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

A unifying framework for continuity and complexity in higher types

We set up a parametrised monadic translation for a class of call-by-value functional languages, and prove a corresponding soundness theorem. We then present a series of concrete instantiations of our translation, demonstrating that a number of fundamental notions concerning higher-order computation, including termination, continuity and complexity, can all be subsumed into our framework. Our main goal is to provide a unifying scheme which brings together several concepts which are often treated separately in the literature. However, as a by-product, we also obtain (i) a method for extracting moduli of continuity for closed functionals of type $(\mathbb{N}\to\mathbb{N})\to\mathbb{N}$ definable in (extensions of) System T, and (ii) a characterisation of the time complexity of bar recursion

A unifying framework for continuity and complexity in higher types

We set up a parametrised monadic translation for a class of call-by-value functional languages, and prove a corresponding soundness theorem. We then present a series of concrete instantiations of our translation, demonstrating that a number of fundamental notions concerning higher-order computation, including termination, continuity and complexity, can all be subsumed into our framework. Our main goal is to provide a unifying scheme which brings together several concepts which are often treated separately in the literature. However, as a by-product, we also obtain (i) a method for extracting moduli of continuity for closed functionals of type $(\mathbb{N}\to\mathbb{N})\to\mathbb{N}$ definable in (extensions of) System T, and (ii) a characterisation of the time complexity of bar recursion