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Intersection Types and Lambda Models
, 2005
"... Invariance of interpretation by #conversion is one of the minimal requirements for any standard model for the #calculus. With the intersection type systems being a general framework for the study of semantic domains for the #calculus, the present paper provides a (syntactic) characterisation of t ..."
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Cited by 11 (1 self)
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Invariance of interpretation by #conversion is one of the minimal requirements for any standard model for the #calculus. With the intersection type systems being a general framework for the study of semantic domains for the #calculus, the present paper provides a (syntactic) characterisation of the above mentioned requirement in terms of characterisation results for intersection type assignment systems.
Compositional Characterisations of λterms using Intersection Types
, 2003
"... We show how to characterise compositionally a number of evaluation properties of λterms using Intersection Type assignment systems. In particular, we focus on termination properties, such as strong normalisation, normalisation, head normalisation, and weak head normalisation. We consider also the p ..."
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We show how to characterise compositionally a number of evaluation properties of λterms using Intersection Type assignment systems. In particular, we focus on termination properties, such as strong normalisation, normalisation, head normalisation, and weak head normalisation. We consider also the persistent versions of such notions. By way of example, we consider also another evaluation property, unrelated to termination, namely reducibility to a closed term. Many of these characterisation results are new, to our knowledge, or else they streamline, strengthen, or generalise earlier results in the literature. The completeness parts of the characterisations are proved uniformly for all the properties, using a settheoretical semantics of intersection types over suitable kinds of stable sets. This technique generalises Krivine's and Mitchell's methods for strong normalisation to other evaluation properties.
Behavioural Inverse Limit λModels
, 2003
"... We construct two inverse limit λmodels which completely characterise sets of terms with similar computational behaviours: the sets of normalising, head normalising, weak head normalising λterms, those corresponding to the persistent versions of these notions, and the sets of closable, closable nor ..."
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We construct two inverse limit λmodels which completely characterise sets of terms with similar computational behaviours: the sets of normalising, head normalising, weak head normalising λterms, those corresponding to the persistent versions of these notions, and the sets of closable, closable normalising, and closable head normalising λterms. More precisely, for each of these sets of terms there is a corresponding element in at least one of the two models such that a term belongs to the set if and only if its interpretation (in a suitable environment) is greater than or equal to that element. We use the finitary logical description of the models, obtained by defining suitable intersection type assignment systems, to prove this.
Resource control and strong normalisation
, 2012
"... We introduce the resource control cube, a system consisting of eight intuitionistic lambda calculi with either implicit or explicit control of resources and with either natural deduction or sequent calculus. The four calculi of the cube that correspond to natural deduction have been proposed by Kesn ..."
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We introduce the resource control cube, a system consisting of eight intuitionistic lambda calculi with either implicit or explicit control of resources and with either natural deduction or sequent calculus. The four calculi of the cube that correspond to natural deduction have been proposed by Kesner and Renaud and the four calculi that correspond to sequent lambda calculi are introduced in this paper. The presentation is paramatrized with the set of resources (weakening or contraction), which enables a uniform treatment of the eight calculi of the cube. The simply typed resource control cube, on the one hand, expands the CurryHoward correspondence to intuitionistic natural deduction and intuitionistic sequent logic with implicit or explicit structural rules and, on the other hand, is related to substructural logics. We propose a general intersection type system for the resource control cube calculi. Our main contribution is a characterisation of strong normalisation of reductions in this cube. First, we prove that typeability implies strong normalisation in the “natural deduction base ” of the cube by adapting the reducibility method. We then prove that typeability implies strong normalisation in the “sequent base ” of the cube by using a combination of wellorders and a suitable embedding in the “natural deduction base”. Finally, we prove that strong normalisation implies typeability in the cube using head subject expansion. All proofs are general and can be made specific to each calculus of the cube