Submitted for publication to Information and Computation. A summary of this paper appeared in TACS '97.

We develop a domain theory for treating recursive types with respect to contextual equivalence. The principal approach taken here deviates from classical domain theory in that we do not produce the recursive types via the usual inverse limits constructions- we have it for free by working directly with the operational semantics. By extending type expressions to endofunctors on a ‘syntactic ’ category, we establish algebraic compactness. To do this, we rely on an operational version of the minimal invariance property. In addition, we apply techniques developed herein to reason about FPC programs. Key words: Operational domain theory, recursive types, FPC, realisable functor, algebraic compactness, generic approximation lemma, denotational semantics 1

The method of logical relations assigns a relational interpretation to types that expresses operational invariants satisfied by all terms of a type. The method is widely used in the study of typed languages, for example to establish contextual equivalences of terms. The chief di#culty in using logical relations is to establish the existence of a relational interpretation. For simple language this is often justified by a straightforward induction on the structure of types, but in the presence of impredicative polymorphism and unrestricted recursive types, it is much more di#cult to carry out the construction. Standard methods rely on denotational semantics, building first a domain model of the language, then constructing relations over the model. Building on Freyd and Pitts work on universal properties of domain models, Birkedal and Harper gave a purely operational account of logical relations for a language with a single recursive type. We extend their work to impredicative (second-order) polymorphism and general recursive types, and apply it to establishing parametricity and representation independence properties in a purely operational setting. We compare our methods to the bisimulation methods introduced by Sumii and Pierce for proving such properties in an operational setting. We argue that, once the existence of a relational interpretation has been established, it is straightforward to use it to establish properties of interest.

The results reported in Part III consist of joint work with Martín Escardó [14]. All the other results reported in this thesis are due to the author, except for background results, which are clearly stated as such. Some of the results in Part IV have already appeared as [28]. Note This version of the thesis, produced on October 31, 2006, is the result of completing all the minor modifications as suggested by both the examiners in the viva report (Ref: CLM/AC/497773). We develop an operational domain theory to reason about programs in sequential functional languages. The central idea is to export domaintheoretic techniques of the Scott denotational semantics directly to the study of contextual pre-order and equivalence. We investigate to what extent this can be done for two deterministic functional programming languages: PCF (Programming-language for Computable Functionals) and FPC (Fixed Point Calculus).