A fully abstract model for sequential computation

In 1977, G. Plotkin pointed out the problem of finding a fully abstract model for the sequential programming language PCF [16], which had been originally developed by D. Scott [19]. This question turned out to be one of the most enduring problems of semantics. A very nice description of the different approaches to this subject with many references can be found in [5]. In this rather brief overview, we mainly focus on those articles which are related to the work in this paper. The first fully abstract model for PCF was presented by R. Milner [11]. However, it was purely syntactic and did not provide a semantic description of sequentiality. In 1992, K. Sieber presented a syntax free characterisation of sequentiality for types up to order two and proved a full abstraction result for PCF types up to order three [20]. Two years later, P. O’Hearn and J. Riecke used Sieber’s technique and gave the first domain theoretic description of a fully abstract model for PCF [14] by applying the concept of Kripke relations [6]. A different concept was used by several research groups elaborating fully abstract models for PCF by using game semantics, see [2,4,13]. G. McCusker extended this approach to sum types and recursive types [10]. The domain theoretic approach was refined by J. Riecke and A. Sandholm who modified the O’Hearn/Riecke model and presented a fully abstract model for FPC, an extension of call-by-value PCF with sum types and recursive types [18]. After an introduction to some domain theoretic concepts and notations (Section 2) we construct a cartesian closed category SD of domains to capture the concept of sequentiality (Section 3). This category is cartesian closed, besides, it has smash products making the subcategory SD⊥ with strict functions ? This paper is an extended abstract of a major part of my PhD thesis. It contains all important results of a technical report that I wrote during a one year stay at the University of Birmingham, however, most of the proofs and many details are omitted. For a more detailed version please look at [8].