We show that a type system based on the intuitionistic modal logic S4 provides an expressive framework for specifying and analyzing computation stages in the context of functional languages. Our main technical result is a conservative embedding of Nielson & Nielson's two-level functional language in our language Mini-ML, which in

A language of constructions for minimal logic is the -calculus, where cut-elimination is encoded as fi-reduction. We examine corresponding languages for the minimal version of the modal logic S4, with notions of reduction that encodes cut-elimination for the corresponding sequent system. It turns out that a natural interpretation of the latter constructions is a -calculus extended by an idealized version of Lisp's eval and quote constructs. In this Part IIIa, we examine the termination and confluence properties of the evQ and evQ H -calculi. Most results are negative: the typed calculi do not terminate, the subsystems \Sigma and \Sigma H that propagate substitutions, quotations and evaluations downwards do not terminate either in the untyped case, and the untyped evQ H -calculus is not confluent. However, the typed versions of \Sigma and \Sigma H do terminate, so the typed evQ-calculus is confluent. It follows that the typed evQ-calculus is a conservative extension of the typed S4-cal...

Starting from the idea that cut elimination is the precise meaning of program execution, we design two languages of constructions for the minimal logic S4, yielding -calculi with idealized versions of Lisp's eval and quote. The first, the S4 -calculus, is based on Bierman and De Paiva's proposal, and has all desirable logical properties, except for its non-operational flavor. The second, the evQ-calculus, is more complicated, but has a clear operational meaning: it is a tower of interpreters in the style of Lisp's reflexive tower. Remarkably, this language was developed from purely logical principles, but nonetheless provides some operational insight into eval/quote mechanisms. 1 Introduction Let's consider two dual questions. The first is: is there a proofs-as-programs, formulasas -types correspondence for the modal logic S4? There is one between minimal and intuitionistic logics and - calculi [How80], and also for classical logic [Gri90] or linear logic [Abr93], so why not S4? A...

We propose a few models of proof terms for the intuitionistic modal propositional logic S4. Some of them are based on partial orders, or cpos, or dcpos, some of them on a suitable category of topological spaces and continuous maps. A structure that emerges from these interpretations is that of augmented simplicial sets. This leads to so-called combinatorial models, where simplices play an important role: the point is that the simplicial structure interprets the 2 modality, and that the category of augmented simplicial sets is itself already a model of intuitionistic propositional S4 proof terms. In fact, this category is an elementary topos, and is therefore a prime candidate to interpret all proof terms for intuitionistic S4 set theory. Finally, we suggest that geometric-like realizations functors provide a recipe to build other models of intuitionistic propositional S4 proof terms.

: We show that reducing any simply-typed oe-term by applying the rules in oe eagerly always terminates, by a translation to the simply-typed -calculus, and similarly for oe * -terms with oe * -eager rewrites. This holds even with term and substitution metavariables. In fact, every reduction terminates provided that (fi)-redexes are only contracted under so-called safe contexts. The previous results follow because in oe, resp. oe *-normal forms, all contexts around terms of sort T are safe. Key-words: oe-calculus, explicit substitutions, termination, -calculus, simple types (R'esum'e : tsvp) Jean.Goubault@inria.fr Unit'e de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) T'el'ephone : (33 1) 39 63 55 11 -- T'el'ecopie : (33 1) 39 63 53 Une preuve de terminaison faible du oe-calcul simplement typ'e R'esum'e : Nous montrons que r'eduire n'importe quel oe-terme simplement typ'e en appliquant toujours les r`egles de oe le plus...

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