# A Characterization of Displayable Logics Extending Update Logic

Abstract : 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.]
Mots-clés :
Type de document :
Rapport
[Research Report] Université de Rennes 1. 2016
Domaine :

Littérature citée [51 références]

https://hal.archives-ouvertes.fr/hal-01276927
Contributeur : Guillaume Aucher <>
Soumis le : dimanche 26 juin 2016 - 20:31:37
Dernière modification le : mercredi 16 mai 2018 - 11:23:29

### Fichier

ResearchReport2016v2.pdf
Fichiers produits par l'(les) auteur(s)

### Licence

Distributed under a Creative Commons Paternité - Pas d'utilisation commerciale - Pas de modification 4.0 International License

### Identifiants

• HAL Id : hal-01276927, version 1

### Citation

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

### Métriques

Consultations de la notice

## 323

Téléchargements de fichiers