Characterising Strongly Normalising Intuitionistic Terms

Abstract

This paper gives a characterisation, via intersection types, of the strongly normalising proof-terms of an intuitionistic sequent calculus (where LJ easily embeds). The soundness of the typing system is reduced to that of a well known typing system with intersection types for the ordinary λ-calculus. The completeness of the typing system is obtained from subject expansion at root position. Next we use our result to analyze the characterisation of strong normalisability for three classes of intuitionistic terms: ordinary λ-terms, ΛJ-terms (λ-terms with generalised application), and λx-terms (λ-terms with explicit substitution). We explain via our system why the type systems in the natural deduction format for ΛJ and λx known from the literature contain extra, exceptional rules for typing generalised application or substitution; and we show a new characterisation of the β-strongly normalising λ-terms, as a corollary to a PSN-result, relating the λ-calculus and the intuitionistic sequent calculus. Finally, we obtain variants of our characterisation by restricting the set of assignable types to sub-classes of intersection types, notably strict types. In addition, the known characterisation of the β-strongly normalising λ-terms in terms of assignment of strict types follows as an easy corollary of our results.

Intesection types and resource control in the intuitionistic sequent lambda calculus

“…Najvažnija osobina koju tipski sistemi sa presekom treba da zadovoljavaju, Karakterizacija jake normalizacije, je za sistem λ Gtz Likavec. Originalni doprinosi predstavljeni u Glavi 4 su ostvareni u saradnji sa José Espírito Santom, Silviom Gilezan i Silviom Likavec, i objavljeni u radovima [24,31,41,42,25]. U Glavi 5, osnovni predmet proučavanja je λ Gtz -račun (resource control lambda Gentzen calculus), intuicionistički sekventni formalni račun sa kontrolom resursa, tj.…”

“…Section 4.1 and Subsection 4.2.1 revisit Espirito Santo's results on the untyped and the simply typed λ Gtz -calculus, while the results involving intersection type systems and the characterisation of strong normalisation represent the original contribution of the thesis. These results are developed by José Espírito Santo, Silvia Ghilezan, Silvia Likavec and myself, and are published in [24,31,41,42,25].…”

“…The intersection types were introduced into the λ Gtz -calculus by Espírito Santo, Ghilezan and Ivetić in [24] in order to obtain a type assignment system in which all strongly normalising terms would be typeable. The detailed account on this system is given in [25].…”

“…This system assigns a restricted form of intersection types, namely strict types, to λ Gtz -expressions. Strict types were proposed by van Bakel in [70] and already used by Espírito Santo et al in [25] for an alternative characterisation of strong normalisation in the λ Gtz -calculus.…”

“…The contributions presented in the Chapter 4 are obtained in collaboration with José Espírito Santo, Silvia Ghilezan and Silvia Likavec, and are published in [24,31,41,42,25].…”