Martin Stepnicka

On the Suitability of the Bandler–Kohout Subproduct as an Inference Mechanism

Fuzzy relational inference (FRI) systems form an important part of approximate reasoning schemes using fuzzy sets. The compositional rule of inference (CRI), which was introduced by Zadeh, has attracted the most attention so far. In this paper, we show that the FRI scheme that is based on the Bandler-Kohout (BK) subproduct, along with a suitable realization of the fuzzy rules, possesses all the important properties that are cited in favor of using CRI, viz., equivalent and reasonable conditions for their solvability, their interpolative properties, and the preservation of the indistinguishability that may be inherent in the input fuzzy sets. Moreover, we show that under certain conditions, the equivalence of first-infer-then-aggregate (FITA) and first-aggregate-then-infer (FATI) inference strategies can be shown for the BK subproduct, much like in the case of CRI. Finally, by addressing the computational complexity that may exist in the BK subproduct, we suggest a hierarchical inferencing scheme. Thus, this paper shows that the BK-subproduct-based FRI is as effective and efficient as the CRI itself.

Analysis and prediction of time series using fuzzy transform

A new methodology for forecasting of time series is proposed. It is based on combination of two techniques: fuzzy transform and perception-based logical deduction on the basis of learned linguistic description.

Numerical solution of partial differential equations with help of fuzzy transform

The paper is devoted to a fuzzy approach to numerical solutions of partial differential equations. Three main types of partial differential equations have been considered to demonstrate the algorithms with help of the fuzzy transform. We have introduced an example of a reasonable application of the fuzzy transform in this area. The justification of our approach including the convergence theorem has been presented as well

Arithmetic Fuzzy Models

It is well known that a fuzzy rule base can be interpreted in different ways. From a logical point of view, the conjunctive interpretation is preferred, while from a practical point of view, the disjunctive interpretation has been dominantly present. Each of these interpretations results in a specific fuzzy relation that models the fuzzy rule base. Basic interpolation requirements naturally suggest a corresponding inference mechanism: the direct image for the conjunctive interpretation and the subdirect image for the disjunctive interpretation. Interpolation then corresponds to solvability of some system of fuzzy relational equations. In this paper, we show that other types of fuzzy relations, which are closely related to Takagi-Sugeno (T-S) models, are of major interest as well. These fuzzy relations are based on addition and multiplication only, from which we get the name arithmetic fuzzy models. Under some mild requirements, these fuzzy relations turn out to be solutions of the same systems of fuzzy relational equations. The impact of these results is both theoretical and practical: There exist simple solutions to systems of fuzzy relational equations, other than the extremal solutions that have received all the attention so far, which are, moreover, easy to implement.

Implication-based models of monotone fuzzy rule bases

Abstract In many modelling problems, there is some inherent monotone relationship between one or more of the input variables and the output variable. We consider the prototypical case of an increasing relationship between each of the input variables and the output variable. When using fuzzy rule-based models, this desired monotonicity is reflected in the rule base, given an appropriate ordering on the fuzzy sets involved in the respective input and output domains. More specifically, the larger the antecedent fuzzy sets, the larger the consequent fuzzy set. However, fuzzy rule-based modelling involves a final defuzzification step, possibly resulting in a function that is no longer monotone. In the context of Mamdani–Assilian conjunctive fuzzy models, ample attention has been paid to this problem, both for the centre-of-gravity defuzzification and mean-of-maxima defuzzification methods. In this paper, we show that for implicative fuzzy models, the non-monotonicity problem can be circumvented by making explicit the semantics of the fuzzy rules by subjecting the antecedent and consequent fuzzy sets to the at-least and/or at-most modifiers.

A Plea for the Usefulness of the Deductive Interpretation of Fuzzy Rules in Engineering Applications

This contribution is intended as a position paper that favors the viewpoint that inference based on deductive rules (i.e., the rules are interpreted using fuzzy implication) can indeed be considered as a valuable inference scheme in real-world applications. For this purpose, we highlight the basic concepts behind the most common fuzzy inference schemes and demonstrate their interpretation by means of illustrative examples. We conclude that, under some reasonable conditions, deductive inference is able to compete with or even outperform the well-known Mamdani-Assilian inference.

F-transform and Fuzzy Natural logic in Time Series Analysis

This paper continues the development of the innovative method for time series analysis and forecasting using special soft-computing techniques: fuzzy (F-) transform and Fuzzy Natural Logic. We will demonstrate that the F-transform is a proper technique for extraction of the trend-cycle of time series. Furthermore, we will elaborate in more detail automatic generation of linguistic evaluation of its behavior in arbitrary time slots. Thanks to the firstdegree F-transform (F1-transform), this works even if the graph of the time series visually does not suggest a clear tendency.

Monotonicity of implicative fuzzy models

Frequent practical problems from decision-making as well as automatic control lead to intuitively monotone fuzzy rule bases. let us assume an appropriate ordering of fuzzy sets is defined. Then by the monotone fuzzy rule base we mean a rule base consisting of such fuzzy rules expressing the monotone dependence of consequent fuzzy sets on antecedent fuzzy sets. In other words the “bigger” antecedent fuzzy is present in a fuzzy rule the “bigger” consequent fuzzy set appears on the right hand side of the same fuzzy rule. Very often real-world applications require some defuzzification to be employed at the end of the inference process. The problem is that after the defuzzification we obtain a crisp input-output function which is not necessarily monotone anymore. Obviously, such behavior is not only counterintuitive but also dangerous. Most of the attention has been paid to the Mamdani-Assilian conjunctive kind of models of fuzzy rule bases built with help of particular t-norms. This paper focuses on the implicative approach for arbitrary residual implication.

Redundancies in systems of fuzzy/linguistic IF-THEN rules

In this paper we will investigate which fuzzy/linguistic rules are redundant in systems of such rules called linguistic descriptions. We present a formal definition of redundancy and show that rules which are seemingly redundant can be in fact indispensable. These results apply for IF-THEN rules which use evaluative linguistic expressions (e.g., small, very big, etc.) and inference method called perception-based logical deduction (PbLD). However, they are also valid for inference systems which use compatible design choices with PbLD.

Recognition of damaged letters based on mathematical fuzzy logic analysis

This paper reports a real application whose task was to recognize characters printed on metal ingots. The problem is that surface of ingots is very uneven - ingots are hot or cold, cut by rough instrument, the printing machine can be worn down, etc. In this paper, we present two original recognition methods: the first one is based on application of mathematical fuzzy logic and the second one is based on representation of an image by a fuzzy-valued function. Results of these methods are compared with a simple neural network classifier and few other common methods.