A proof of strong normalisation using domain theory

Ulrich Berger presented a powerful proof of strong normalisation using domains, in particular it simplifies significantly Tait's proof of strong normalisation of Spector's bar recursion. The main contribution of this paper is to show that, using ideas from intersection types and Martin-Lof's domain interpretation of type theory one can in turn simplify further U. Berger's argument. We build a domain model for an untyped programming language where U. Berger has an interpretation only for typed terms or alternatively has an interpretation for untyped terms but need an extra condition to deduce strong normalisation. As a main application, we show that Martin-L\"{o}f dependent type theory extended with a program for Spector double negation shift.Comment: 16 page

Reducing the Cost of Precise Types

Programs involving precise types enforce more properties via type checking, but precise types also prevent the reuse of functions throughout a program since no single precise type is used throughout a large program. My work is a step toward eliminating the underlying dilemma regarding type precision versus function reuse. It culminates in a novel traversal operator that recovers the reuse by automating most of each conversion between "similar" precise types, for a notion of similarity that I characterize in both the intuitive and technical senses. The benefits of my techniques are clear in side-by-side comparisons; in particular, I apply my techniques to two definitions of lambda-lifting. I present and implement my techniques in the Haskell programming language, but the fundamental ideas are applicable to any statically- and strongly-typed programming functional language with algebraic data types