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Explicit Substitutitions for Constructive Necessity



1 Introduction This paper introduces a *-calculus with explicit substitutions, corresponding to an S4 modal logic of constructive necessity. As well as being semantically well motivated, the calculus can be used (a) to develop abstract machines, and (b) as a framework for specifying and analysing computation stages in the context of functional languages. We first provide a full Curry-Howard isomorphism for the S4 constructive modal logic. This entails giving a calculus for annotating natural deduction proofs in the logic with typed *-calculus terms. The annotations must be such that types correspond to propositions, terms correspond to proofs, and proof normalisation corresponds to term reduction. (The calculus we provide can be given a categorical model, but this is not described in this paper). We then add explicit substitutions to the calculus. Explicit substitutions are a theoretical way of making *-calculi closer to their implementation through making substitution part of the calculus itself, rather than a meta-theoretical operation

Year: 2008
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