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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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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...
A Proof of Weak Termination of Typed λσCalculi
, 1998
"... We show that reducing any simplytyped oeterm (resp. oe*) by applying the rules in oe (resp. oe *) eagerly always terminates, by a translation to the simplytyped calculus. This holds even with term and substitution metavariables. In fact, every reduction terminates provided that (fi)redexes ar ..."
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We show that reducing any simplytyped oeterm (resp. oe*) by applying the rules in oe (resp. oe *) eagerly always terminates, by a translation to the simplytyped calculus. This holds even with term and substitution metavariables. In fact, every reduction terminates provided that (fi)redexes are only contracted under socalled safe contexts; and in oe, resp. oe *normal forms, all contexts around terms of sort T are safe. The result is then extended to secondorder type systems.