy system, the proposed algorithms and their reversibility are universal.

PROPERTY ANALYSIS OF TRIPLE IMPLICATION METHOD FOR APPROXIMATE REASONING ON ATANASSOVS INTUITIONISTIC FUZZY SETS

Firstly, two kinds of natural distances between intuitionistic fuzzy sets are generated by the classical natural distance between fuzzy sets under a unified framework of residual intuitionistic implication operators. Secondly, the continuity and approximation property of a method for solving intuitionistic fuzzy reasoning are defined. It is proved that the triple implication method for intuitionistic fuzzy modus ponens has both continuity and approximation property with respect to these two kinds of natural distances based on {L}ukasiewicz implication, meanwhile the triple implication method for intuitionistic fuzzy modus tollens possesses unconditional continuity and conditional approximation property. Finally, some robustness results about the triple implication method of intuitionistic fuzzy reasoning are given.

Implication-based fuzzy ternary subsemigroups in ternary semigroups

The aim of this study is to introduce the concept of implication-based fuzzy ternary subsemigroups in ternary semigroups. Using the four implication operators, that is, Gaines-Rescher implication operator, Godel implication operator, the contraposition of Godel implication operator and the Luckasiewicz implication operator, the implication-based fuzzy ternary subsemigroups are considered. Furthermore, based on this novel idea, relationships between fuzzy [resp.(ϵ,ϵ Vԛ)-fuzzy] ternary subsemigroups and implication-based fuzzy ternary subsemigroups are discussed. Finally, conditions for a fuzzy ternary subsemigroup with thresholds 0 and 0.5 to be an implication-based fuzzy ternary subsemigroups under the Luckasiewicz implication operator are provided.

Degree of (L, M)-Fuzzy Semi-Precontinuous and (L, M)-Fuzzy Semi-Preirresolute Functions

Abstract The aim of this paper is to present the degree of semi-preopenness, semi-precontinuity, and semi-preirresoluteness for functions in (L, M)-fuzzy pretopology with the help of implication operation and (L, M)-fuzzy semi-preopen operator introduced by [Ghareeb A., L-fuzzy semi-preopen operator in L-fuzzy topological spaces, Neural Comput. & Appl., 2012, 21, 87-92]. Further, we generalize the properties of semi-preopenness, semi-precontinuity and semi-preirresoluteness to (L, M)-fuzzy pretopological setting relying on graded concepts. Also, we discuss their relationships with the corresponding degrees of semiprecompactness, semi-preconnectedness and semi-preseparation axioms.

Hesitant fuzzy Lukasiewicz implication operation and its application to alternatives’ sorting and clustering analysis

Hesitant fuzzy set (HFS) takes several possible values as the membership degree of an element to a set to express the decision makers’ hesitance when making decisions. Since its appearance, the HFS has been widely used in many fields, such as decision making, clustering analysis. Lukasiewicz implication operator, an indispensable part of implication operators, can grasp more nuances compared with the others. In this paper, we shall combine the Lukasiewicz implication operator with HFSs to realize a direct clustering analysis algorithm and a novel alternative sorting method in decision making under hesitant fuzzy environment. To do that, we first apply the Lukasiewicz implication operator to deal with HFEs by getting a hesitant fuzzy Lukasiewicz implication operator, and then construct a hesitant fuzzy triangle product and a hesitant fuzzy square product based on the new implication operator. After that, the hesitant fuzzy square product is applied to define the similarity degree between HFSs, and based on which, we develop a direct clustering algorithm for hesitant fuzzy information. Meanwhile, the hesitant fuzzy triangle product is used to induce a new alternative sorting method. Finally, two numerical examples are given to illustrate the effectiveness and practicability of our method and algorithm, one of which involves the evaluation analysis of the Arctic development risk.

Free Modal Pseudocomplemented De Morgan Algebras

Modal pseudocomplemented De Morgan algebras (or mpM-algebras) were investigated in A. V. Figallo, N. Oliva, A. Ziliani, Modal pseudocomplemented De Morgan algebras, Acta Univ. Palacki. Olomuc., Fac. rer. nat., Mathematica 53, 1 (2014), pp. 65–79, and they constitute a proper subvariety of the variety of pseudocomplemented De Morgan algebras satisfying xΛ(∼x)* = (∼(xΛ(∼x)*))* studied by H. Sankappanavar in 1987. In this paper the study of these algebras is continued. More precisely, new characterizations of mpM-congruences are shown. In particular, one of them is determined by taking into account an implication operation which is defined on these algebras as weak implication. In addition, the finite mpM-algebras were considered and a factorization theorem of them is given. Finally, the structure of the free finitely generated mpM-algebras is obtained and a formula to compute its cardinal number in terms of the number of the free generators is established. For characterization of the finitely-generated free De Morgan algebras, free Boole-De Morgan algebras and free De Morgan quasilattices see: [16, 17, 18].

Fuzzy logic-based FMEA robust design: a quantitative approach for robustness against groupthink in group/team decision-making

Group/team decision-making is an integral part of almost all failure mode and effects analysis (FMEA) projects. A dysfunctional aspect of this decision-making fashion in fuzzy FMEA is that group/team members’ designs for membership functions and IF-THEN rules may be overshadowed by a member’s design. This problem is caused by groupthink, a pitfall known by the Organisational Behaviour science. This study aims to develop a fuzzy FMEA approach which is robust to the problem. We applied the Taguchi’s robust parameter design and investigated the effects of various control parameters namely Defuzzification, Aggregation, And and Implication operators for the fuzzy inference system (FIS). Our experiments illustrate that the control parameters, in the above-mentioned order, have the most effect on the signal-to-noise ratio (SNR). These factors’ optimal setting consists of the Centroid, Sum, Minimum and Minimum levels, respectively.

Realizability in ordered combinatory algebras with adjunction

In this work, we continue our consideration of the constructions presented in the paperKrivine's Classical Realizability from a Categorical Perspectiveby Thomas Streicher. Therein, the author points towards the interpretation of the classical realizability of Krivine as an instance of the categorical approach started by Hyland. The present paper continues with the study of the basic algebraic set-up underlying the categorical aspects of the theory. Motivated by the search of a full adjunction, we introduce a new closure operator on the subsets of the stacks of an abstract Krivine structure that yields an adjunction between the corresponding application and implication operations. We show that all the constructions from ordered combinatory algebras to triposes presented in our previous work can be implemented,mutatis mutandis, in the new situation and that all the associated triposes are equivalent. We finish by proving that the whole theory can be developed using the ordered combinatory algebras with full adjunction or strong abstract Krivine structures as the basic set-up.

A new representation theorem of L-subsets

In the paper, the implication and tensor operations of strong L-nested systems are found, and further we show that the family of all strong L-nested systems on a universal set forms a complete Girard residuated lattice provided the underlying complete residuated lattice is Girard. Finally, an isomorphism of complete Girard residuated lattices between the family of all strong L-nested systems and the L-power set for any fixed universal set is established, and thus a new representation theorem of L-subsets is obtained.

Stateful Logic Operations Implemented With Graphite Resistive Switching Memory

Memristor-based logic operation has attracted a lot of interest ever since the material implication and NAND logic operations were successfully implemented with several memristive devices. Here, we present a memory device fabricated with graphite membrane, which has a simple two terminal planar structure and high ON/OFF ratio over $10^{2}$ . We use the memory device to establish logic gates, and stateful logic operations were successfully carried out. This letter paves the way for carbon electronics to be used in new computation technology.

Unified forms of the CDR method of approximate reasoning on Antanassov's intuitionistic fuzzy sets and its property analysis

AbstractTwo basic models of fuzzy reasoning are fuzzy modus ponens and fuzzy modus tollens. Correspondingly, the key point of intuitionistic fuzzy reasoning is to solve the problems of intuitionistic fuzzy modus ponens (IFMP) and intuitionistic fuzzy modus tollens (IFMT). In many important algorithms of fuzzy reasoning, the Consequent Dilation Rule (CDR) method possesses the virtue of unconditional reductivity. This paper proposed an intuitionistic CDR (ICDR) method for IFMP and IFMT problems by extending the CDR method to the intuitionistic fuzzy reasoning. It was proved that the intuitionistic CDR methods, both for IFMP and IFMT, are unconditionally reductive in the circumstance of using residual intuitionistic implication operators. At the same time, the continuity, approximation properties and robustness of the CDR method in Łukasiewicz intuitionistic fuzzy reasoning space have been studied by using the new defined average natural distance between intuitionistic fuzzy sets.

On a correspondence between probabilistic and fuzzy choice functions

Probabilistic and fuzzy choice functions are used to describe decision situations in which some degree of uncertainty or imprecision is involved. We propose a way to equate these two formalisms by means of residual implication operations. Furthermore, a set of new rationality conditions for probabilistic choice functions is proposed and proved to be sufficient to ensure that the associated fuzzy choice function is rational.

«YOU MISUNDERSTAND ME»: IMPLICATION AND INTERFERENCE AS SENSE FORMING OPERATIONS IN THE PROCESS OF COMMUNICATION

Any type of speech interaction consists not only in generating utterances, but also in the listener’s adequate understanding of what is being said. The degree of speech eff ectiveness is much determined by the addressee’s equivalent interpretation of the message transmitted in the process of dialogue interplay. The given work is focused upon the problem of the listener’s adequate perception of speech messages sent by the speaker. The urgency of the research undertaken follows from the absence of universal theoretical conceptions of decoding the sense of the utterance and is also motivated by the practical needs of optimising speech communication. Our object of study is the original English literary dialogue, and the subject of investigation is verbal and non-verbal means indirectly infl uencing the sense of the message. The objective of this paper is uncovering intra- and extra-lingual factors determining the individual perception of the utterance. In the course of study, we have used such methods as linguistic observation and description, contextual and communicative approaches, quantitative and qualitative analysis. Since it is an often case that a greater part of the intended message remains unsaid and requires mental restoration from the listener, we found it necessary to study the logical operation of implication. It was found out that extra-lingual interference, such as background knowledge of the interlocutor’s identity and the topic of conversation, plays a domineering role in deducing implicatures and decoding the sense of the utterance on the level of personal interpretation of the message.

Memcomputing (Memristor + Computing) in Intrinsic SiOx-Based Resistive Switching Memory: Arithmetic Operations for Logic Applications

In this paper, implication (IMP) operations are demonstrated in a circuit with two SiO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</sub> -based memristors and a CMOS transistor. Specifically, a circuit with two one-diode and one-resistor (1D1R) memory elements and a transistor are designed to perform the IMP operations. A circuit consisting of a 4×4 crossbar 1D1R memristor array together with selection transistors is proposed and used to realize the functionality of a one-bit, full adder in 43 steps. Compared with CMOS logic circuits, the advantages and disadvantages of memristor-enabled logic circuits are discussed. This result suggests that the memristor-enabled logic circuit is most suitable for low-power and high-density applications, as well as a simple and robust approach to realize programmable memcomputing chips compatible with large-scale CMOS manufacturing technology.

Quantum algorithm to solve function inversion with time–space trade-off

In general, it is a difficult problem to solve the inverse of any function. With the inverse implication operation, we present a quantum algorithm for solving the inversion of function via using time---space trade-off in this paper. The details are as follows. Let function $$f(x)=y$$f(x)=y have k solutions, where $$x\in \{0, 1\}^{n}, y\in \{0, 1\}^{m}$$x?{0,1}n,y?{0,1}m for any integers n, m. We show that an iterative algorithm can be used to solve the inverse of function f(x) with successful probability $$1-\left( 1-\frac{k}{2^{n}}\right) ^{L}$$1-1-k2nL for $$L\in Z^{+}$$L?Z+. The space complexity of proposed quantum iterative algorithm is O(Ln), where L is the number of iterations. The paper concludes that, via using time---space trade-off strategy, we improve the successful probability of algorithm.