Analysis of Case-Based Representability of Boolean Functions by Monotone Theory

Abstract

Classification is one of major tasks in case-based reasoning(CBR) and many studies have been done for analyzing properties of case-based classification [ 1 , 14 , 10 , 15 , 12 , 9 , 13 , 7 ]. However, these studies only consider numerical similarity measures whereas there are other kinds of similarity measure for diffierent tasks. Among these measures, HYPO system [ 2 , 3 ] in a legal domain uses a similarity measure based on set inclusion of diffierences of attributes in cases. In this paper, we give an analysis of representability of boolean functions in case-based classification using the above set inclusion based similarity. We show that such case-based classification has a strong connection between monotone theory studied in [ 4 , 11 ]. Monotone theory is originated from computational learning theory and is used to show learnability of boolean function with polynomial DNF size and polynomial CNF size [ 4 ] and is used for deductive reasoning as well [ 11 ]. In this paper, we analyze a case-based representability of boolean functions by using the above relationship between the case-based classification by set inclusion based similarity and the monotone theory. We show that any boolean function is representable by a casebase whose size is bounded in polynomial of its DNF size and its CNF size and thus, k -term DNF, k -clause CNF can be efficiently representable in a casebase using set inclusion similarity.