# Completeness and Incompleteness of Synchronous Kleene Algebra

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## Abstract

*Synchronous Kleene algebra* (*SKA*), an extension of Kleene algebra (KA), was proposed by Prisacariu as a tool for reasoning about programs that may execute synchronously, i.e., in lock-step. We provide a countermodel witnessing that the axioms of SKA are incomplete w.r.t. its language semantics, by exploiting a lack of interaction between the *synchronous product* operator and the Kleene star. We then propose an alternative set of axioms for SKA, based on Salomaa’s axiomatisation of regular languages, and show that these provide a sound and complete characterisation w.r.t. the original language semantics.

## Notes

### Acknowledgements

The first author is grateful for discussions with Hans-Dieter Hiep and Benjamin Lion.

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