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Rapport (Rapport De Recherche) Année : 2016

A Characterization of Displayable Logics Extending Update Logic

Résumé

Correspondence results for substructural logics are proved and a series of correspondence algorithms are introduced for relating analytic inference rules of display calculi and first-order frame conditions. These results and algorithms are obtained thanks to update logic, which is a generalization of the non-associative Lambek calculus. We characterize all the properly displayable logics without (truth) constant extending update logic (and thus the Lambek calculus). Our characterization tells us that a logic without constant extending update logic is properly displayable if, and only if, the class of pointed substructural frames on which the logic is based can be defined by some finite set of specific primitive first-order formulas called prototypic formulas. In that case, we provide algorithms to compute the prototypic formulas defining the class of pointed substructural frames that correspond to the analytic inference rules of the proper display calculus and, vice versa, we also provide algorithms to compute the analytic inference rules of the display calculus that correspond to the prototypic formulas defining the class of substructural frames. Our proofs and algorithms resort to a specific multi-modal tense logic and they use extensively Sahlqvist's as well as Kracht's results and techniques developed for tense logics. [N.B.: The results of this Research Report were presented for discussion at the Lorentz center workshop ``unified correspondence'' on the 16th of February 2016 (https://www.lorentzcenter.nl/lc/web/2016/757/info.php3?wsid=757). The previous version of this report contains errors which are corrected in this version.]
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Dates et versions

hal-01276927 , version 1 (26-06-2016)

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Paternité - Pas d'utilisation commerciale - Pas de modification

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  • HAL Id : hal-01276927 , version 1

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Guillaume Aucher. A Characterization of Displayable Logics Extending Update Logic. [Research Report] Université de Rennes 1. 2016. ⟨hal-01276927⟩
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