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A Modal Analysis of Staged Computation
 JOURNAL OF THE ACM
, 1996
"... 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 twolevel functional la ..."
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Cited by 208 (22 self)
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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 twolevel functional language in our language MiniML, which in
On Computational Interpretations of the Modal Logic S4 IIIa. Termination, Confluence, Conservativity of λevQ
 INSTITUT FUR LOGIK, KOMPLEXITAT UND DEDUKTIONSSYSTEME, UNIVERSITAT
, 1996
"... A language of constructions for minimal logic is the calculus, where cutelimination is encoded as fireduction. We examine corresponding languages for the minimal version of the modal logic S4, with notions of reduction that encodes cutelimination for the corresponding sequent system. It turns o ..."
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Cited by 8 (4 self)
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A language of constructions for minimal logic is the calculus, where cutelimination is encoded as fireduction. We examine corresponding languages for the minimal version of the modal logic S4, with notions of reduction that encodes cutelimination 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 evQcalculus is confluent. It follows that the typed evQcalculus is a conservative extension of the typed S4cal...
OrderTheoretic, Geometric and Combinatorial Models of Intuitionistic S4 Proofs
 IN INTUITIONISTIC MODAL LOGICS AND APPLICATIONS (IMLA’99)
, 1999
"... 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 augme ..."
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Cited by 7 (3 self)
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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 socalled 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 geometriclike realizations functors provide a recipe to build other models of intuitionistic propositional S4 proof terms.
Logical Foundations of Eval/Quote Mechanisms, and the Modal Logic S4
 IN PRESS S15708683(05)000431/FLA AID:71 Vol.•••(•••) [DTD5] P.12 (112) JAL:m1a v 1.40 Prn:15/07/2005; 8:08 jal71 by:SL p. 12 12 N. Alechina, D. Shkatov / Journal of Applied Logic
, 1997
"... 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 ..."
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Cited by 5 (0 self)
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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 nonoperational flavor. The second, the evQcalculus, 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 proofsasprograms, 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...
On Computational Interpretations of the Modal Logic S4
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A Proof of Weak Termination of the SimplyTyped λσCalculus
, 1997
"... : We show that reducing any simplytyped oeterm by applying the rules in oe eagerly always terminates, by a translation to the simplytyped calculus, and similarly for oe * terms with oe * eager rewrites. This holds even with term and substitution metavariables. In fact, every reduction termina ..."
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: We show that reducing any simplytyped oeterm by applying the rules in oe eagerly always terminates, by a translation to the simplytyped 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 socalled safe contexts. The previous results follow because in oe, resp. oe *normal forms, all contexts around terms of sort T are safe. Keywords: oecalculus, 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 oecalcul simplement typ'e R'esum'e : Nous montrons que r'eduire n'importe quel oeterme simplement typ'e en appliquant toujours les r`egles de oe le plus...