Fuzzy Relation Equations and Their Applications to Knowledge Engineering

This Ch. can be regarded as an introduction to this book that originates from the primordial need to collect results from several papers on fuzzy relation equations. First of all we would like to emphasize some considerations of a general character in fuzzy set theory. This will enable the reader to get a certain perspective on this field of research. Moreover these remarks could suggest a philosophy behind the methodology of fuzzy sets.

We would like to underline the following statement of Goguen [5]: “The importance of relations is almost self-evident. Science is, in a sense, the discovery of relations between observables... Difficulties arise in the so-called “soft” sciences because the relations involved do not appear to be “hard”, as they are, say, in classical physics. A thoroughgoing application of probability theory has relieved many difficulties, but it is clear that others remain. We suggest that further difficulties might be cleared up through a systematic exploitation of fuzziness”.

The whole of this Ch. is devoted to the study of the lower solutions of max-min fuzzy equations. In all the Secs., except Secs. 5 and 6, we assume L to be a linear lattice with universal bounds 0 and 1 and the domains, on which fuzzy sets and fuzzy relations are defined, to be finite sets, we denote by |X| the cardinality of a finite set X. In Secs. 1 and 2, we deal with max-min fuzzy equations of type (2.5) and (2.6), respectively. Further lattice results in the set S are given in Sec.3 and interesting properties of a particular fuzzy relation of S are pointed out in Sec.4. Secs.5 and 6 are devoted to the study of lower solutions of max-min fuzzy equations defined on complete Brouwerian lattices and on complete completely distributive lattices, respectively.

In this Ch. we continue to study the max-min fuzzy relation equation (3.3) defined on finite sets and assuming L to be a linear lattice with universal bounds 0 and 1 but endowed with an additional structure L’ of a complete linear lattice with universal bounds 0’ and 1’, tied to the foregoing one by accurate and reasonable requirements. These are basic preliminaries in order to solve some important optimization problems in the set of solutions of a fuzzy equation. Strictly speaking, introducing a suitable functional which measures the “fuzziness content” of a fuzzy relation, we characterize all the solutions of a max-min composite fuzzy equation possessing the smallest and the greatest value of such fuzziness.

This Ch. is entirely dedicated to the study of the Boolean (i.e. non-fuzzy) solutions of a max-min fuzzy equation defined on a complete Brouwerian lattice L with universal bounds 0 and 1. In Sec.1, we give a simple necessary and sufficient condition for the existence of the greatest Boolean solution of S and in Sec.2, using some results of Ch.3, we determine the minimal Boolean elements of S if L is a linear lattice. In Sec.3, if Boolean solutions do not exist, an algorithm is presented to find the elements of S with maximum Boolean degree, i.e. with the maximum number of 0 and 1. The referential sets are assumed to be necessarily finite in Secs.2 and 3.

In this Ch., we investigate a new type of fuzzy relation equation defined on finite sets and with membership functions on a complete Brouwerian lattice. We characterize the entire set of the solutions when the fuzzy equation is assigned on a linear lattice. We also study when a fuzzy relation is pointwise decomposable in the intersection (resp. union) of two fuzzy sets and we show that such relations are max-min transitive (resp. compact). Finally, properties of convergence of powers of these fuzzy relations are established.

In this Ch., we solve a decomposition problem, more complicated than the problem mentioned in Sec.6.5, and precisely, we present a numerical algorithm, illustrated by a flowchart, which assures the existence of a fuzzy relation Z∈F(X×X) such that Z⊙Z=R, where R∈F(X×X) is an assigned fuzzy relation defined on a referential set X and assuming values in a linear lattice L with universal bounds 0 and 1. Connections with results already existing in recent literature are also given.

In this Ch. we present another generalization of the fuzzy relation equations which have been considered in the previous chapters. Here we focus our attention on a broad class of logical connectives applied in fuzzy set theory, i.e. triangular norms (for short, t-norms) and conorms (for short, s-norms) which in turn enable us to consider max-t and min-s compositions. In Sec.8.1 we present essentially the concept of the logical connectives modelled by t-norms and related s-norms. In Sec.8.2 we analyze max-t fuzzy relation equations and related dual min-s fuzzy relation equations, assuming that t (resp s) is lower (resp. upper) semicontinuous. In Sec.8.3 we pay attention to equations of complex structure and a related adjoint fuzzy relation equation is studied in Sec.8.4. In the last Sec.8.5, max-t fuzzy relation equations under upper semicontinuous t-norms are studied. All the equations considered in this Ch. are assigned on finite referential sets.

In this Ch. we will introduce other composition operators which have a plausible logical interpretation. They will be called equality and difference operator, respectively. We first define a notion of equality and difference of any two grades of membership of a fuzzy set and of a fuzzy relation. In the sequel this concept will be utilized to form the respective composition operators and the related fuzzy equations. Afterwards, we provide a method of resolution of these equations, characterizing completely the set of the solutions. All the fuzzy sets involved are defined on finite sets and with membership values in [0,1].

In the previous Chs. we have discussed various problems concerning solutions of fuzzy relation equations. Obviously an underlying assumption is that there exists a nonempty set of solutions. A situation, may occurr, and indeed it is quite common, in which no solution exists. Nevertheless even in this case one might be interested to obtain an approximate solution and know to which extent it can be viewed as a solution. This stream of investigations is particularly interesting for applicational purposes. Contrary to the topics already discussed in the previous Chs., this is a field of research which has not been developed enough so far. It concerns studies on solvability properties of fuzzy relation equations. In this Ch., we shall try to answer, by using several techniques, the following basic question: how difficult is it to attain a situation in which the system of equations has solutions and then how to measure this property?

In this Ch., as well as in the following, we will study a unified approach for handling and processing sources of uncertainty in knowledge-based systems. This goal is achieved in the framework of fuzzy relation equations. We point out how the mechanisms of the theory developed in the previous Chs. of this book can be treated as a convenient platform for construction of knowledge-based systems. More precisely, it will be indicated how fuzzy equations contribute to each of the conceptual levels recognized in the construction of these systems (viz. knowledge representation, meta-knowledge, inference techniques, etc.) as well as how they are directly used in formation of the particular elements of the problem-oriented expert systems. It is assumed that the reader has a certain background concerning Knowledge Engineering and Artificial Intelligence, at least on fundamentals of architecture of knowledge-based systems. It is also expected that he is familiar with some of the well-known expert systems, especially those broadly documented in literature (e.g. PROSPECTOR, MYCIN) and mechanisms involved there which are capable of coping with uncertainty, no matter how it has been introduced. This Ch. must be viewed as a concise prerequisite for the successive Chs. and it indicates many problems occurring in knowledge engineering when factors of uncertainty have to be processed.

This Ch. is devoted to central issues arising in any knowledge-based system. Concisely speaking, having already a specified scheme of knowledge representation, we are interested in getting the knowledge concerning the area of interest and, with the aid of the format dictated by the knowledge representation, coding it and indicating a way of effective utilization.

This Ch. summarizes some common techniques of inference utilized for fuzzy data. Special attention has been paid to the implementation of modus ponens (which realizes a data-driven mode of reasoning) and modus tollens (corresponding to a goal-driven mode of reasoning). The detachment principle (corresponding to a means of expressing a similarity between fuzzy statements) is also investigated. We discuss how different forms of fuzzy relation equations are used to handle each of these modes of inference. Also the question of a direct link between the relevancy of the KB and the length of the inference chain leading to meaningful conclusions is considered. This is of primordial importance; it has to be analyzed to interpret the results of inference and, in particular, to visualize precision. A proper reformulation of the problem in terms of fuzzy equations makes it possible to consider this knowledge transformation in a greater detail.

The schemes of inference, introduced in Sec.2, are used in Sec.3 to discuss the special case of an expert system oriented towards control of an industrial process consisting of a steam engine and a boiler. We consider here only max-min fuzzy equations and α-fuzzy equations (cfr. Ch.6) defined on finite sets.

We conclude this book exhibiting an useful list of papers on fuzzy relation equations which, to the best of our knowledge, have been published on several books and international journals. This list (of course to be considered as not exhaustive) also includes papers dealing with applications of fuzzy relation equations and close topics. Thus the reader can have a global point of view on the present literature, which, of course, this book could not entirely cover. We avoid recalling the papers and the books already cited in the references of the previous Chs. An additional list of papers covering several topics close to the theory of fuzzy relations and their applications is also enclosed. For some papers, unfortunately, an updated reference is not complete (we have only related preprints). Due to the abundance of the present literature, some authors could have been involuntarily omitted.