On the Approximation of Denotational Mu-Semantics

Abstract

A signature Σ gives rise to a language LΣ(Var) by extending Σ with variables x ∈ Var and binding constructs μ x and ν x, corresponding to least and greatest fixed points respectively. The natural denotational models for such languages are bicomplete dcpos as monotone Σ-algebras. We prove that several approximating denotational semantics have the usual compositional semantics as their limit. These results provide techniques for relating syntactic and semantic concepts such as in full abstraction or completeness proofs. In the presence of an involutive antitone map on a bicomplete dcpo D we may translate the language LΣ(Var) into one with least fixed points only such that meanings are preserved. This allows an approximative semantics where least and greatest fixed points are simultaneously approximated by ‘unwindings’ in the syntax, provided that the limit semantics is substitutive. We discuss the principal difficulties of simultaneous unwindings in the absence of such semantic negations.