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Comparison of rough-set and interval-set models for uncertain reasoning. (English) Zbl 0858.68107
Summary: In the rough-set model, a set is represented by a pair of ordinary sets called the lower and upper approximations. In the interval-set model, a pair of sets is referred to as the lower and upper bounds which define a family of sets. A significant difference between these models lies in the definition and interpretation of their extended set-theoretic operators. The operators in the rough-set model are not truth-functional, while the operators in the interval-set model are truth-functional. Within the framework of possible-worlds analysis, we show that the rough-set model corresponds to the modal logic system $$S_5$$, while the interval-set model corresponds to Kleene’s three-valued logic system $$K_3$$. It is argued that these two models extend set theory in the same manner as the logic systems $$S_5$$ and $$K_3$$ extend standard propositional logic. Their relationships to probabilistic reasoning are also examined.

##### MSC:
 68T27 Logic in artificial intelligence 68T30 Knowledge representation 03E47 Other notions of set-theoretic definability
##### Keywords:
vagueness; rough-set model; modal logic system