Fuzzy Approximation as a Theory Stemming from Fuzzy IF-THEN Rules

The notion of a linguistic variable [17] together with fuzzy IF-THEN rules suggested a new formalism in the description of a man-machine interaction. Theoretical analysis of this formalism led to creation of a number of fuzzy theories: fuzzy logics, fuzzy measures, theory of t-norms etc. Successful applications of fuzzy IF-THEN rules in control, decision-making and expert systems [16] challenged justification of this technique. A new direction called universal approximation arose in order to fulfil this requirement. We will observe some milestones of this approach and show its relevance for a new branch of the theory of approximation which can be called fuzzy approximation. In brief, fuzzy approximation relates not only to approximation of imprecisely given functions, but also to approximation of more general correspondences expressed by relations and fuzzy relations. In fact, fuzzy IF-THEN rules constitute an informal characterization of some dependence which may be further formalized in a plenty of ways. However, we can distinguish two principally different ways of formalizations which originate to two well known logical formulas — the disjunctive (DNF) and conjunctive normal form (CNF). The first way of formalization is usually connected with the name of E. H. Mamdani, while the second one came from logical analysis. There exist many universal approximation results which are derived from Mamdani-style DNF formulas; first such results were formulated and proved, almost simultaneously, in 1990–92 papers by J. Buckley, Z. Cao, E. Czogala, D. Dubois, M. Grabisch, J. Han, Y. Hayashi, C.-C. Jou, A. Kandel, B. Kosko, H. Prade, and L.-X. Wang for a survey, see, e.g., [5] and reference therein. There also exist several universal approximation results for implication-style CNF formulas [2, 8, 9]. The fascinating ability of logical formulas to accomplish universal approximation inspired some authors to justify the approximation technique from pure logical point of view. P. Hájek considered in [3] special deduction rules and showed that they are consistent with the first order fuzzy predicate calculus. I. Perfilieva proved in [11] a conditional equivalence between logical DNF/CNF formulas and extensional formulas. V. Novák in [7] creates fuzzy type theory as a higher order fuzzy logic; its expressive power has potential to develop logical theory of human way of reasoning that is based on the use of natural language. Formal fuzzy logic based on continuous t-norms [3] has been demonstrated to be a good basis for approximate description of different dependencies. Among advantages of particular approximation models is the possibility to work with them in further computations. As known from the theory of numerical methods, approximation models can replace original complex functions in some computations, such as solving differential equations and others. In [13], a new type of approximation model has been obtained after two fuzzy transforms of the original function: direct and inverse. This model is universal in the sense that it is represented by a special form, unique for each approximating function. The way how the parameters of the approximation model are computed is analogous to that used in the theory of generalized functions, Fourier and Laplace transforms. Based on this analogy, this technique is called fuzzy transform, or shortly F-transform. Is is shown in [13] that precise and approximate value of a definite integral as well as approximate solution to ordinary differential equations can be obtained on the basis of F-transform can. The technique of F-transform has further been extended in [14, 15] to obtain approximate solution to partial differential equations. Therefore, we call this type of technique numerical methods based on fuzzy approximation models.