Fuzzy Implication Operators Research Articles

On contra-symmetry and MPT conditionality in fuzzy logic

This article deals with the N-contrapositive symmetry of fuzzy implication operators J verifying either Modus Ponens or Modus Tollens inequalities, in a similar and complementary framework to the one in which Fodor (“Contrapositive symmetry of fuzzy implications.” Fuzzy Set Syst 1995;69:141–156) did begin with the subject in fuzzy logic, that is, with the verification of J(a, b) = J(N(b), N(a)) for all a, b in [0,1] and some strong-negation function N. This property corresponds to the classical p → q = ¬q → ¬p. The aim of this article is to study that property in relation to either Modus Ponens or Modus Tollens meta-rules of inference when the functions J are taken among those that belong to the usual families of implications in fuzzy logic. That is, the contra-positive of S implications, R implications, Q implications, and Mamdani–Larsen operators, verifying either Modus Ponens or Modus Tollens inequalities or both, the conditionality's aspect on which lies the complementarity with Fodor. Within this study new types of implication functions are introduced and analyzed. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 313–326, 2005.

INTUITIONISTIC FUZZY IMPLICATION OPERATORS — AN EXPRESSION AND MAIN PROPERTIES

In this paper we present the definition and properties of intuitionistic fuzzy implication operators. We study the expression obtained from said operators when fuzzy implication and coimplication operators are applied to different aggregations of degrees of truth and non-truth of the propositions.

Computationally Efficient Mining for Fuzzy Implication-Based Association Rules in Quantitative Databases

Association rule mining is one of the focal points in research on knowledge discovery. While conventional approaches usually deal with databases with binary values, this paper presents an approach to discovering association rules (such as ) from quantitative datasets, which are commonly seen in real-world applications. Primary attention is paid to association rules with degrees of support and implication (ARsi), taking a more logic-oriented viewpoint for X-to-Y relationships. Fuzzy logic is applied to “discretization” of quantitative domains as well as to logic implication operations so as to remedy possible boundary problems due to sharp partitioning and facilitate fuzzy implication, respectively. In doing so, a mining algorithm (FIAR) is proposed to discover ARsi, in that several properties of t-norms and fuzzy implication operators (FIOs) are investigated to reduce times of scanning databases. Furthermore, simple rules are discussed and incorporated as optimization strategies into the algorithm. Finally, experiments with synthetic data as well as with real datasets are carried out to show the performance of the proposed algorithm.

Fuzzy controllers based on some fuzzy implication operators and their response functions *

Abstract The fuzzy controllers constructed by 23 fuzzy implication operators based on CRI algorithm and their response functions are discussed. The conclusions show that the fuzzy controllers constructed by 9 fuzzy implication operators are universal approximators to continuous functions and can be used in practical fuzzy control systems. And these 9 fuzzy implication operators except for Einstein operator intersection are all the adjoint pairs of some fuzzy implication operators. Besides, there are 3 other fuzzycontrollers formed by fuzzy implication operators being regarded approximately as fitted functions.

Flexible neuro-fuzzy systems

In this paper, we derive new neuro-fuzzy structures called flexible neuro-fuzzy inference systems or FLEXNFIS. Based on the input-output data, we learn not only the parameters of the membership functions but also the type of the systems (Mamdani or logical). Moreover, we introduce: 1) softness to fuzzy implication operators, to aggregation of rules and to connectives of antecedents; 2) certainty weights to aggregation of rules and to connectives of antecedents; and 3) parameterized families of T-norms and S-norms to fuzzy implication operators, to aggregation of rules and to connectives of antecedents. Our approach introduces more flexibility to the structure and design of neuro-fuzzy systems. Through computer simulations, we show that Mamdani-type systems are more suitable to approximation problems, whereas logical-type systems may be preferred for classification problems.

Parameterized R-implications

A residuated implication operator (R-implications) is a fuzzy material implication operator defined in terms of its corresponding T-norm. This paper is concerned with a family of R-implications derived from the Schweizer–Sklar family of parameterized T-norms. Analysis of fuzzy and defuzzified outputs from two- and three-rule systems includes proper and improper fuzzy set outputs and exact defuzzified solutions for important special cases such as Gödel-Brouwer, Goguen, and Łukasiewicz implications. A rule can be characterized as strong or weak in its interaction with a neighboring rule. If two rules both interact with one another weakly, there exists a range of y values all of which are 100% compatible with both rules given any x in their domain of interaction. The paper concludes with a discussion of the joint effect of the strength of rule interaction and the value of the Schweizer–Sklar parameter.

Fuzzy reasoning based on the extension principle

According to the operation of decomposition (also known as representation theorem) (Negoita CV, Ralescu, DA. Kybernetes 1975;4:169–174) in fuzzy set theory, the whole fuzziness of an object can be characterized by a sequence of local crisp properties of that object. Hence, any fuzzy reasoning could also be implemented by using a similar idea, i.e., a sequence of precise reasoning. More precisely, we could translate a fuzzy relation “If A then B” of the Generalized Modus Ponens Rule (the most common and widely used interpretation of a fuzzy rule, A, B, are fuzzy sets in a universe of discourse X, and of discourse Y, respectively) into a corresponding precise relation between a subset of P(X) and a subset of P(Y), and then extend this corresponding precise relation to two kinds of transformations between all L-type fuzzy subsets of X and those of Y by using Zadeh's extension principle, where L denotes a complete lattice. In this way, we provide an alternative approach to the existing compositional rule of inference, which performs fuzzy reasoning based on the extension principle. The approach does not depend on the choice of fuzzy implication operator nor on the choice of a t-norm. The detailed reasoning methods, applied in particular to the Generalized Modus Ponens and the Generalized Modus Tollens, are established and their properties are further investigated in this paper. © 2001 John Wiley & Sons, Inc.

Inclusion grade and fuzzy implication operators

In this paper we propose a family of fuzzy implication operators, which generalises the Luckasiewicz's one, and to analyse the impacts of Smets and Magrez properties on these operators. The result of this approach will be a characterisation of a proposed family of inclusion grade operators that satisfies the axioms of Divyendu and Dogherty.

Reducing computation overhead in MISO fuzzy systems

Replacing each multi-input single-output (MISO) fuzzy rule by an equivalent collection of single-input and single-output (SISO) rules effects a considerable saving in computation time. In this paper, we show that the conclusion inferred from a MISO rule can be computed as either the union or intersection of the conclusions inferred from the equivalent SISO rules depending on whether the fuzzy implication operator is non-increasing or non-decreasing with respect to its first argument. This is of the utmost importance in fuzzy reasoning applied to fuzzy systems with several MISO rules.

Some considerations on input and output partitions to produce meaningful conclusions in fuzzy inference

A large number of papers have been devoted to point out which combinations of the several fuzzy implication, composition and aggregation operators are necessary to satisfy a natural requirement for meaningful reasoning, that is, given a system observation which matches the antecedent of a rule, the conclusion of the inference process matches the consequent of that rule. Nevertheless, only few of these papers have analysed, once fixed a combination, which constraints on the reciprocal position of input and output fuzzy sets should be satisfied. In this paper, we consider fuzzy implication operators which are extensions of the two-valued logic implication operator and are non-decreasing with respect to their second argument, generic Sup-T composition operators, and minimum as aggregation operator. First, we analyse some features (namely, uniform level of indetermination, perfect matching with the consequent of a rule, and complete indetermination) of the conclusions inferred by those fuzzy implication operators with regard to fuzzy reasoning with one rule. Then, as regards approximate reasoning with multiple rules, we demonstrate that, if the fundamental requirement for fuzzy reasoning is satisfied, then the fuzzy sets which partition the input and output universes have to meet appropriate constraints. Finally, we provide a sufficient condition defined on input fuzzy sets to obtain a reasonable inference result.

FUZZY LOGICAL OPERATORS ON FINITE CHAINS

The extension of classical binary logical operators to multi-valued operators is a very interesting topic, both from a theoretical and a practical point of view. In this paper we pay special attention to fuzzy implication operators on the unit interval and on finite chains. We discuss some properties and define several special classes, in particular the class of Lukasiewicz-like implicators. On the unit interval we prove a nice characterization for this class, while we have to replace the continuity condition on [0,1] by a smoothness condition on finite chains to obtain a similar characterization. In contrast to implicators on [0,1], there is no direct way to construct implicators on a finite chain. Therefore, we discuss several algorithms that generate different types of implicators. In this way we also obtain the number of implicators on a finite chain, which gives us an idea of the restrictivity of the involved properties.

Subsethood measure: new definitions

We comment on the subsethood measure and weak subsethood measure defined by Young and give some new definitions of subsethood measure. We present some subsethood measure formulas from the point of view of set-theoretic approach, and from fuzzy implication operator. Finally an application of subsethood measure to cluster validity is shown.

A new class of fuzzy implications. Axioms of fuzzy implication revisited

Many different fuzzy implication operators have been proposed; most of them fit into one of the two classes: implication operations that are based on an explicit representation of implication A → B in terms of &, ∨, and ¬ (e.g., S-implications that are based on the formula B ∨ ¬ A), and R-implications that are based on an implicit representation of implication A → B as the weakest C for which C&B implies A. However, some fuzzy implication operations (such as b a ) cannot be naturally represented in this form. To describe such operations, we propose a new (third) class of implication operations called A-implications whose relation to &, ∨, and ¬ is described by (implicit) axioms.

Genetic algorithm based methodology to generate automatically optimal fuzzy systems

Fuzzy reasoning models are model-free estimators of control systems, which makes them very powerful tools in control applications. Their performances depend on several factors such as the completeness of the rule base, the subjective definitions of the fuzzy subset parameters or fuzzy partitioning, the fuzzy implication operator and the defuzzification method. In most cases these factors are decided by the subjective experience of the human expert. The authors attempt to improve the fuzzy systems performances by means of a genetic-based learning mechanism. They propose a new methodology based upon two genetic algorithm (GA) loops; the inner GA loop enables the generation of an optimal set of parameters of the fuzzy reasoning matrix based upon their selection by particular optimal fuzzy partitioning generated by an outer GA loop. Simulation of typical control problems shows the effectiveness and high performance of such a method. The effects of GA parameter settings on convergence and total performance are also discussed.

Analysis of the operators involved in the definition of the implication functions and in the fuzzy inference process

This paper analyzes the performance of some fuzzy implications proposed in the bibliography together with the operators needed for their definition and for the fuzzy inference process. Examining the specialized literature, it is clear that the selection of the best fuzzy implication operator has become one of the main question in the design of a fuzzy system, being occasionally contradictory (at presently there are more than 72 fuzzy implication proposed and investigated). An approach to the problem from a different perspective is given. The question is to determine whether the selection of the fuzzy implication operator is more important with respect to the behaviour of the fuzzy system than the operators (mainly T-norm, T-conorm and defuzzification method) involved in the definition of the implication function and in the rest of the inference process. The relevance and relative importance of the operators involved in the fuzzy inference process are investigated by using a powerful statistical tool, the ANalysis Of the VAriance (ANOVA) [Box et al., Statistics for experiments: an introduction to design, data analysis and model building, Wiley, New York, 1978; Montgomery, Design and Analysis of Experiments, Wiley, New York, 1984].

Fuzzy logic or Lukasiewicz logic: A clarification

We discuss some issues on fuzzy logic developed by Pavelka [Z. Math. Logic 25 (1979) 45–52, 119–134, 447–464] vs. the many-valued Lukasiewicz's logic. The focus is on the proper choice of fuzzy implication operations, a question which has been addressed many times in the fuzzy research literature. Pavelka had shown in 1979 that the only natural way of formalizing fuzzy logic for truth values in the unit interval [0, 1] is by using Lukasiewicz's implication operator a → b = min {1, 1 - a + b} or some isomorphic form of it. Many other papers around the same time had attempted to formulate alternative definitions for a → b by giving intuitive justifications. There continues to be some confusion, however, even today about the right notion of fuzzy logic. Much of this confusion lies in the use of improper “an” (“or”) and the “not” operations and a misunderstanding of some of the key differences between “proofs” or inferencing in fuzzy logic and those in Lukasiewicz's logic. We point out the need for defining the strong conjunction operator “⊗” in connection with fuzzy Modus-ponens rule and why we do not need the fuzzy Syllogism inference rule. We formulate two requirements of the fuzzy implication operator, which are satisfied by Lukasiewicz's a → b, but which fail for many of the alternative definitions for “→”.

Applicability analysis of fuzzy inference by means of generalized Dempster-Shafer theory

The generalization of Dempster-Shafer (D-S) theory to fuzzy sets provides means for evidential reasoning on basis of fuzzy information. Fuzzy implication operators are used to turn partial fuzzy knowledge about a physical system into an inference engine. Presently there are more than 72 fuzzy implication operators known and investigated, evoking demand for strategies of finding the best performing fuzzy implication operator for the physical system under investigation. In this paper the authors propose such a concept based on generalized D-S theory. A convention to update prior knowledge regarding the applicability of fuzzy implication operators, with respect to fuzzy evidences pertaining to the physical system under consideration, is presented.

A characterization of the Hamacher family of t-norms

Abstract Starting from a fuzzy conjunction, there are two main ways of defining a fuzzy implication operator. A functional equation for t-norms is obtained and studied which describes the equality of these two implications when the underlying conjunction has a particular form defined from a t-norm. The solution makes clear that this functional equation, together with an additional condition, provides a complete characterization of the Hamacher family of t-norms having positive parameter.

LEVELS OF SIGNIFICANCE FOR FUZZY IMPLICATION

Fuzzy implication operators have been proposed as a tool for measuring the strength of connections between recorded concepts. Hasse diagrams are used to graphically illustrate the sometimes complicated relationships between such concepts. Establishing measures of significance for fuzzy implications is the focus of this paper. Lower and upper bounds for mean implication using the Kleene-Dienes operator are established. Examples of the intended use of these measures are given.

Logical operation based fuzzy MLP for classification and rule generation

A fuzzy layered neural network for classification and rule generation is proposed using logical neurons. It can handle uncertainty and/or impreciseness in the input as well as the output. Logical operators, namely, t-norm T and t-conorm S involving And and Or neurons, are employed in place of the weighted sum and sigmoid functions. Various fuzzy implication operators are introduced to incorporate different amounts of mutual interaction during the back propagation of erros. In case of partial inputs the model is capable of querying the user for the more important feature information, if and when required. Justification for an inferred decision may be produced in rule form. The built-in And-Or structure of the network enables the generation of appropriate rules expressed as the disjunction of conjunctive clauses. The effectiveness of the model is tested on a speech recognition problem and on some artificially generated pattern sets.