Discovering Critical Cases in Case-Based Reasoning

In this paper, we discuss a method to nd a critical casebase to classify boolean concepts. By a critical casebase we mean that any subset of the casebase cannot represent a concept correctly. This notion is important for reduction of not only the size of casebase but also classi cation speed. In this paper, we consider set-inclusion based similarity which is originated from a legal reasoning system, HYPO [Ashley90, Ashley94]. In HYPO, a case c is more similar to c 1 than c 2 if the set of di erent attributes between c and c 1 is included by the set of di erent attributes between c and c 2 . A case c is regarded as a positive case if there is a positive case such that no negative case is more similar to c than the positive case. In [Satoh98], by using relationship between the above case-based reasoning and monotone theory [Bshouty93, Khardon96], we have shown that we can represent any boolean function f in a casebase which consists of at most jDNF (f )j (1 + jCNF (f)j) cases where jDNF (f)j(jCNF (f )j, respectively) is the size of a minimal DNF(CNF respectively) representation of f . Speci cally, we have shown that a boolean function de ned by a casebase with our similarity measure is a complement of a monotone extension [Bshouty93] such that a set of positive cases in the casebase is called basis in [Bshouty93] and negative cases are assignments in the monotone extension. In this paper, by extending the above results, we rstly give a construction method of a critical casebase if a boolean function is known. Then, we propose an approximation method of nding a critical casebase if a boolean function is unknown and analyze the approximation method in PAC (probably approximately correct) learning framework. In the approximation method, we use not only cases obtained from the domain, but also a membership query which asks the classi cation