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On Computational Interpretations of the Modal Logic S4 I. Cut Elimination
 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 ..."
Abstract

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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 first part, we analyze how cutelimination works in the standard sequent system for minimal S4, and where problems arise. Bierman and De Paiva's proposal is a natural language of constructions for this logic, but their calculus lacks a few rules that are essential to eliminate all cuts. The S4  calculus, namely Bierman and De Paiva's proposal extended with all needed rules, is confluent. There is a polynomialtime algorithm to compute principal typings of given terms, or answer that the given terms are not typable. The typed S4calculus te...