Convolution and Concurrency

02/06/2020

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by   James Cranch, et al.

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We show how concurrent quantales and concurrent Kleene algebras arise as convolution algebras Q^X of functions from structures X with two ternary relations that satisfy relational interchange laws into concurrent quantales or Kleene algebras Q. The elements of Q can be understood as weights; the case Q= corresponds to a powerset lifting. We develop a correspondence theory between relational properties in X and algebraic properties in Q and Q^X in the sense of modal and substructural logics, and boolean algebras with operators. As examples, we construct the concurrent quantales and Kleene algebras of Q-weighted words, digraphs, posets, isomorphism classes of finite digraphs and pomsets.