Axiomatic Domain Theory

The denotational semantics approach to the semantics of programming languages interprets the language constructions by assigning elements of mathematical structures to them. The structures form so-called categories of domains and the study of their closure properties is the subject of domain theory Sco70, Sco82, Plo83, GS90, AJ94]. Typically, categories of domains consist of suitably complete partially ordered sets together with continuous maps. But, what is a category of domains? The main aim of axiomatic domain theory is to answer this question by axiomatising the structure needed on a mathematical universe so that it can be considered a category of domains. Criteria required from categories of domains can be of the most varied sort. For example, we could ask them to have a rich collection of type have a Stone dual providing a logic of observable properties Abr87, Vic89, Zha91]. An additional aim of the axiomatic approach is to relate these mathematical criteria with computational criteria. As we indicate below an axiomatic treatment of various of the above aspects is now available but much research remains to be done. Developments In the beginning, the axiomatic treatment of domain theory was scattered; it concentrated on xed-points, mainly of endofunctors but also of endomorphisms. Concerning xed-points for endofunctors, already in Sco72], Scott mentions a suggestion by Lawvere aiming at providing a categorical framework for performing the D 1 construction. But it was not until Wan79] that the solution of recursive type equations in categories of domains was rst treated abstractly, in the sense that no commitment to a particular category of domains was required. Subsequently this approach was developed in SP82]. The approach was very much appreciated as a uniication of the techniques for solving recursive type equations in categories of domains, but its axio-matic character remained overlooked. For instance, it lead Lehmann and Smyth LS81] Appears in FJM + 96, x Axiomatic Domain Theory].