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On the Lambek Calculus with an Exchange Modality
- Source :
- Electronic Proceedings in Theoretical Computer Science, Vol 292, Iss Proc. Linearity-TLLA 2018, Pp 43-89 (2019)
- Publication Year :
- 2019
- Publisher :
- arXiv, 2019.
-
Abstract
- In this paper we introduce Commutative/Non-Commutative Logic (CNC logic) and two categorical models for CNC logic. This work abstracts Benton's Linear/Non-Linear Logic by removing the existence of the exchange structural rule. One should view this logic as composed of two logics; one sitting to the left of the other. On the left, there is intuitionistic linear logic, and on the right is a mixed commutative/non-commutative formalization of the Lambek calculus. Then both of these logics are connected via a pair of monoidal adjoint functors. An exchange modality is then derivable within the logic using the adjunction between both sides. Thus, the adjoint functors allow one to pull the exchange structural rule from the left side to the right side. We then give a categorical model in terms of a monoidal adjunction, and then a concrete model in terms of dialectica Lambek spaces.<br />Comment: In Proceedings Linearity-TLLA 2018, arXiv:1904.06159
- Subjects :
- FOS: Computer and information sciences
Computer Science - Logic in Computer Science
Computer Science - Programming Languages
Structural rule
lcsh:Mathematics
F.3.2
F.4.1
Adjunction
lcsh:QA1-939
Linear logic
lcsh:QA75.5-76.95
Logic in Computer Science (cs.LO)
Mathematics::Category Theory
Computer Science::Logic in Computer Science
Dialectica interpretation
Calculus
Monoidal adjunction
lcsh:Electronic computers. Computer science
Adjoint functors
Commutative property
Modality (semiotics)
Mathematics
Programming Languages (cs.PL)
Subjects
Details
- Database :
- OpenAIRE
- Journal :
- Electronic Proceedings in Theoretical Computer Science, Vol 292, Iss Proc. Linearity-TLLA 2018, Pp 43-89 (2019)
- Accession number :
- edsair.doi.dedup.....f9b6c266920f7c1fd777402126c4d98b
- Full Text :
- https://doi.org/10.48550/arxiv.1904.06847