Deductive Approximate Reasoning | a Comparison of Two Approaches

This paper deals with deductive approximate reasoning techniques. Approximate reasoning is concerned with the appropriate processing of vague and uncertain knowledge, given in a rule based manner. The role model for dealing with vague and uncertain knowledge is the human mind. The ability to build abstractions and vague concepts lies at the heart of the immense capabilities of the human brain. Furthermore, approximate reasoning techniques allow us to formalize and process human expert knowledge. In order to realize an approximate reasoning mechanism we are faced with a lot of important design decisions. A very basic decision is concerned with the choice of an appropriate implication relation. In this paper we examine two complementary choices, namely the Mamdani relation and the G odel relation. We present empirical results with single rules, with groups of rules and rule chaining. We show that when complex knowledge is involved G odel implication yields better results than the Mamdani approach. I Introduction and Goals Deductive reasoning enables to infer some usable result called conclusion from a known rule and an input fact, where the rule describes some relation from inputs (premises) to outputs (conclusions). Deductive inference can be performed by means of classical logic modus ponens, which can be schematically described by: given the rule: IF A THEN B given the fact: A infer the result: B (1) For processing vague information and using linguistic variables and fuzzy or vague rules a generalized modus ponens has been proposed and studied 8, 5]. It enables an approximate deductive reasoning with the following general scheme: given the rule: IF x is ~ A THEN y is ~ B given the fact: x is ~ A 0 infer the result: y is ~ B 0 (2) Here, x and y are linguistic variables and ~ A, ~ B, ~ A 0 , and ~ B 0 are fuzzy sets describing linguistic values or attributes of x and y. They are deened on universes U x and U y : Classical reasoning using modus ponens draws conclusion B if and only if the fact is identical to the premise. Otherwise the modus ponens is not applicable. In everyday human reasoning where linguistic variables are used the equality of fact and premise, (i.e. ~ A 0 = ~ A) is rather the exception than the rule. Excluding the cases when fact and premise are completely diierent or identical, there is …