We introduce a class of applicative structures called bi-BDI-algebras. Bi-BDI-algebras are generalizations of partial combinatory algebras and BCI-algebras, and feature two sorts of applications (left and right applications). Applying the categorical realizability construction to bi-BDI-algebras, we obtain monoidal bi-closed categories of assemblies (as well as of modest sets). We further investigate two kinds of comonadic applicative morphisms on bi-BDI-algebras as non-symmetric analogues of linear combinatory algebras, which induce models of exponential and exchange modalities on non-symmetric linear logics.
@InProceedings{tomita:LIPIcs.CSL.2022.35, author = {Tomita, Haruka}, title = {{Planar Realizability via Left and Right Applications}}, booktitle = {30th EACSL Annual Conference on Computer Science Logic (CSL 2022)}, pages = {35:1--35:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-218-1}, ISSN = {1868-8969}, year = {2022}, volume = {216}, editor = {Manea, Florin and Simpson, Alex}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2022.35}, URN = {urn:nbn:de:0030-drops-157558}, doi = {10.4230/LIPIcs.CSL.2022.35}, annote = {Keywords: Realizability, combinatory algebra, monoidal bi-closed category, exponential modality, exchange modality} }
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