Rough Sets and Rough Logic: A KDD Perspective [chapter]

Zdzisław Pawlak, Lech Polkowski, Andrzej Skowron
2000 Studies in Fuzziness and Soft Computing  
Basic ideas of rough set theory were proposed by Zdzis law Pawlak [85, 86] in the early 1980's. In the ensuing years, we have witnessed a systematic, world-wide growth of interest in rough sets and their applications. The main goal of rough set analysis is induction of approximations of concepts. This main goal is motivated by the basic fact, constituting also the main problem of KDD, that languages we may choose for knowledge description are incomplete. A fortiori, we have to describe concepts
more » ... of interest (features, properties, relations etc.) not completely but by means of their reflections (i.e. approximations) in the chosen language. The most important issues in this induction process are: -construction of relevant primitive concepts from which approximations of more complex concepts are assembled, -measures of inclusion and similarity (closeness) on concepts, -construction of operations producing complex concepts from the primitive ones. Basic tools of rough set approach are related to concept approximations. They are defined by approximation spaces. For many applications, in particular for KDD problems, it is necessary to search for relevant approximation spaces in the large space of parameterized approximation spaces. Strategies for tuning parameters of approximation spaces are crucial for inducing concept approximations of high quality. Methods proposed in rough set approach are kin to general methods used to solve Knowledge Discovery and Data Mining (KDD) problems like feature selection, feature extraction (e.g. discretization or grouping of symbolic value), data reduction, decision rule generation, pattern extraction(templates, association rules), or decomposition of large data tables. In this Chapter we examine rough set contributions to Knowledge Discovery from the perspective of KDD as a whole. This Chapter shows how several aspects of the above problems are solved by the classical rough set approach and how they are approached by some recent extensions to the classical theory of rough sets. We point out the role of Boolean reasoning in solving discussed problems. Rough sets induce via its methods a specific logic, which we call rough logic. In the second part of this Chapter, we discuss rough logic and related incomplete logics from a wider perspective of logical approach in KDD. Keywords: indiscernibility, lower and upper approximations, rough sets, boundary region, positive region, rough membership function, decision rules, patterns, dependencies in a degree, rough mereology, logic, propositional logic, predicate logic, modal logic, belief and belief revision logics, default logic, para-consistent logic, non-monotonic logic, fuzzy logic, rough logic. their negations) such that if the values of these literals are true under an arbitrary valuation v of variables then the value of the function f under v is also true. A prime implicant is a minimal implicant. Here we are interested in implicants of monotone Boolean functions only, i.e. functions constructed without negation.
doi:10.1007/978-3-7908-1840-6_13 fatcat:vy35ujzaszcwvevywvjcms4wzy