Abstract
In this paper we study proof translations between labelled and label-free calculi for the logic of Bunched Implications (\(\mathrm {BI}\)). We first consider the bunched sequent calculus \(\mathrm {LBI}\) and define a labelled sequent calculus, called \(\mathrm {GBI}\), in which labels and constraints reflect the properties of a specifically tailored Kripke resource semantics of \(\mathrm {BI}\) with two total resource composition operators and explicit internalization of inconsistency. After showing the soundness of \(\mathrm {GBI}\) w.r.t. our specific Kripke frames, we show how to translate any \(\mathrm {LBI}\)-proof into a \(\mathrm {GBI}\)-proof. Building on the properties of that translation we devise a tree property that every \(\mathrm {LBI}\)-translated \(\mathrm {GBI}\)-proof enjoys. We finally show that any \(\mathrm {GBI}\)-proof enjoying this tree property (and not only \(\mathrm {LBI}\)-translated ones) can systematically be translated to an \(\mathrm {LBI}\)-proof.
Work supported by the TICAMORE project (ANR grant 16-CE91-0002).
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Galmiche, D., Marti, M., Méry, D. (2019). Relating Labelled and Label-Free Bunched Calculi in BI Logic. In: Cerrito, S., Popescu, A. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2019. Lecture Notes in Computer Science(), vol 11714. Springer, Cham. https://doi.org/10.1007/978-3-030-29026-9_8
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