Fuzzy Logic Theory and Applications

The following sections are included:

• Dedication
• Preface
• About the Authors
• About the Contributors
• Contents

A fuzzy set is a class of objects with a continuum of grades of membership. Such set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one. The notions of inclusion, union, intersection, complement, relation, convexity, etc., are extended to such sets, and various properties of these notions in the context of fuzzy sets are described. In particular, a separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint [501].

The term fuzzy logic (FL) is used in this chapter to describe an imprecise logical system, in which the truth-values are fuzzy subsets of the unit interval with linguistic labels such as true, false, not true, very true, quite true, not very rue and not very false, etc. The truth-value set, $\mathcal{T}$, of FL is assumed to be generated by a context-free grammar, with a semantic rule providing a means of computing the meaning of each linguistic truth-value in $\mathcal{T}$ as a fuzzy subset of [0,1]…

A precise definition of truth is formulated in this chapter. The theory outlined in the chapter is a departure from traditional theories of truth and meaning. In the suggested theory, truth values are allowed to be described in the natural language. For example, more or less true, almost true, usually true, etc. Such truth values are referred in this theory as linguistic truth values. Linguistic truth values are not allowed in traditional logical systems, but are routinely used by humans in everyday reasoning and everyday discourse. The centerpiece of this theory is a deceptively simple concept — the concept of a restriction. The suggested theory opens the door to the construction of mathematical solutions of computational problems, which are stated in the natural language.

The conventional approaches to decision analysis are based on the assumption that the probabilities which enter into the assessment of the consequences of a decision are known numbers. In most realistic settings, this assumption is of questionable validity since the data from which the probabilities must be estimated are usually incomplete, imprecise or not totally reliable…

Classical crisp functions are important in mathematical modeling. Fuzzy modeling is based on fuzzy functions. Fuzzy functions may be obtained as an extension of a crisp function to map fuzzy sets to fuzzy sets. Fuzzy functions may be described by using methods such as the extension principle and the alpha cuts-based method. Different properties such as differentiability and integrability of a fuzzy function are considered in this chapter.

Many of the systems encountered in the real-world; are too complex and/or too ill-defined to be susceptible of exact analysis. The concept of a fuzzy set, that is, a class which of intermediate admits grades of membership in it, opens the possibility of analyzing such systems both qualitatively and quantitatively by allowing the input and/or the output and/or the state of the system to range over fuzzy sets. In this chapter, several basic concepts relating to the characterization of fuzzy systems are considered.

The concept of a Z-number has a potential for many applications, especially in the realms of computation with probabilities and events described in NL. Of particular importance are applications in economics, decision analysis, risk assessment, prediction, anticipation, planning, biomedicine and rule-based manipulation of imprecise functions and relations…

Uncertainty is an attribute of information. The path-breaking work of Shannon has led to a universal acceptance of the thesis that information is statistical in nature. Concomitantly, existing theories of uncertainty are based on probability theory. The generalized theory of uncertainty (GTU) departs from existing theories in essential ways [542]…

In restriction-based semantics, the meaning of propositions is defined as follows. It is needed to answer: what the restricted variable is; what restricting relation is; and how restricting relation restricts the restricted variable. In natural languages, restrictions are predominantly possibilistic. In this chapter, a comprehensive study of restriction-based semantics as a generalization of truth-conditional and possible-world semantics is given. Precisiation of propositions through the use of restriction-based semantics is discussed.

Granular Computing is concerned with constructing and processing carried out at the level of information granules. Using information granules, we comprehend the world and interact with it, no matter which intelligent endeavor this may involve. The landscape of Granular Computing is immensely rich and involves set theory (interval mathematics), fuzzy sets, etc. in a highly synergetic environment. This chapter is a comprehensive study of this emerging paradigm and embraces its fundamentals, underlying methodological framework and a sound algorithmic environment.

Fuzzy logic, introduced by Zadeh along with his introduction of fuzzy sets, is a continuous multi-valued logic system. Hence, it is a generalization of the classical logic and the classical discrete multi-valued logic (e.g., Łukasiewicz’ three/many-valued logic). Throughout the years Zadeh and other researches have introduced extensions to the theory of fuzzy setts and fuzzy logic. Notable extensions include linguistic variables, type-2 fuzzy sets, complex fuzzy numbers, and Z-numbers. Another important extension to the theory, namely the concepts of complex fuzzy logic and complex fuzzy sets, has been investigated by Kandel et al. This extension provides the basis for control and inference systems relating to complex phenomena that cannot be readily formalized via type-1 or type-2 fuzzy sets. Hence, in recent years, several researchers have used the new formalism, often in the context of hybrid neuro-fuzzy systems, to develop advanced complex fuzzy logic-based inference applications. In this chapter we reintroduce the concept of complex fuzzy sets and complex fuzzy logic and survey the current state of complex fuzzy logic, complex fuzzy sets theory, and related applications.

Recently, fuzzy control systems have been widely used for automatic control of complex processes. This type of control systems, being a qualitatively new class of control systems, proved to be very effective in control of complex nonlinear uncertain dynamic objects for which the classical deterministic and stochastic controllers are unable to operate. Fuzzy controller is a knowledge-based controller in which fuzzy logic is used to represent knowledge and logical inference…

The existing decision models have been successfully applied to solving many decision problems in management, business, economics, and other fields, but nowadays arises a need to develop more realistic decision models. The main drawback of the existing utility theories starting from von Neumann–Moregnstern expected utility to the advanced nonexpected models is that they are designed for laboratory examples with simple, well-defined gambles which do not adequately enough reflect real decision situations. In real-life decision-making problems preferences are vague and decision-relevant information is imperfect as described in natural language (NL). Vagueness of preferences and imperfect decisionrelevant information require using suitable utility model which would be fundamentally different to the existing precise utility models. Precise utility models cannot reflect vagueness of preferences, vagueness of objective conditions and outcomes, imprecise beliefs, etc. The time has come for a new generation of decision theories. In this study, we propose a decision theory, which is capable to deal with vague preferences and imperfect information. The theory discussed here is based on a fuzzyvalued non-expected utility model representing linguistic preference relations and imprecise beliefs.

Two conflicting goals are often involved in the design of fuzzy rule-based systems: accuracy maximization and interpretability maximization. A number of approaches have been proposed for finding a fuzzy rule-based system with a good accuracy–interpretability tradeoff. Formulation of the accuracy maximization is usually straightforward in each application area of fuzzy rule-based systems such as classification, regression and forecasting. Formulation of the interpretability maximization, however, is not so easy. This is because various aspects of fuzzy rule-based systems are related to their interpretability. In this chapter, we activate discussions on how to measure the interpretability of fuzzy rule-based systems which are used for classification purposes.

In this chapter, we discuss fuzzy reinforcement learning (FRL). The field of reinforcement learning (RL) can be classified into two generations RL1 and RL2 based on the invention of Q-Learning by Watkins [466]. RL1 was suffering from a major short coming that here was no analytical proof of convergence for RL. Using Q-Learning, this problem was solved when Tsitsiklis and Konda [250] analytically proved convergence of RL. We describe FRL1 which was the first generation of fuzzy RL. The most significant result of FRL1 was the generalized approximate-reasoning-based intelligent control (GARİC) [65] architecture. GARIC combined the state of the art in reinforcement learning with fuzzy logic. We then discuss how GARIC can be used for performance evaluation [68]. FRL1 was followed by FRL2. The most significant development of FRL2 was the invention of the actor-critic-based fuzzy reinforcement learning algorithm (ACFRL) [70]. The ACFRL algorithm has been followed by fuzzy Q-learning to solve fuzzy dynamic programming (FDP).

System modeling based on conventional mathematical tools (e.g., differential equations) is not well suited for dealing with ill-defined and uncertain systems. By contrast, a fuzzy inference system employing fuzzy if-then rules can model the qualitative aspects of human knowledge and reasoning processes without employing precise quantitative analyses. In this chapter, we consider a novel architecture called the Adaptive Network-based Fuzzy Inference System, or simply ANFIS, which can serve as a basis for constructing a set of fuzzy if-then rules with appropriate membership functions to generate the stipulated input–output pairs. Modeling of nonlinear functions and prediction of chaotic time series are considered.

Expert system (ES) is a computer-based intelligent system that emulates the mental abilities of a human expert. It performs in all respects like a human expert. Usually, human experts’ belief and knowledge are imprecise and uncertain, and described by fuzzy concepts. Widespread interest in fuzzy ESs can be attributed to these systems’ ability to aid various institutions in solving important practical, real-world problems. In this chapter, we discuss different types of ESs, emphasizing on the fuzzy ES ESPLAN created by the author.

In this chapter, we will consider the model construction of complex systems with fuzzy cognitive maps, which provide us with a quite simple method for simulating the diversified phenomena of the real world. Since several mathematical methods for examining these maps are available already, we will in turn focus on the statistical aspects and in particular on the logistic regression models.

The characteristics of the human body are quite different from person to person. This fact makes it difficult to examine, diagnose, care and cure humans. The complexity of their clinical practice and health care should be manipulated by a dynamic and human-centered theory in computers. The most appropriate one is fuzzy logic. Fuzzy logic would play a primary role in future medical and health care systems. We consider applications of fuzzy logic to process the signal of brain imaging and evaluate medical checkup data.

The following sections are included:

• Bibliography
• Index