Fuzzy Abductive Reasoning for Diagnostic Problems

Fuzzy abduction is a procedure for deriving fuzzy sets of hypotheses which explain a given fuzzy set of events using a set of implications with a truth value. The derived fuzzy sets of hypotheses are called fuzzy explanations. This paper starts with discussion for diagnosis using conventional expert systems and with fuzzy relational equations. Then, it proposes an approach using a fuzzy abduction, which is devised for diagnosis based on a fuzzy abduction system we propose. Application of the system to diagnosis of a maritime diesel engine is described. 1. NTRODUCTION Diagnosis using knowledge-based systems has long been a major topic of research. However, the difficulty of knowledge acquisition has not yet been overcome, though various approaches have been intensively studied. Limiting the problem to diagnosis, one reason for the difficulty of knowledge acquisition is the "directionality" of production rules [1]. Most expert knowledge takes the form: “if some fault happens then certain symptoms will arise,” while conventional expert systems use the opposite directionality: “if then .” This means that constructing a knowledge base is almost equivalent to creating knowledge to solve inverse problems in a general manner. To avoid this difficulty, another approach, called abduction [1]-[4], has been proposed. It is a type of reasoning that derives a set of hypotheses (causes) which explain a given set of events (symptoms) using some knowledge (causal relations). Adoption of this approach prevents us from having to solve the inverse problems, and enables us to create a knowledge base more easily. Another problem in conventional expert systems is the treatment of “intensities”, or degrees of symptoms. One way to deal with the problem is to assign different symbols to different degrees of a symptom, as if they were different symptoms. This approach, however, produces a large number of combinations of “pseudo symptoms” for a cause, and eventually forces us to construct a vast number of rules. An approach based on fuzzy theory apparently solves the above two problems. This approach uses fuzzy relational equations [5]-[9]. This approach expresses causes, symptoms and their causalities as a fuzzy set P on possible causes P = {Pi} (i=1,...,n), a fuzzy set Q on possible symptoms Q = {Qj} (j=1,...,m) and a fuzzy relation R on P × Q, respectively. Then, the fuzzy relational equation is given in the following Max-Min composition. q j = i ∨(pi ∧ rij) , (1) where pi, qj and rij are memberships of Pi in P, Qj in Q and in R, respectively. Then, the inverse problem of fuzzy relational equations–which is a problem to derive pi given qj and rij–could be considered as a kind of abductive reasoning. Furthermore, the memberships pi, qj and rij seem to be interpreted as intensities of causes, symptoms and causal relations, respectively. However, this interpretation raises a question about the relation between pi and qj. In diagnosis, the more (less) the intensity of a cause is, the more (less) that of its symptoms tends to be. Considering an "i" in eq. (1), however, the value of qj can not be greater than that of rij, even if pi is large. Such characteristics of the equation might not be adequate for expressing intensities of causes and symptoms. Based on possibility theory, the fuzzy relational equations should be considered as dealing with possibilities rather than intensities of occurrences. If rij is regarded as a conditional possibility π(Qj | Pi), pi ∧ rij is a combinational possibility of “Pi” and “Qj when Pi is present” [10]. Then, the obtained qj should be considered as the possibility that Qj arises, not as the intensity of occurrence of Qj. In the paper, we describe a fuzzy abduction devised for diagnosis. The approach employs rules to express causal relations between causes and symptoms. Those rules have a truth value between 0 and 1 that is interpreted as a degree of causal intensity. We detail application to a diesel engine diagnostic problem. 2. FUZZY ABDUCTION Fuzzy Abduction Proposed Before [4] First, we briefly describe a fuzzy abduction proposed before. We suppose that there are three sets, P = {Pi} (i=1,...,n), Q = {Qj} (j=1,...,m) and R = {Rij}, where P ∩ Q = ∅ and Rij is a rule given as: