This paper analyses the computational behaviour of λterm applications. The properties we are interested in are weak normalisation (i.e. there is a terminating reduction) and strong normalisation (i.e. all reductions are terminating). One can prove that the application of a λ-term M to a fixed number n of copies of the same arbitrary strongly normalising λ-term is strongly normalising if and onl...

We study normalising reduction strategies for infinitary Combinatory Reduction Systems (iCRSs). We prove that all fair, outermost-fair, and needed-fair strategies are normalising for orthogonal, fully-extended iCRSs. These facts properly generalise a number of results on normalising strategies in first-order infinitary rewriting and provide the first examples of normalising strategies for infin...

The synchronous reactive data flow language Lustre is an expressive language, equipped with a suite of tools for modelling, simulating and model-checking wide variety safety-critical systems. A critical intermediate step in the formally certified compilation involves translation to well-behaved sub-language called “Normalised Lustre” (NLustre). Recently, we proposed simple Denning-style lattice...

We construct two inverse limit λ-models which completely characterise sets of terms with similar computational behaviours: the sets of normalising, head normalising, weak head normalising λ-terms, those corresponding to the persistent versions of these notions, and the sets of closable, closable normalising, and closable head normalising λ-terms. More precisely, for each of these sets of terms ...

Contrary to all expectations, the-calculus, the canonical simply-typed lambda-calculus with explicit substitutions, is not strongly normalising. This result has led to a proliferation of calculi with explicit substitutions. This paper shows that the reducibility method provides a general criterion when a calculus of explicit substitution is strongly normalising for all untyped lambda-terms that...

We present a quantitative basis-independent analysis of combinatory logic. Using a general argument regarding plane binary trees with labelled leaves, we generalise the results of David et al. (see [9]) and Bendkowski et al. (see [6]) to all Turing-complete combinator bases proving, inter alia, that asymptotically almost no combinator is strongly normalising nor typeable. We exploit the structu...