Some issues of linguistic approximation

Summary form only given. Two of the most exemplary capabilities of the human mind are the capability of using perceptions in purposeful ways and the capability of approximating perceptions by statements in natural language. Understanding these capabilities and emulating them by machines is the crux of intelligent systems. To construct intelligent systems, we need to develop appropriate methodological tools for dealing with perceptions in machines. A feasible way to deal with perceptions in machines is to approximate them by statements in natural language and, then, to use fuzzy logic to represent these statements and deal with them as needed. This approach to developing perception-based machines, which is currently a subject of current research, was initiated under the name "computing with words" by Zadeh (1996, 1999). Once perceptions are approximated in the context of a given application by statements in natural language and the latter are approximated, in turn, by appropriate propositions of fuzzy logic, we can utilize all available resources of fuzzy logic to formalize approximate, human-like reasoning. The usual outcome of this reasoning is a fuzzy set. For some purposes (such as control), we need to replace this fuzzy set with a single value that, in some sense is its best representative. This replacement (or a single-valued approximation) of the given fuzzy set is called defuzzification. For other purposes (such as communication of intelligent machines with human beings), we need to approximate the given fuzzy set by an appropriate linguistic term, a term that has an understandable interpretation expressed by another fuzzy set. We thus approximate one fuzzy set by another fuzzy set that, in the context of a given application, represents a specific linguistic expression. Both types of approximation of fuzzy sets may be viewed as special cases of the same problem category. Another special case in this problem category is the problem of approximating a fuzzy set by a crisp set. The term "linguistic approximation" may be thus viewed as term that subsumes the three special cases of approximating fuzzy sets. It is important to realize that these special cases are not independent of one another and may be combined in various ways. Defuzzification methods, whi- ch are important in fuzzy control, have been investigated quite extensively. The other two special cases of linguistic approximation have been discussed in the literature, but are far less developed at this time. There are of course various views about what the terms "good approximation" or "best approximation" are supposed to mean. An epistemological position taken here is that these terms should always be viewed in information-theoretic terms. That is, a good approximation should be one in which the loss of information is small and, similarly, the best approximation (not necessarily unique in this case) should be one of those in which the loss of information is minimal. This requires of course that we can measure in a justifiable way the loss of relevant information. The issue of measuring information in terms of reduction of uncertainty has been the subject of generalized information theory |3]. A unique measure of uncertainty-based information is now well established for fuzzy sets defined on finite domains and its counterpart for fuzzy sets on infinite domains is also well justified even though its uniqueness has not been proven as yet (1999). The information-theoretic approach to linguistic approximation is thus feasible. There is of course more than one way in which the approach can be applied to any of the three special cases of the linguistic approximation problem. It is the purpose of this presentation to examine the various possibilities.