Planar Realizability via Left and Right Applications

Author Haruka Tomita

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Haruka Tomita
  • Research Institute for Mathematical Sciences, Kyoto University, Japan


I would like to thank Masahito Hasegawa for a lot of helpful discussions and comments. I am also grateful to Naohiko Hoshino for useful advice. Thanks also to anonymous reviewers for their helpful feedback.

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Haruka Tomita. Planar Realizability via Left and Right Applications. In 30th EACSL Annual Conference on Computer Science Logic (CSL 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 216, pp. 35:1-35:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Categorical semantics
  • Realizability
  • combinatory algebra
  • monoidal bi-closed category
  • exponential modality
  • exchange modality


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