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...

We present typed pattern calculus with explicit pattern matching and explicit substitutions, where both the typing rules and the reduction rules are modeled on the same logical proof system, namely Gentzen sequent calculus for minimal logic. Our calculus is inspired by the Curry-Howard Isomorphism, in the sense that types, both for patterns and terms, correspond to propositions, terms correspond to proofs, and term reduction corresponds to sequent proof normalization performed by cut elimination. The calculus enjoys subject reduction, confluence, preservation of strong normalization w.r.t a system with meta-level substitutions, and strong normalization for well-typed terms, and, as a consequence, can be seen as an implementation calculus for functional formalisms using meta-level operations for pattern matching and substitutions.

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.

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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...

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