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 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).