Intersection Types for the Resource Control Lambda Calculi

Abstract

Abstract. We propose intersection type assignment systems for two resource control term calculi: the lambda calculus and the sequent lambda calculus with explicit operators for weakening and contraction. These resource control calculi, λ and λ Gtz , respectively, capture the computational content of intuitionistic natural deduction and intuitionistic sequent logic with explicit structural rules. Our main contribution is the characterisation of strong normalisation of reductions in both calculi. We first prove that typability implies strong normalisation in λ by adapting the reducibility method. Then we prove that typability implies strong normalisation in λ Gtz by using a combination of well-orders and a suitable embedding of λ Gtz -terms into λ -terms which preserves types and enables the simulation of all its reductions by the operational semantics of the λ -calculus. Finally, we prove that strong normalisation implies typability in both systems using head subject expansion.

Non-idempotent intersection types and strong normalisation

“…We are not aware of any intersection type system featuring this property, which is here a consequence of dropping the implicit AC properties of intersections, and a clear advantage over the system in [BL11a]. Similar properties can however be found in systems such as those of [GILL11], which avoids having to explicitly type a term by an intersection type. Remark 2.14.…”

Intesection types and resource control in the intuitionistic sequent lambda calculus

“…Za razliku od prethodna tri prikazana računa koji su prethodili λ Gtz -računu i uticali na njegovo zasnivanje, ovaj račun je nastao kasnije, kao pomoćni sistemčija je osnovna svrha bila da se kreira jednostavnije okruženje x sa kontrolom resursa i tipovi-ma sa presekom, u koje bi potom sistem λ Gtz ∩ bio preveden. Sintaksa i operacijska semantika ovog sistema predstavljaju varijantu λ cw -računa, jednog od konstitutivnih računa Kesner and Renaudovog Prizmoida resursa [46,47], dok je tipski sistem sa striktnim tipovima uveden u radu Gilezan et al [32].…”

“…Formalni račun sa kontrolom resursa predstavljen u Glavi 5 i njegovi tipski sistemi, sa osnovnim i striktnim tipovima, uključujući i sve dokazane osobine, predstavljaju originalan doprinos ove disertacije. Oni su nastali u saradnji sa Pierreom Lescanneom, Silviom Gilezan, Silviom Likavec i Dragišom Žunićem, i objavljeni su u radovima [35,32]. U Glavi 6 su generalizovani formalni računi i tipski sistemi uvedeni u prethodne dve glave.…”

“…Finally, some technical results regarding the strong normalisation of the system λ Gtz ∩ (Section 5.3) rely on the analogous results obtained for λ ∩ -the λcalculus with intersection types. This system, proposed by Ghilezan et al in [32], was initially proposed as an auxiliary system with the goal of creating a simpler resource control setting in which λ Gtz ∩ could be translated.…”

“…The calculus presented in Chapter 5 and corresponding type systems, both with simple and intersection types, including all its properties, represent the original results of the thesis. Contributions are the result of the joint work with Pierre Lescanne, Silvia Ghilezan, Silvia Likavec and Dragiša Žunić and are published in [35,32].…”