A domain-specific embedded language (DSEL) is a domain-specific programming language with no concrete syntax of its own. Defined as a set of combinators encapsulated in a module, it borrows the syntax and tools (such as type-checkers and compilers) of its host language; hence it is economical to design, introduce, and maintain. Unfortunately, this economy is counterbalanced by a lack of room for growth. DSELs cannot match sophisticated domain-specific languages that offer tools for domainspecific error-checking and optimisation. These tools are usually based on syntactic analyses, so they do not work on DSELs. Abstract interpretation is a technique ideally suited to the analysis of DSELs, due to its semantic, rather than syntactic, approach. It is based upon the observation that analysing a program is equivalent to evaluating it over an abstract semantic domain. The mathematical properties of the abstract domain are such that evaluation reduces to solving a mutually recursive set of equations. This dissertation shows how abstract interpretation can be applied to a DSEL by replacing it with an abstract implementation of the same interface; evaluating a program with the abstract implementation yields an analysis result, rather than an executable. The abstract interpretation of DSELs provides a foundation upon which to build sophisticated error-checking and optimisation tools. This is illustrated with three examples: an alphabet analyser for CSP, an ambiguity test for parser combinators, and a definedness test for attribute grammars. Of these, the ambiguity test for parser combinators is probably the most important example, due to the prominence of parser combinators and their rather conspicuous lack of support for the well-known LL(k) test. In this dissertation, DSELs and their signatures are encoded using the polymorphic lambda calculus. This allows the correctness of the abstract interpretation of DSELs to be proved using the parametricity theorem: safety is derived for free from the polymorphic type of a program. Crucially, parametricity also solves a problem commonly encountered by other analysis methods: it ensures the correctness of the approach in the presence of higher-order functions.

Cette thèse porte sur l’adaptation de la réalisabilité et la paramétricité au cas des types dépendants dans le cadre des Systèmes de Types Purs. Nous décrivons une méthode systématique pour construire une logique à partir d’un langage de programmation, tous deux décrits comme des systèmes de types purs. Cette logique fournit des formules pour exprimer des propriétés des programmes et elle offre un cadre formel adéquat pour développer une théorie de la réalisabilité au sein de laquelle les réalisateurs des formules sont exactement les programmes du langage de départ. Notre cadre permet alors de considérer les théorèmes de représentation pour le système T de Gödel et le système F de Girard comme deux instances d'un théorème plus général.Puis, nous expliquons comment les relations logiques de la théorie de la paramétricité peuvent s'exprimer en terme de réalisabilité, ce qui montre que la logique engendrée fournit un cadre adéquat pour développer une théorie de la paramétricité du langage de départ. Pour finir, nous montrons comment cette théorie de la paramétricité peut-être adaptée au système sous-jacent à l'assistant de preuve Coq et nous donnons un exemple d'application original de la paramétricité à la formalisation des mathématiques. / This thesis focuses on the adaptation of realizability and parametricity to dependent types in the framework of Pure Type Systems. We describe a systematic method to build a logic from a programming language, both described as pure type systems. This logic provides formulas to express properties of programs and offers a formal framework that allows us to develop a theory of realizability in which realizers of formulas are exactly programs of the starting programming language. In our framework, the standard representation theorems of Gödel's system T and Girard's system F may be seen as two instances of a more general theorem. Then, we explain how the so-called « logical relations » of parametricity theory may be expressed in terms of realizability, which shows that the generated logic provides an adequate framework for developping a general theory of parametricity. Finally, we show how this parametricity theory can be adapted to the underlying type system of the proof assistant Coq and we give an original example of application of parametricity theory to the formalization of mathematics.