On Proof Terms and Embeddings of Classical Substructural Logics

Abstract

There is an intimate connection between proofs of the natural deduction systems and typed lambda calculus. It is well-known that in simply typed lambda calculus, the notion of formulae-as-types makes it possible to find fine structure of the implicational fragment of intuitionistic logic, i.e., relevant logic, BCK-logic and linear logic. In this paper, we investigate three classical substructural logics (GL, GLc, GLw) of Gentzen's sequent calculus consisting of implication and negation, which contain some of the right structural rules. In terms of Parigot's λμ-calculus with proper restrictions, we introduce a proof term assignment to these classical substructural logics. According to these notions, we can classify the λμ-terms into four categories. It is proved that well-typed GLx-λμ-terms correspond to GLx proofs, and that a GLx-λμ-term has a principal type if stratified where x is nil, c, w or cw. Moreover, we investigate embeddings of classical substructural logics into the corresponding intuitionistic substructural logics. It is proved that the Godel-style translations of GLx-λμ-terms are embeddings preserving substructural logics. As by-products, it is obtained that an inhabitation problem is decidable and well-typed GLx-λμ-terms are strongly normalizable.