Kaoru Hirota

Approximate reasoning by linear rule interpolation and general approximation

Abstract The problem of sparse fuzzy rule bases is introduced. Because of the high computational complexity of the original compositional rule of inference (CRI) method, it is strongly suggested that the number of rules in a final fuzzy knowledge base is drastically reduced. Various methods of analogical reasoning available in the literature are reviewed. The mapping style interpretation of fuzzy rules leads to the idea of approximating the fuzzy mapping by using classical approximation techniques. Graduality, measurability, and distance in the fuzzy sense are introduced. These notions enable the introduction of the concept of similarity between two fuzzy terms, by their closeness derived from their distance. The fundamental equation of linear rule interpolation is given, its solution gives the final formulas used for interpolating pairs of rules by their α-cuts, using the resolution principle. The method is extended to multiple dimensional variable spaces, by the normalization of all dimensions. Finally, some further methods are shown that generalize the previous idea, where various approximation techniques are used for the α-cuts and so, various approximations of the fuzzy mapping R : X → Y .

Interpolative reasoning with insufficient evidence in sparse fuzzy rule bases

Abstract Rule based fuzzy approximate reasoning uses various techniques of modified modus ponens . The observation is in most cases not identical with any of the antecedents in the rules. However, a conclusion still can be computed by using some combination of all consequents where an overlapping of observation and antecedent is present. If the rule base is sparse, i.e., it contains insufficient information on the total state space, it might occur that an observation has absolutely no overlapping with any of the antecedents and so not even a single rule is fired, i.e., no conclusion can be computed on the basis of modus ponens . In such a case, interpolative reasoning in the strict sense can be applied: some kind of (weighted) average of the flanking rules is calculated. This technique can be extended to a form of extrapolation, when the observation is not flanked from both sides. Linear interpolation and extrapolation is presented, and then the idea is extended to arbitrary approximation.

Construction of fuzzy models through clustering techniques

Abstract In this note we will discuss fuzzy models designed with the aid of fuzzy clustering. The clustering method utilizes a generalized objective function involving a collection of linear varieties. In this way the model is distributed and consists of a series of ‘local’ linear-type models (based on the revealed clusters). In comparison to the algorithms existing in the literature and producing function-like models, the proposed one is of a relational character allowing for multidirectional accessibility. A detailed identification algorithm is given and illustrated with the aid of numerical examples.

Ordering, distance and closeness of fuzzy sets

Abstract In dense rule bases where the observation usually overlaps with several antecedents in the rule base, various algorithms are used for approximate reasoning and control. If the antecedents are located sparsely, and the observation does not overlap as a rule with any of the antecedents, function approximation techniques combined with the Resolution Principle lead to applicable conclusions. This kind of approximation is possible only if a new concept of ordering and distance, i.e. a metric in the state space, and a partial ordering among convex and normal fuzzy sets (CNF sets) is introduced. So, the fuzzy distance of two CNF sets can be defined, and by this distance, closeness and similarity of CNF sets, as well.

OR/AND neuron in modeling fuzzy set connectives

The paper introduces a neural network-based model of logical connectives. The basic processing unit consists of two types of generic OR and AND neurons structured into a three layer topology. Due to the functional integrity we will be referring to it as an OR/AND neuron. The specificity of the logical connectives is captured by the OR/AND neuron within its supervised learning. Further analysis of the connections of the neuron obtained in this way provides a better insight into the nature of the connectives applied in fuzzy sets by emphasizing their features of locality and interactivity. Afterward, we will study several architectures of neural networks comprising these neurons treated as their basic functional components. The numerical studies embrace both the structures formed by single OR/AND neurons and aimed at modeling logical connectives (including the Zimmermann-Zysno data set, 1980) and the networks representing various decision-making architectures. We will also propose a realization of a pseudo median filter in which the OR/AND neurons play an ultimate role. >

The concept of fuzzy flip-flop

The authors propose and define a fuzzy flip-flop that is an extended form of an ordinary binary flip-flop, specifically, a J-K flip-flop. A truth table for a J-K flip-flop is fuzzified, extending binary NOT, AND, and OR operations to fuzzy negation, t-norm and s-norm, respectively. Two types of fundamental characteristic equations of the fuzzy flip-flop are introduced: the reset- and the set-type equations, both of which are fuzzy extensions of a characteristic equation of a J-K flip-flop. Their characteristics are demonstrated graphically, especially in the case in which fuzzy negation, t-norm and s-norm relate to complementation, min, and max operations, respectively. Other fundamental fuzzy operations are examined, and their characteristics are demonstrated graphically. Both of the above types are unified in the case of complementation, min, and max operations, and a fundamental characteristic equation for a min-max-type fuzzy flip-flop negation, t-norm and s-norm gates is proposed. A circuit is presented and tested. >

Interpolation in hierarchical fuzzy rule bases

A major issue in the field of fuzzy applications is the complexity of the algorithms used. In order to obtain efficient methods, it is necessary to reduce complexity without losing the easy interpretability of the components. One of the possibilities to achieve complexity reduction is to combine fuzzy rule interpolation with the use of hierarchical structured fuzzy rule bases, as proposed by Sugeno et al. (1991). For interpolation, the method of Koczy and Hirota (1993) is used, but other techniques are also suggested. The difficulty of applying this method is that it is often impossible to determine a partition of any subspace of the original state space so that in all elements of the partition the number of variables can be locally reduced. Instead of this, a sparse fuzzy partition is searched for and so the local reduction of dimensions will be usually possible. In this case however, interpolation in the sparse partition itself, i.e. interpolation in the meta-rule level is necessary. This paper describes a method how such a multilevel interpolation is possible.

Algebraic fuzzy flip-flop circuits

Abstract Algebraic fuzzy flip-flop circuits in discrete and continuous mode are presented. The algebraic fuzzy flip-flop is one example of the general fuzzy flip-flop concept which has been defined as an extension of the binary J - K flip-flop. Two types of the algebraic fuzzy flip-flop, which are reset type and set type, are defined using complementation, algebraic product, and algebraic sum operations for fuzzy negation, t-norm, and s-norm, respectively. A unified equation of the reset type and set type of an algebraic fuzzy flip-flop is derived for the purpose of realization of hardware circuits. Based on the equation, two types of hardware circuits, in discrete mode and continuous mode, are constructed. Moreover the characteristics of various fuzzy flip-flops presented previously are investigated such as min-max fuzzy flip-flop in both discrete and continuous mode, and the algebraic fuzzy flip-flop presented in this paper.

Fuzzy controlled robot arm playing two-dimensional ping-pong game

Abstract A robot-arm system equipped with a CCD camera which is able to play two-dimensional ping-pong game has been built. The system is able to play in the form of both robot versus robot and robot versus human. In the robot control part, 25 fuzzy production rules are used. Ambiguous instructions in terms of membership functions are generated by the robot itself using imagery data from a CCD-camera. Each of these instructions consists of three fuzzy items. Two of them are input (ambiguous) information concerning the fuzzy point, that is where the ball was hit, and the fuzzy angle, that is from which direction the ball is coming, and the other is output information which shows the point that the ball will reach. This output information, which also indicates the point the robot should move to, is calculated based on a fuzzy inference method. The whole system is controlled by only one 16-bit personal computer, and works in real time. The advantage of the proposed method are the reduction of processing time and the availability of low level devices, which have not been realized by other methods.

Analysis and synthesis of fuzzy systems by the use of probabilistic sets

The paper deals with an application of probabilistic sets in system theory, especially in identification problems in systems described by means of max-min fuzzy relational equations. The identification procedures discussed are based on some ideas of iterative clustering techniques (ISODATA and FUZZY C-MEANS) which lead to a concrete method of determination of probabilistic sets. A vagueness function associated with the fuzzy relation of the system forms a validity indicator of the identification algorithm. Numerical examples containing fuzzy and nonfuzzy data form an illustration of the methods provided.