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Relational models for the Lambek calculus with intersection and unit
Multi angle
Authors : Kuznetsov, Stepan (Author of the conference)
CIRM (Publisher )
03B47 - Substructural logics
08A02 - Relational systems, laws of composition
Additional resources :
https://www.cirm-math.fr/RepOrga/2398/Slides/Kuznetsov.pdf
CIRM (Publisher )
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Abstract :
We consider the Lambek calculus extended with intersection (meet) operation. For its variant which does not allow empty antecedents, Andreka and Mikulas (1994) prove strong completeness w.r.t. relational models (R-models). Without the antecedent non-emptiness restriction, however, only weak completeness w.r.t. R-models (so-called square ones) holds (Mikulas 2015). Our goals are as follows. First, we extend the calculus with the unit constant, introduce a class of non-standard R-models for it, and prove completeness. This gives a simpler proof of Mikulas' result. Second, we prove that strong completeness does not hold. Third, we extend our weak completeness proof to the infinitary setting, to so-called iterative divisions (Kleene star under division).
Keywords : Lambek calculus; relational semantics; completeness
MSC Codes : Keywords : Lambek calculus; relational semantics; completeness
03B47 - Substructural logics
08A02 - Relational systems, laws of composition
Additional resources :
https://www.cirm-math.fr/RepOrga/2398/Slides/Kuznetsov.pdf
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Information on the Video
Film maker : Recanzone, LucaLanguage : English Available date : 26/11/2021 Conference Date : 05/11/2021 Subseries : Research talks arXiv category : Logic ; Logic in Computer Science Mathematical Area(s) : Computer Science ; Logic and Foundations Format : MP4 (.mp4) - HD Video Time : 00:29:45 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2021-11-5_Kuznetsov.mp4 
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Information on the Event
Event Title : 19th International Conference on Relational and Algebraic Methods in Computer Science / 19ème Conférence internationale de méthodes relationnelles et algébriques en informatiqueEvent Organizers : Fahrenberg, Uli ; Gehrke, Mai ; Santocanale, Luigi ; Winter, Michael Dates : 02/11/2021 - 06/11/2021 Event Year : 2021 Event URL : https://conferences.cirm-math.fr/2398.html
Citation Data
DOI : 10.24350/CIRM.V.19828603Cite this video as: Kuznetsov, Stepan (2021). Relational models for the Lambek calculus with intersection and unit. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19828603 URI : http://dx.doi.org/10.24350/CIRM.V.19828603 |
See Also
- [Multi angle] Quantified constraint satisfaction problem: towards the classification of complexity / Author of the conference Zhuk, Dmitriy.
- [Multi angle] Time warps, from algebra to algorithms / Author of the conference Santschi, Simon.
- [Multi angle] Fixpoint games / Author of the conference König, Barbara.
- [Multi angle] Relational tight field bounds for distributed analysis of programs / Author of the conference Frias, Marcelo.
Bibliography
- ANDRÉKA, Hajnal et MIKULÁS, Szabolcs. Lambek calculus and its relational semantics: completeness and incompleteness. Journal of Logic, Language and Information, 1994, vol. 3, no 1, p. 1-37. - https://doi.org/10.1007/BF01066355
- LAMBEK, Joachim. The mathematics of sentence structure. The American Mathematical Monthly, 1958, vol. 65, no 3, p. 154-170. - https://doi.org/10.1080/00029890.1958.11989160
- MIKULÁS, Szabolcs. The equational theories of representable residuated semigroups. Synthese, 2015, vol. 192, no 7, p. 2151-2158. - https://doi.org/10.1007/s11229-014-0513-3
- MIKULÁS, Szabolcs. Lower semilattice-ordered residuated semigroups and substructural logics. Studia Logica, 2015, vol. 103, no 3, p. 453-478. - https://doi.org/10.1007/s11225-014-9574-z
- KUZNETSOV, Stepan L. Relational Models for the Lambek Calculus with Intersection and Unit. In : International Conference on Relational and Algebraic Methods in Computer Science. Springer, Cham, 2021. p. 258-274. - https://doi.org/10.1007/978-3-030-88701-8_16
