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

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...

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SKIn and SKInT are two first-order languages that have been proposed recently by Healfdene Goguen and the author. While SKIn encodes lambda-calculus reduction faithfully, standardizes and is confluent even on open terms, it normalizes only weakly in the simply-typed case. On the other hand, SKInT normalizes strongly in the simply-typed case, standardizes and is confluent on open terms, and also encodes lambda-calculus reduction faithfully, although in a less direct way. This report has two goals. First, we show that the natural simple type system for SKInT, seen as a natural deduction system, is not exactly a proof system for intuitionistic logic, but for a very close fragment of the modal logic S4, in which intuitionistic logic is easily coded. This explains why the SKIn and SKInT typing rules are different, and why SKInT encodes lambda-calculus in a less direct way than SKIn. Second, we show that SKInT, like AE and a few other calculi of explicit substitutions, preserves strong nor...

: 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...