Abstract
We study convolution and residual operations within convolution quantales of maps from partial abelian semigroups and effect algebras into value quantales, thus generalising separating conjunction and implication of separation logic to quantitative settings. We show that effect algebras lift to Girard convolution quantales, but not the standard partial abelian monoids used in separation logic. It follows that the standard assertion quantales of separation logic do not admit a linear negation relating convolution and its right adjoint. We consider alternative dualities for these operations on convolution quantales using boolean negations, some old, some new, relate them with properties of the underlying partial abelian semigroups and outline potential uses.
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Notes
- 1.
Most results on partial abelian monoids, more generally relational monoids, and (convolution) quantales can be found in the Archive of Formal Proofs [11, 24]. The complete formalisation can be found online http://hoefner-online.de/ramics21.
- 2.
CBI models are relational monoids, deterministic means that results of compositions are singletons, partial deterministic that they are singletons or empty.
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Acknowledgments
This research was supported in part by Australian Research Council (ARC) Grant DP190102142. The third author acknowledges sponsorship by Labex DigiCosme for an invited professorship at Laboratoire d’informatique de l’École polytechnique. The authors are grateful for helpful comments by the anonymous reviewers.
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Bannister, C., Höfner, P., Struth, G. (2021). Effect Algebras, Girard Quantales and Complementation in Separation Logic. In: Fahrenberg, U., Gehrke, M., Santocanale, L., Winter, M. (eds) Relational and Algebraic Methods in Computer Science. RAMiCS 2021. Lecture Notes in Computer Science(), vol 13027. Springer, Cham. https://doi.org/10.1007/978-3-030-88701-8_3
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