On computational interpretations of the modal logic S4 I. Cut elimination (1996) [8 citations — 5 self]
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@TECHREPORT{Goubault-larrecq96oncomputational,author = {Jean Goubault-larrecq},
title = {On computational interpretations of the modal logic S4 I. Cut elimination},
institution = {Institut fur Logik, Komplexitat und Deduktionssysteme, Universitat},
year = {1996}
}
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Abstract:
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 first part, we analyze how cut-elimination 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 polynomial-time algorithm to compute principal typings of given terms, or answer that the given terms are not typable. The typed S4-calculus terminates, and normal forms are exactly constructions for cut-free proofs. Finally, modulo some notion � � of equivalence, there is a natural Curry-Howard style isomorphism between typed S4-terms and natural deduction proofs in minimal S4. However, the S4-calculus has a non-operational flavor, in that the extra rules include explicit garbage collection, contraction and exchange rules. We shall propose another language of constructions to repair this in Part II. 1
