# Indexed Rough Approximations, A Polymodal System, and Generalized Possibility Measures

Chapter
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 95)

## Abstract

Indexed rough approximations that generalize fuzzy rough sets are proposed. A family of indexed relations between objects with the set of indices being a lattice is considered. Relations in the family are ordered by the inclusion, and moreover the ordering is assumed to be consistent with the ordering of the lattice. Thus, a collection of rough approximations, each of which is induced from a relation in the family, is obtained. A polymodal system in which the modal operators with the indices are defined; the completeness between the axiomatic system and the Kripke model which is the above collection of rough approximations is proved. A possibility and necessity measures for sentences that takes the values of the lattice are derived from the polymodal system. These measures are proved to be equivalent to the ordinary possibility and necessity measures when the lattice is the unit interval.

## Keywords

Modal Logic Inference Rule Unit Interval Axiomatic System Kripke Model
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
B. F. Chellas, B.F. (1980) Modal Logic.Cambridge University Press.Google Scholar
2. 2.
Dubois, D. and Prade, H. (1988) Possibility Theory: An Approach to Computerized Processing of Uncertainty.Plenum.Google Scholar
3. 3.
Dubois D. and Prade, H. (1990) Rough fuzzy sets and fuzzy rough sets. Int. J. General Systems, 17, 191 - 209.
4. 4.
Friedman N. and Halpern, J. Y. (1996) Plausibility measures and default reasoning. Proc. of the 13th National Conf. on Artificial Intelligence, 2, 1297 - 1304.
5. 5.
Goguen, J. A. (1967) L-fuzzy sets. J. of Math. Anal. and Appl., 18, 145 - 174.
6. 6.
MacLane S. and Birkoff, G. (1979) Algebra, 2nd ed. Macmillan.Google Scholar
7. 7.
Pawlak, Z. (1982) Rough sets. International Journal of Computer and Information Sciences, 11, 341 - 356.
8. 8.
Pawlak, Z. (1991) Rough Sets. Kluwer Academic Publishers, Dordrecht.
9. 9.
Popkorn, S. (1994) First Steps in Modal Logic. Cambridge University Press.Google Scholar
10. 10.
Vakarelov, D. (1991) A modal logic for similarity relations in Pawlak knowledge representation systems. Fondamenta Informaticae, 15, 61 - 79.
11. 11.
Zadeh, L. A. (1965) Fuzzy sets. Information and Control, 8, 338 - 353.
12. 12.
Zadeh, L. A. (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems, 1, 3 - 28.