A duality between LM-fuzzy possibility computations and their logical semantics

Abstract

Let X be a dcpo and let L be a complete lattice. The family σL(X) of all Scott continuous mappings from X to L is a complete lattice under pointwise order, we call it the L-fuzzy Scott structure on X. Let E be a dcpo. A mapping g : σL(E) −> M is called an LM-fuzzy possibility valuation of E if it preserves arbitrary unions. Denote by πLM(E) the set of all LM-fuzzy possibility valuations of E. The denotational semantics assigning to an LM-fuzzy possibility computation from a dcpo D to another one E is a Scott continuous mapping from D to πLM(E), which is a model of non-determinism computation in Domain Theory. A healthy LM-fuzzy predicate transformer from D to E is a sup-preserving mapping from σL(E) to σM(D), which is always interpreted as the logical semantics from D to E. In this paper, we establish a duality between an LM-fuzzy possibility computation and its LM-fuzzy logical semantics.