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# Decidable Logics with Associative Binary Modalities

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LIPIcs.CSL.2017.15.pdf
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## Cite As

Joseph Boudou. Decidable Logics with Associative Binary Modalities. In 26th EACSL Annual Conference on Computer Science Logic (CSL 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 82, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.CSL.2017.15

## Abstract

A new family of modal logics with an associative binary modality, called counting logics is proposed. These propositional logics allow to express finite cardinalities of sets and more generally to count the number of subsets satisfying some properties. We show that these logics can be seen both as specializations of the Boolean logic of bunched implications and as generalizations of the propositional dependence logic. Moreover, whereas most logics with an associative binary modality are undecidable, we prove that some counting logics are decidable, in particular the basic counting logic bCL. We conjecture that this interesting result is due to the valuation constraints in counting logics' semantics and prove that the logic corresponding to bCL without these constraints is undecidable. Finally, we give lower and upper bounds for the complexity of bCL's validity problem.
##### Keywords
• modal logics
• abstract separation logics
• team semantics
• resource logics
• substructural logics

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