Modular Implementation of Programming Languages and a Partial-Order Approach to Infinitary Rewriting

This dissertation is a collection of nine research papers pertaining to two independent areas of research. In the first part, entitled Modular Implementation of Programming Languages, we develop techniques for implementing programming languages in a modular fashion. Within this problem domain, we focus on operations on typed abstract syntax trees with the goal of developing a framework that facilitates the definition, manipulation and composition of such operations. The result of our work is a comprehensive combinator library that provides these facilities. What sets our approach apart is the use of recursion schemes derived from tree automata in order to implement operations on abstract syntax trees. In the first two papers we illustrate the power of this approach by showcasing tree homomorphisms – a very limited form of tree automata – as basic building blocks for simple tree transformations. Their simplicity allows us to combine them with monadic effects, manipulate and combine them in a flexible manner, and perform optimisations in the form of deforestation. In the third paper, we move to more powerful tree automata. Usually, these more powerful automata are cumbersome to define as they combine different computational aspects. We show, however, that these automata can be constructed from simpler ones, viz. tree homomorphisms and simple state machines. In the second paper, we focus on the important issue of representing variable names and variable binders using a carefully restricted form of higher-order abstract syntax. In the fourth paper, we presents a comprehensive and realistic application of our library: a prototype implementation of a novel enterprise resource planning system built around a family of domain-specific languages that make it possible to customise the system in a highly flexible manner. The system combines several highly integrated domain-specific languages, which are implemented using our library. The second part of this dissertation, entitled A Partial-Order Approach to Infinitary Rewriting, is concerned with infinitary rewriting, a field that studies transfinite rewrite sequences. We extend the established theory of infinitary rewriting in two directions: (1) a novel approach to convergence in infinitary rewriting that replaces convergence in a metric space with the limit inferior in a partially ordered set; (2) extending infinitary term rewriting to infinitary term graph rewriting. For the first item, we show correspondences between the established calculi based on metric convergence and the newly developed calculi based on partial orders. We also study both approaches on an abstract level and show the advantages of our partial order approach in terms of better confluence and normalisation properties of infinitary term rewriting as well as in terms of better completeness properties for infinitary term graph rewriting. For the second item, we explore several approaches to convergence on term graphs and analyse the resulting calculi. We distinguish two calculi among them – based on a metric space respectively a partially ordered set – by showing that they satisfy strong soundness and completeness properties w.r.t. infinitary term rewriting.