) Eike Ritter and Valeria de Paiva ? School of Computer Science, University of Birmingham Abstract. Calculi with explicit substitutions have found widespread acceptance as a basis for abstract machines for functional languages. In this paper we investigate the relations between variants with de Bruijnnumbers, with variable names, with reduction based on raw expressions and calculi with equational judgements. We show the equivalence between these variants, which is crucial in establishing the correspondence between the semantics of the calculus and its implementations. 1 Introduction Explicit substitution calculi (or oe-calculi for short) first appeared in a seminal paper by Abadi et al. [1]. The basic idea is that instead of having substitutions as a meta-level operation, as in traditional -calculus, we should make them part of the object-level calculus. The advantages of this approach are twofold. Firstly, it makes it possible to design much more efficient abstract machines as we a...

provide a formal but intuitive way of presenting and reasoning about programs, which is widely used in practice, although in an informal or semi-formal fashion. In this thesis, we investigate categorical models of programming languages based on a graphical presentation. In the first part, we use a graphical presentation of processes to motivate a categorical model of processes which provides process types and constructors similar to those available in categories of graphs. The model is parametrised on a base category of processes, and may therefore be used to model a variety of process calculi or languages. We present a concrete instance of this model, based on the process calculus CCS, and show that it arises as a syntactic category of an extension of the base calculus. In the second part of the thesis, we use a graphical semantics due to Jeffrey to model and prove correct a step in the compilation of higher-order functional programming languages: closure conversion -- a program tra