A Curry-Howard Correspondence for the Minimal Fragment of Łukasiewicz Logic

09/12/2018
by   Rob Arthan, et al.
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In this paper we introduce a term calculus B which adds to the affine λ-calculus with pairing a new construct allowing for a restricted form of contraction. We obtain a Curry-Howard correspondence between B and the sub-structural logical system which we call "minimal Łukasiewicz logic", also known in the literature as the logic of hoops (a generalisation of MV-algebras). This logic lies strictly in between affine minimal logic and standard minimal logic. We prove that B is strongly normalising and has the Church-Rosser property. We also give examples of terms in B corresponding to some important derivations from our work and the literature. Finally, we discuss the relation between normalisation in B and cut-elimination for a Gentzen-style formulation of minimal Łukasiewicz logic.

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