Complex fuzzy logic

A novel framework for logical reasoning, termed complex fuzzy logic, is presented in this paper. Complex fuzzy logic is a generalization of traditional fuzzy logic, based on complex fuzzy sets. In complex fuzzy logic, inference rules are constructed and "fired" in a manner that closely parallels traditional fuzzy logic. The novelty of complex fuzzy logic is that the sets used in the reasoning process are complex fuzzy sets, characterized by complex-valued membership functions. The range of these membership functions is extended from the traditional fuzzy range of [0,1] to the unit circle in the complex plane, thus providing a method for describing membership in a set in terms of a complex number. Several mathematical properties of complex fuzzy sets, which serve as a basis for the derivation of complex fuzzy logic, are reviewed in this paper. These properties include basic set theoretic operations on complex fuzzy sets - namely complex fuzzy union and intersection, complex fuzzy relations and their composition, and a novel form of set aggregation - vector aggregation. Complex fuzzy logic is designed to maintain the advantages of traditional fuzzy logic, while benefiting from the properties of complex numbers and complex fuzzy sets. The introduction of complex-valued grades of membership to the realm of fuzzy logic generates a framework with unique mathematical properties, and considerable potential for further research and application.     

Prolog-ELF incorporating fuzzy logic

Prolog-ELF incorporating fuzzy logic and several useful functions into Prolog has been implemented as a basic language for building knowledge systems with uncertainty or fuzziness. Prolog-ELF inherits all the desirable basic features of Prolog. In addition to assertions with truth-values between 1.0 and 0.5 (0 for exceptional cases), fuzzy sets can be very easily manipulated. An application of fuzzy logical database is illustrated.     

Neural fuzzy logic programming

A foundational development of propositional fuzzy logic programs is presented. Fuzzy logic programs are structured knowledge bases including uncertainties in rules and facts. The precise specifications of uncertainties have a great influence on the performance of the knowledge base. It is shown how fuzzy logic programs can be transformed to neural networks, where adaptations of uncertainties in the knowledge base increase the reliability of the program and are carried out automatically.     

Toward a better self-regulation: degree of certainty through fuzzy logic in a formative assessment

In the present paper, we plan to introduce a new procedure for learner’s assessment in the learning environment in a way true to reality by using fuzzy logic. In this evaluation, learner’s responses are accompanied by a degree of certainty expressed by him. This method allows detection of problems encountered by the learner and also to fix the concepts mastered and those that are not. It is a diagnostic procedure that improves the process of content adaptation and self-adjustment on the one hand and makes the knowledge model clearly interpretable and more understandable to learner and tutor/teacher/head teacher on the other hand.     

Medical diagnosis with the aid of using fuzzy logic and intuitionistic fuzzy logic

The objective of the present study is to develop/establish a web-based medical diagnostic support system (MDSS) by which health care support can be provided for people living in rural areas of a country. In this respect, this research provides a novel approach for medical diagnosis driven by integrating fuzzy and intuitionistic fuzzy (IF) frameworks. Subsequently, based on the proposed approach a web-based MDSS is developed. The proposed MDSS comprises of a knowledge base (KB) and intuitionistic fuzzy inference system (IFIS). Based on the observation that medical data cannot be described with both precision and certainty, a medical KB is constructed in the form of a set of if-then decision rules by employing both fuzzy and IF logics. After constructing the medical KB, a new set of patients is considered for diagnosing the diseases. For each patient, linguistic values of the patients’ symptoms are considered as inputs of the proposed IFIS and modeled by using the generalized triangular membership functions. Subsequently, integrated fuzzy and IF rule-based inference system is used to find a valid conclusion for the new set of patients. In a nutshell, in this paper fuzzy rule-based and IFS based inference systems are combined for better and more realistic representation of uncertainty of the medical diagnosis problem and for more accurate diagnostic result. The method is composed of following four steps: (1) the modeling of antecedent part of the rules, which consist of linguistic assessments of the patients’ symptoms provided by the doctors/medical experts with their corresponding confidence levels, by using generalized fuzzy numbers; (2) the modeling of consequent part, which reveals the degree of association and the degree of non-association of diseases into the patient, by using IFSs; (3) the use of IF aggregation operator in inference process; (4) the application of relative closeness function to find the final crisp output for a given diagnosis. Finally, the applicability of the proposed approach is illustrated with a suitable case study. This article has also justified the proposed approach by using similarity measurement.     

On fuzzy equality and approximation in fuzzy logic

The paper is a contribution to the theory of fuzzy logic in narrow sense with evaluated syntax (FLn). We show that the concepts of fuzzy equality and the provability degree enable to generalize the concept of fuzzy approximation. In the second part of the paper we return to the Mamdani-Assilian formula, which is formed on the basis of the so called totally bounded fuzzy equality and using which we can approximate any function with the prescribed accuracy.This paper has been supported by Grant A1187901/99 of the GA AV R and the project VS96037 of MMT of the Czech Republic.     

On contradiction in fuzzy logic

 We clarify which space of functions in [0, 1] ^E would be reasonable in fuzzy logic in order to avoid self-contradiction.     

Implication operators in fuzzy logic

The choice of fuzzy implication as well as other connectives is an important problem in the theoretical development of fuzzy logic, and at the same time, it is significant for the performance of the systems in which fuzzy logic technique is employed. There are mainly two ways in fuzzy logic to define implication operators: (1) an implication operator is considered as the residuation of conjunction operator; and (2) it is directly defined in terms of negation, conjunction, and disjunction operators. The purpose of this paper is to determine the number of implication operators defined in the second way for some usual negation, conjunction and disjunction operators in fuzzy logic     

A PWM fuzzy logic controller

Researchers usually implement fuzzy inference systems in software on digital computers or microprocessors. This approach copes with most problems, however real-time systems often require very short time responses. In this case, a hardware implementation becomes the only solution. This 1.5-μm CMOS implementation uses a current mode circuit to generate input membership functions and processes inferences using pulse width modulation     

A complementary fuzzy logic system

This article discusses complementary (C) fuzzy logic system that is one of continuous multiary logic systems that satisfies a complementary law differently from usual fuzzy logic systems. This article includes formulation of the C fuzzy logic system, derivation of tautologies, and mentions an example that typically shows a difference in inference computation between the C fuzzy logic system and a usual fuzzy logic system.     

Soft computing and fuzzy logic

What is soft computing? What is fuzzy computing? What is the relationship between them? This paper intends to provide clear answers to these questions. We focus on exploring the notions of the fuzzy coordinate system and the related transformations between qualitative and quantitative information. These notions are considered to be the core ideas of fuzzy computing. Then the three novel theories of fuzzy computing and soft computing developed by the first author of this paper, namely, the Falling Shadow Representation of fuzzy theory, the Factors Space theory and the Truth Value Flow Inference theory are introduced.     

Knowledge representation in fuzzy logic

The author presents a summary of the basic concepts and techniques underlying the application of fuzzy logic to knowledge representation. He then describes a number of examples relating to its use as a computational system for dealing with uncertainty and imprecision in the context of knowledge, meaning, and inference. It is noted that one of the basic aims of fuzzy logic is to provide a computational framework for knowledge representation and inference in an environment of uncertainty and imprecision. In such environments, fuzzy logic is effective when the solutions need not be precise and/or it is acceptable for a conclusion to have a dispositional rather than categorical validity. The importance of fuzzy logic derives from the fact that there are many real-world applications which fit these conditions, especially in the realm of knowledge-based systems for decision-making and control     

Basic fuzzy logic and BL-algebras

 The many-valued propositional logic BL (basic fuzzy logic) is investigated. It is known to be complete for tautologies over BL-algebras (particular residuated lattices). Each continuous t-norm on [0,1] determines a BL-algebra; such algebras are called t-algebras. Two additional axioms B1, B2 are found such that BL+(B1,B2) is complete for tautologies over t-algebras. It remains open whether B1, B2 are provable in BL.     

Systems dynamics with fuzzy logic

Systems dynamics has been used to model and simulate a variety of environments, e.g. economic, social and political, which require quantification or some types of human behaviour. The lack of empirical verification of the relationships in the systems dynamics models has often been criticised. Nevertheless, the methodology is effective in dealing with time-varying (dynamic) interactions among components of the analysed system. The effectiveness of systems dynamics as a methodology for modelling, simulating and analysing real-life systems can be significantly increased if it is extended to deal with imprecise and vague variables or events. Such an extension requires: (1) treatment of imprecise and vague input variables as fuzzy variables: (2) use of fuzzy arithmetic in the level, rate and auxiliary equations when fuzzy numbers are involved; and (3) replacement of some of the relationships in the systems dynamics models either with conditional statements including fuzzy variables, or with fuzzy algorithms.     

Type-2 fuzzy logic systems

We introduce a type-2 fuzzy logic system (FLS), which can handle rule uncertainties. The implementation of this type-2 FLS involves the operations of fuzzification, inference, and output processing. We focus on “output processing,” which consists of type reduction and defuzzification. Type-reduction methods are extended versions of type-1 defuzzification methods. Type reduction captures more information about rule uncertainties than does the defuzzified value (a crisp number), however, it is computationally intensive, except for interval type-2 fuzzy sets for which we provide a simple type-reduction computation procedure. We also apply a type-2 FLS to time-varying channel equalization and demonstrate that it provides better performance than a type-1 FLS and nearest neighbor classifier     

Soft computing and fuzzy logic

Discusses soft computing, a collection of methodologies that aim to exploit the tolerance for imprecision and uncertainty to achieve tractability, robustness, and low solution cost. Its principal constituents are fuzzy logic, neurocomputing, and probabilistic reasoning. Soft computing is likely to play an increasingly important role in many application areas, including software engineering. The role model for soft computing is the human mind     

Word recognition using fuzzy logic

This paper presents an offline word-recognition system based on structural information in the unconstrained written word. Oriented features in the word are extracted with the Gabor filters. We estimate the Gabor filter parameters from the grayscale images. A two-dimensional fuzzy word classification system is developed where the spatial location and shape of the membership functions are derived from the training words. The system achieves an average recognition rate of 74% for the word being correctly classified in the top position and an average of 96% for the word being correctly classified within the top five positions