Shadows of Fuzzy Sets { A Natural Way to Describe 2 - D and Multi - DFuzzy Uncertainty in Linguistic

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 1-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 1-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 2-D and multi-D cases, when we are interested in several quantities y1 ; : : : ; 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 1966, to use words to describe fuzzy information about diierent combinations y = f (y1; : : : ; 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 diierent combinations y is lossless. 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 deened on a set X by the corresponding membership function 1