Rough membership functions: a tool for reasoning with uncertainty
A variety of numerical approaches for reasoning with uncertainty have been investigated in the literature. We propose rough membership functions, rm-functions for short, as a basis for such reasoning. These functions have values in the interval [0,1] and are computable on the basis of the observable information about the objects rather than on the objects themselves. We investigate properties of the rm-functions. In particular, we show that our approach is intensional with respect to the class of all information systems [P91]. As a consequence we point out some differences between the rm-functions and the fuzzy membership functions [Z65], e.g. the rm-function values for X ∪ Y (X ∩ Y) cannot be computed in general by applying the operation max(min) to the rm-function values for X and Y.
- [DP80] D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press, 1980.
- [P91] Z. Pawlak, Rough Sets: Theoretical Aspects of Reasoning about Data, Kluwer, 1991.
- [PS91] Z. Pawlak and A. Skowron, Rough membership functions, ICS Research Report 10/91, Warsaw Univ. of Technology.
- [Sc82] D. Scott, Domains for denotational semantics, a corrected and expanded version of a paper presented at ICALP 82, Aarhus, Denmark, 1982.
- [Sh76] G. Shafer, A Mathematical Theory of Evidence, Princeton Univ. Press, 1976.
- [S91] A. Skowron, The rough set theory as a basis for the evidence theory, ICS Research Report 2/91, 53 pp.
- [Sk91] A. Skowron, Numerical uncertainty measures, lecture delivered at the S. Banach Mathematical Center during the semester 'Algebraic Methods in Logic and their Computer Science Applications', Warszawa, November 1991.
- [SG91] A. Skowron and J. Grzymała-Busse, From the rough set theory to the evidence theory, ICS Research Report 8/91, 49 pp.
- [Z65] L. A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965), 338-353.