Shadows of Fuzzy Sets A Natural Way to Describe D and Multi D Fuzzy Uncertainty in Linguistic Terms

Fuzzy information processing systems start with expert knowledge which is usually formulated in terms of words from natural language This knowledge is then usually reformulated in computer friendly terms of membership functions and the system transform these input membership functions into the membership functions which describe the result of fuzzy data processing It is then desirable to translate this fuzzy information back from the computer friendly membership functions language to the human friendly natural language In a D case when we are interested in a single quantity y it is usually easy to describe the resulting membership function by a word from natural language because most words do describe D case and there are therefore so many of them that the corresponding membership functions form a dense set in the class of all possible membership functions The problem becomes more complicated in D and multi D cases when we are interested in several quantities y ym because there are fewer words which describe the relation between several quantities To describe such fuzzy information in terms of a natural language L Zadeh proposed in to use words to describe fuzzy information about di erent combinations y f y ym of the desired variables This idea is similar to the use of marginal distributions in probability theory The corresponding terms are called shadows of the original fuzzy set The main question is do we lose any information in this translation Zadeh has shown that under certain conditions the original fuzzy set can be uniquely reconstructed from its shadows In this paper we prove that for appropriately chosen shadows the reconstruction is always unique Thus the translation from the original membership function into the linguistic terms which describe di erent combinations y is lossless Membership Functions As a Computer Friendly Translation of Natural Language Terms Humans often describe their knowledge by terms from natural language like young large etc If we want a computer to be able to use this knowledge we must reformulate it in terms which are understandable to a computer One of the main objectives of fuzzy methodology is to provide such a translation Fuzzy logic describes each natural language term t de ned on a set X by the corresponding membership function t x X a function which describes for each element x of the domain X to what extent this element x satis es the property t Fuzzy methodology provides us with the tools t norms t conorms fuzzy inference rules etc which are able to process these functions A typical application of these tools is to the following situation We are interested in the values of some quantities y ym about which we have no direct knowledge e g we may be interested to know how the economy will grow in the next few years What we do know is the relation between these quantities yi and some other quantities x xn about which we have some fuzzy knowledge For example for an economy we may know how it was growing in the past we may know some speci c parameters characterizing its common state etc The rules connecting xi and yj are also typically described not in precise mathematical form but rather by words from natural language Fuzzy methodology enables us to transform a fuzzy knowledge about xi and the fuzzy rules which connect xi and yj into a fuzzy knowledge about yj i e into the membership function y on the set of all possible values of y y ym see e g In short we get the desired information about yj but we get it in terms of membership functions It Is Desirable to Translate The Result of Fuzzy Data Processing Back Into the Natural Language A membership function is not something which is natural for a human to understand and to use it was invented as a way of representing human fuzzy knowledge in a language which is understandable for a com puter From this viewpoint the fact that the result of using traditional fuzzy methodology is a membership function means that this result is not presented in a very user friendly form for the user s convenience we must translate the result of computer s information processing from the computer native language of membership functions into the human friendly natural language Such a Translation Is Relatively Easy in D Cases This translation is relatively easy in a D case when we are interested only in the value of a single quantity y This easiness comes from the fact that most words from natural language characterize a single quantity young small etc so there are plenty of di erent membership functions corresponding to di erent words of this type Since there are many such membership functions stemming from natural language terms these functions form a dense net in the set of all possible membership functions Therefore often for a membership function y produced by the fuzzy system we are able to nd a natural language term t for which the corresponding membership function t y is close enough to y It is then natural to return this term t as the result of fuzzy information processing D and Multi D Cases The Idea of Shadows of a Fuzzy Set For D and multi D problems when we are interested in the values of several quantities y ym m the situation is radically di erent There are much fewer terms from natural language which describe the relation between several quantities than in D case e g we can say that y is much larger than y etc So in such cases when the fuzzy system produces a membership function y y ym we are often unable to nd a natural language term which describes this membership function How can we in this situation describe the membership function in user friendly linguistic terms To describe such fuzzy information in terms of natural language L Zadeh proposed in to use words to describe fuzzy information about di erent combinations y f y ym of the desired variables The corresponding D fuzzy set is called a shadow of the original multi D fuzzy set This idea is similar to the use of marginal distributions in probability theory For example we can use linear combinations