Reasoning with Non-Numeric Linguistic Variables

Where decisions are based on imprecise numeric data and linguistic variables, the development of automated decision aids presents particular difficulties. In such applications, linguistic variables often take their values from a pre-ordered set of vaguely defined linguistic terms. The mathematical structures that arise from the assumption that sets of linguistic terms are pair-wise tolerant are considered. A homomorphism between tolerance spaces, filter bases and fuzzy numbers is shown. A proposal for modeling linguistic terms with an ordered set of fuzzy numbers is introduced. A procedure for structured knowledge acquisition based on the topology of the term sets and the cognitive theory of prototypes is shown to give rise to sparse rule bases. Similarity as a function of “distance” between fuzzy numbers treated as tolerance mappings is used as an inference mechanism in sparse rule bases to give linguistically valued outputs. Measuring the “distance” between fuzzy sets to correspond to intuitive notions of nearness is not straightforward, since the usual metric axioms are not adequate. An alternative way of measuring “distance” between fuzzy numbers is introduced, which reduces to the usual one when applied to crisp numbers.