Generalized Modus Tollens Research Articles

Fuzzy reasoning method based on picture similarity measure and its application in medical diagnosis

In previous studies, based on picture fuzzy sets, full implication algorithm and quintuple implication principle have been studied. The problem is, full implication algorithm did not consider the relationship between the input and antecedent of rule, while quintuple implication principle only considered the inclusion relationship between the rule’s antecedent and input. Both algorithms did not consider the similarity of rule’s antecedent and input. This leads to unreasonable calculation results in some cases. Therefore, we study fuzzy reasoning methods based on picture similarity measure for generalized modus ponens and generalized modus tollens in this paper. First of all, based on picture biresiduation, a new picture similarity measure between picture fuzzy sets is constructed. Then, representations of solutions of fuzzy reasoning methods based on picture similarity measure are given. Moreover, reducibility and robustness of fuzzy reasoning methods based on picture similarity measure are studied. Finally, two issues related to medical diagnosis are addressed by the proposed methods, demonstrate the rationality and effectiveness of our suggested methods.

Investigations of T-power based implications satisfying some functional equations related to reasoning schemes

In fuzzy logic, the fuzzy reasoning rules generalized hypothetical syllogism (GHS), generalized modus tollens (GMT) and generalized reduction to absurdity (GRA) play an indispensable role in selecting appropriate fuzzy implications in specific tasks. In order to enhance the set of fuzzy implications satisfying these three properties to a greater extent, in this work, we investigate in detail under what conditions the T-power based implications will satisfy (GHS), (GMT) and (GRA). Then the necessary and sufficient conditions are given. The results obtained can not only provide a theoretical basis for selecting fuzzy implication satisfying these properties, but also help us further insight the behaviour of this new kind of fuzzy implication.

Interval-valued fuzzy inference based on aggregation functions

Generalized modus ponens (GMP) and generalized modus tollens (GMT), as two basic patterns of approximate reasoning, aim to acquire some reasonable imprecise conclusions from a collection of imprecise premises using some inference rules. To solve the GMP and GMT problems under interval-valued fuzzy setting, an interval-valued A-compositional rule of inference (ACRI) method and quintuple implication principle (QIP) method with interval-valued implication generated by A under any partial order are presented in this paper, where A is an interval-valued aggregation function. In order to develop these methods, we firstly discuss interval-valued negation generated by an interval-valued aggregation function with any partial order. Some properties of interval-valued implications generated by interval-valued aggregation functions with an arbitrary order are then analyzed. We further investigate the ACRI method and quintuple implication principle (QIP) method with interval-valued implication generated by interval-valued aggregations to solve the interval-valued fuzzy modus ponens (IFMP) and interval-valued fuzzy modus tollens (IFMT). Finally, two examples are implemented to illustrate our proposed approaches using some special interval-valued aggregation functions.

Propositional syntax and semantics induced knowledge re-structuring in a fuzzy logic network for ad hoc reasoning

Traditional approaches to fuzzy reasoning usually employ Generalized Modus Ponens, Generalized Modus Tollens and Generalized Hypothetical Syllogisms to derive fuzzy inferences from a given set of fuzzy observations/facts and fuzzy if-then rules. However, the above approaches occasionally fail to generate inferences, although semantically plausible fuzzy inferences follow as a natural consequence of the given set of rules and fuzzy facts/observations. This paper serves to fill this void. It extends propositional syntax and semantics in the context of fuzzy reasoning to transform/re-structure individual rules to a suitable form (by negating and transposing propositions from antecedent to consequent and vice-versa), so as to get the propositions present in the antecedent (consequent) instantiated (refuted) by positive (negative) facts to enable the rules for forward (backward) firing. Next fuzzy compositional rule of inference is used to perform forward/backward reasoning. Consequently, when the rules are embedded in a fuzzy Petri net like structure, the rules apparently fire randomly in forward or backward direction without maintaining any topological order in transition-firing. Such random firing of rules in distributed parts of the network is referred to as ad hoc reasoning. Here, we transform such randomly ordered forward/backward firing sequence of transitions in the network into a fixed topological order of transition-firing in forward direction only by replacing backward firing transitions into equivalent forward firing transitions. The proposed method of automated reasoning has successfully been applied in diagnostic application of an electronic rectifier circuit, and the results are appealing.

Generalized Modus Tollens with Linguistic Modifiers for Inverse Approximate Reasoning

AbstractBased on our previous researchs about generalized modus ponens (GMP) with linguistic modifiers for If … Then rules, this paper proposes new generalized modus tollens (GMT) inference rules with linguistic modifiers in linguistic many–valued logic framework with using hedge moving rules for inverse approximate reasoning.

Similarity-Based Inverse Approximate Reasoning

This paper investigates the inverse problem of ap proximate reasoning [to conclude A' from B' and (A→ B)] from different angles. We develop a method to And a solution to the inverse problem by the use of the law of contrapositive symme try. In the process, generalized modus ponens, generalized modus tollens, and their generating functions are studied extensively. An attempt is made here to establish that the concept of similarity plays an important role in inverse approximate reasoning. We apply the concept of similarity to model inverse approximate reasoning and develop a new methodology that is named similarity-based inverse approximate reasoning for a solution to the problem. Algorithms are presented and illustrated with simple yet concrete examples. We then use Zadeh's compositional rule of inference (CRI) as a combi nation of cylindrical extension and projection and, thereby, model generalized modus tollens in a different way. We, thus, develop a third way to find a solution to the inverse problem of approximate reasoning. A comparison between the result of application of our technique and the real static characteristic of a dc shunt motor is presented.

Ordering relation of fuzzy implications

In this paper we introduce a specific method, which is based on generalized modus ponens and generalized modus tollens, for introducing an ordering relation on the set of fuzzy implications. The existence of an ordering relation between fuzzy implications helps us to order fuzzy implications in a more powerful and/or more functional classes. We believe that, apart from the theoretical interest, this method appears to have interesting applications.

진리함수사상을 이용한 일반화된 대우추론

Baldwin은 진리함수사상을 이용하는 근사추론법을 정의하였다. 우리는 논문[4]에서 기존에 사용되는 진리함수사상 이외에 새로운 진리함수사상 두 가지를 정의하여 일반화된 연역추론에 적용한 결과를 소개하였다. 이 논문에서는 이 진리함수사상을 적용한 일반화된 대우추론에 대하여 알아보고 그 결과를 소개하였다. Baldwin defined the approximate reasoning using truth function mapping. In paper [4], we defined two truth function mappings and applied these truth function mappings to generalized modus ponens. In this paper, we introduce the results of generalized modus tollens using these two truth function mappings.

Computationally efficient reasoning using approximated fuzzy intervals

In this paper we emphasize the use of sigmoid-like membership functions which take values in the open unit interval, and propose the membership driven inference (MDI) reasoning scheme. With sigmoid-like membership functions one can avoid the so-called indetermination part of the conclusion, which occur in reasoning with the compositional rule of inference (CRI). Moreover, the MDI and the min and product based CRI are closed under such membership functions. The axiomatic properties of the MDI reasoning scheme are shown, including not only the generalized modus ponens, but also the generalized modus tollens, the generalized chain rule, and more. As a special sigmoid-like function, we present the so-called squashing function by which piecewise-linear fuzzy intervals can be arbitrarily approximated. We show that by utilizing approximated fuzzy intervals in rules and premises, the MDI reasoning scheme can be efficiently calculated only by the parameters that define the fuzzy sets in the rule and the premise.

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.

Some considerations about a least squares model for fuzzy rules of inference

In this paper, the method of least squares is applied to the fuzzy inference rules. We begin studying the conditions in which from a fuzzy set we can build another through the method of least squares. Then we apply this technique in order to evaluate the conclusions of the generalized modus ponens and the generalized modus tollens. We present different theorems and examples that demonstrate the fundamental advantages of the method studied. We conclude presenting a method for obtaining the conclusion when working with fuzzy conditional propositions that contain two fuzzy propositions joined by the connective “and”.

Fuzzy Modeling based Approach to Facial Expressions Understanding

We consider two problems related to the understanding of facial expressions: (I) modeling the perception of human facial expressions from information extracted from face photographs and (II) reasoning about the transition between facial expressions as modeled in (I). For (I) we develop a hierarchical fuzzy modeling method. The results obtained are satisfactory when compared with those of other approaches: for the six basic facial expressions (happiness, surprise, sadness, anger, disgust and fear), we achieve a recognition ratio of 85 %. More importantly, in addition to the usual advantages of fuzzy rules such as easy interpretation and adaptability in a changing environment, our approach can be applied without additional training to (i) the identification of mixed facial expressions - the recognition ratio of the two strongest facial expressions in the same face being about 75%, and (ii) the recognition in noisy environment (that is, our approach can successfully deal with lack of data). For (II) we propose a model for transition between facial expressions based on a hybrid reasoning scheme combining generalized modus tollens and fuzzy backward reasoning. Appropriate examples are used to illustrate our approach and we conclude with a discussion about future issues, and possible applications to an intelligent humancomputer interface using the transition model.

Inverse approximate reasoning

We consider the problem: Given the conclusion of a fuzzy rule, find in some sense the “best” data (input) that will produce the given conclusion. We investigate the use of the generalized modus tollens to solve this problem.

A least squares model for fuzzy rules of inference

In this paper, the method of least squares which uses the power function is applied to the fuzzy inference rules. These rules, in fuzzy logic, are known as the ‘generalized modus ponens’ and the ‘generalized modus tollens’. The proposed technique gives reasonable evaluations for fuzzy consequences and is simple for computer applications.

Processing Fault-Trees by Approximate Reasoning in Solving the Technical Diagnostic Problem(A Possibilistic Diagnostic Decision-Making Method)

The work may be viewed as a qualitative jump from the use of artificial intelligence (AI) techniques to that of fuzzy logic based methods in the design of diagnost expert systems. In response to the classical AI approaches of solving the technical diagnostic problem by forward and backward processing causality propagation through a probabilistic knowledge tree, a fuzzy logic - based approach is proposed which is able to overcome the main disadvantages of the AI approaches. The main steps of the diagnostic methodology are briefly exposed in the following. A-priori a symptoms-faults fuzzy interrelational diagnostic. model is OFF-LINE designed as a basis for the diagnostic reasoning. The acquired diagnostic information is first organized in diagnostic causal networks. These, via a fuzzification procedure are transformed in diagnostic relevant networks (DRN) . By compressing via fuzzy composition the information from the DRNs the diagnostic model is obtained and encoded within the knowledge base as a collection of fuzzy IF-THEN rules. The inference mechanism performes ON-LINE diagnosis in two steps. First by approximate reasoning on the fuzzy diagnostic model with the detected values of the symptoms the model is adapted to the current diagnostic situation. Then, by aggregating its information according to the connectives of the URNs, the adapted diagnostic model is compressed to a vector containing fault possibilities. As fault candIdates the ones with the maximal fault possibilities are considered by being validated via generalized modus tollens. The proposed diagnostic methodology is developed on a very simple example for a DC motor.

FUZZY LOGICS AND THE GENERALIZED MODUS PONENS REVISITED

After a background on the use of triangular norms in multiple-valued logics, a systematic investigation of implication functions based on triangular norms is carried out; five families of implication functions are considered. Then Zadeh's generalized modus ponens is thoroughly reconsidered in the framework of possibility theory. It is pointed out that the choice of the implication function for representing the rule “if X is A, then Y is B” is completely determined by the choice of the conjunctive operation used for aggregating an a priori possibility distribution with a conditional possibility distribution. The chaining of conditional rules is proved to be transitive when we choose a continuous triangular norm as conjunctive operation. The generalized modus tollens, the treatment of truth-qualified rules and the processing of a collection of conditional rules providing an incomplete description of a causal link, are studied in detail. Finally the generalized modus ponens, as a pattern of deductive reasoning taking into account the fuzziness of premises, is contrasted with patterns of plausible reasoning.

Comparison of fuzzy reasoning methods

L.A. Zadeh, E.H. Mamdani, and M. Mizumoto et al. have proposed methods for fuzzy reasoning in which the antecedent involves a fuzzy conditional proposition 'If x is A then y is B', with A and B being fuzzy concepts. Mizumoto et al. have investigated the properties of their methods in the case of 'generalized modus ponens'. This paper deals with the properties of their methods in the case of 'generalized modus tollens', and investigates the other new fuzzy reasoning methods obtained by introducing the implication rules of many valued logic systems. Finally, the properties of syllogism and contrapositive are investigated under each fuzzy reasoning method.