. We study the longstanding problem of semantics for input /output (I/O) expressed using side-effects. Our vehicle is a small higher-order imperative language, with operations for interactive character I/O and based on ML syntax. Unlike previous theories, we present both operational and denotational semantics for I/O effects. We use a novel labelled transition system that uniformly expresses both applicative and imperative computation. We make a standard definition of bisimilarity and prove it is a congruence using Howe's method. Next, we define a metalogical type theory M in which we may give a denotational semantics to O. M generalises Crole and Pitts' FIX-logic by adding in a parameterised recursive datatype, which is used to model I/O. M comes equipped both with judgements of equality of expressions, and an operational semantics; M itself is given a domain-theoretic semantics in the category CPPO of cppos (bottom-pointed posets with joins of !-chains) and Scott continuous functions...

of real-number computation

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).