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...

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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 (resp. oe*) by applying the rules in oe (resp. oe *) eagerly always terminates, by a translation to the simply-typed -calculus. This holds even with term and substitution meta-variables. In fact, every reduction terminates provided that (fi)-redexes are only contracted under so-called safe contexts; and in oe, resp. oe *-normal forms, all contexts around terms of sort T are safe. The result is then extended to second-order type systems. 1 Introduction The simply-typed oe-calculus does not terminate strongly [8, 9], but it terminates in the weak sense: every typed term has a normal form. We present a proof of this; in fact, we prove the widely believed claim that every reduction where oe steps are applied eagerly is finite, as a consequence of a more general theorem stating that every reduction where (fi)-contraction only occurs under so-called safe contexts terminates. For the sake of generality, we prove this result even in the pres...