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Concurrent Kleene Algebras and Pomset Languages

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A concurrent Kleene algebra (CKA) is essentially a Kleene algebra expanded by an operation and axioms for concurrent composition. Pomsets form a standard model of true concurrency; they have recently attracted some renewed attention in the area of weak memory concurrency. In this lecture I outline some connections between pomset languages and CKA . I characterise the free algebras in the varieties generated by some sublanguages and present two completeness results for classes of pomset languages that generalise the rational languages to the realm of concurrency. More precisely I show that the congruence on series-parallel rational pomset expressions generated by series-parallel rational pomset language identity is axiomatised by the axioms of Kleene algebra plus those of commutative Kleene algebra. A decision procedure is extracted from this proof. A second, more intricate completeness result relates down-closed series-parallel rational pomset languages with the full set of CKA axioms. A decision procedure for the equational theory of CKA can be obtained from this result (joint work with Michael Laurence).

This talk is part of the Logic and Semantics Seminar (Computer Laboratory) series.

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