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Convolution and concurrency

Published online by Cambridge University Press:  23 March 2022

James Cranch ,
Simon Doherty  and
Georg Struth Open the ORCID record for Georg Struth [Opens in a new window]
James Cranch
Affiliation:
University of Sheffield, Sheffield S10 2TN, UK
Simon Doherty
Affiliation:
University of Sheffield, Sheffield S10 2TN, UK
Georg Struth*
Affiliation:
University of Sheffield, Sheffield S10 2TN, UK
*
*Corresponding author. Email: [email protected]
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Abstract

We show how concurrent quantales and concurrent Kleene algebras arise as convolution algebras of functions from relational structures with two ternary relations that satisfy relational interchange laws into concurrent quantales or Kleene algebras, among others. The elements of the quantales can be understood as weights; the case where weights are drawn from the booleans corresponds to languages. We develop a correspondence theory between properties of the relational structures and algebraic properties in the weight and convolution algebras in the sense of modal and substructural logics, or boolean algebras with operators. The resulting correspondence triangles yield in particular general construction principles for models of concurrent quantales and Kleene algebras as convolution algebras from much simpler relational structures, including weighted ones for quantitative applications. As examples, we construct the concurrent quantales and Kleene algebras of weighted words, digraphs, posets, isomorphism classes of finite digraphs and pomsets.

Keywords

Relational monoidsquantalesKleene algebrasconvolution algebrasconcurrency theory
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 31 , Issue 8 , September 2021 , pp. 918 - 949
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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