Pomset languages and concurrent Kleene algebras

Séminaire automates in Paris, November 2017

Concurrent Kleene algebras (CKA) and bi-Kleene algebras support equational reasoning about computing systems with concurrent behaviours. Their natural semantics is given by series(-parallel) rational pomset languages, a standard true concurrency semantics, which is often associated with processes of Petri nets. In the first part of the talk, I will present an automaton model designed to describe such languages of pomset, which satisfies a Kleene-like theorem. The main difference with previous constructions is that from expressions to automata, we use Brzozowski derivatives. In a second part, I will use Petri nets to reduce the problem of containment of languages of pomsets to the equivalence of finite state automata. In doing so, we prove decidabilty as well as provide tight complexity bounds. I will finish the presentation by briefly presenting a recent proof of completeness, showing that two series-rational expressions are equivalent according to the laws of CKA exactly when their pomset semantics are equal. Joint work with Damien Pous, Georg Struth, Tobias Kappé, Bas Luttik, Alexandra Silva, and Fabio Zanasi.

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