Three Inadequate Models

Abstract

Abstract. The connection between operational and denotational semantics is of longstanding interest in the study of programming languages. The emphasis has been on positive results, whether for adequacy or full abstraction. One normally considers the standard solution of an evident natural domain equation for the language; this is generally adequate but not fully abstract if one uses any of the usual categories of domains. One then tries other categories to get improved results. Here we restrict ourselves to a standard category of domains and show, for an untyped λ-calculus with arithmetic, that inadequate models exist if one considers non-standard solutions to the domain equation. One model is inadequate, simpliciter; a second is adequate but inadequate when the language is extended by a “parallel or” construct; the third is adequate in the latter sense, but in it the Y -combinator does not denote the least fixed point operator. We also consider whether it is possible to do better than the standard solution as regards full abstraction. Surprisingly this question only makes sense for solutions which are adequate for the extended language. For these the standard solution is indeed closest to full abstraction, justifying the use of non-standard categories.