Interpolation and Approximation with Fuzzy Logic Systems and t-Logic Systems
DOI:
https://doi.org/10.15837/ijccc.2026.3.7526Keywords:
interpolation, approximation, logic system, t-norm, fuzzy logic system, dense set, isopleth mappingAbstract
In many engineering, computer science and economic applications, the model or the system design has to satisfy the condition that it precisely (or approximately) goes through a set of specified points. This means that the system has to perform as an (approximate) interpolator (interpolating function, also named interpolants). The first question asked is if there is a simple way to design good interpolators with FLSs. We answer this first question in a positive way by providing a systematic and simple method for interpolation with FLSs and proving that FLSs are universal exact interpolators. Fuzzy logic systems are known to be universal approximators; that guarantees that they can also perform as universal approximate interpolators. Finding a good approximator with fuzzy logic systems (FLSs) may be difficult and computationally demanding. A procedure is provided for automatically building guaranteed approximators for real valued functions defined on the line. Also, a method for building interpolators for functions with jump discontinuities is shown.
The next question asked is: Is it possible to define interpolators with logic systems under various other logics determined by t-norms and related s-co-norms? A method for interpolation and approximation in Rn is proposed, based on t-Logic Systems (TLS). t-Logic Systems are systems from an n-dimensional space to the real line, based on a logic defined by t-norms and co-norms. They include a first stage of assigning to the real-valued inputs a t-distributions, a second stage where an inference is performed in the framework of the chosen logic, and an estimation stage, where the result of the inference is converted into a real number. Typical examples include fuzzy systems, Z-number based system, and probabilistic systems. Similar to neural networks (NN), fuzzy logic systems (FLSs) with center of gravity (c.o.g.) estimator (named defuzzifier for FLSs) are universal approximators. Even more, FLSs with a c.o.g. defuzzifier are interpolators for well-behaved functions. The general t-logic systems (tLS) have similar properties. These general properties have wide applicability in economics, engineering, and decision making. In many cases, the type of t-logic, in particular the type of fuzzy logic used in the FLS does not play an important role in the universal approximation or interpolation properties, thus leaving much space for the tLS and FLSs optimization. The paper provides results about interpolation, and in subsidiary, for approximation with tLSs, in particular with FLSs, and discusses various applications. The results may find applications in various fields.
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