Deciding Synchronous Kleene Algebra with Derivatives ⋆ Sabine Broda, S´ ılvia Cavadas, Miguel Ferreira, and Nelma Moreira CMUP & DCC, Faculdade de Ciˆ encias da Universidade do Porto Rua do Campo Alegre, 4169-007 Porto, Portugal sbb@dcc.fc.up.pt,silviacavadas@gmail.com, miguelferreira108@gmail.com,nam@dcc.fc.up.pt Abstract. Synchronous Kleene algebra (SKA) is a decidable framework that combines Kleene algebra (KA) with a synchrony model of concur- rency. Elements of SKA can be seen as processes taking place within a fixed discrete time frame and that, at each time step, may execute one or more basic actions or then come to a halt. The synchronous Kleene algebra with tests (SKAT) combines SKA with a Boolean algebra. Both algebras were introduced by Prisacariu, who proved the decidability of the equational theory, through a Kleene theorem based on the classical Thompson ε-NFA construction. Using the notion of partial derivatives, we present a new decision procedure for equivalence between SKA terms. The results are extended for SKAT considering automata with transi- tions labeled by Boolean expressions instead of atoms. This work con- tinous previous research done for KA and KAT, where derivative based methods were used in feasible algorithms for testing terms equivalence. Keywords: Synchronous Kleene Algebra, Concurrency, Equivalence, Derivative 1 Introduction Synchronous Kleene algebra (SKA) combines Kleene algebra (KA) with the syn- chrony model of concurrency of Milner’s Synchronous Calculus of Communica- tion Systems (SCCS) [20]. Synchronous here means that two concurrent pro- cesses execute a single action simultaneously at each time instant of a unique global clock. Although this synchrony model seems to be a very weak model of concurrency when compared with asynchronous interleaving models, its equa- tional theory is powerful and the SCCS calculus includes the Calculus of Com- munication Systems (CCS) as a sub-calculus. It also models the Esterel pro- gramming language [5], a tool used by the industry [29]. ⋆ This work was partially supported by CMUP (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020, and through the programme COMPETE and by the Portuguese Government through the FCT under project FCOMP-01-0124-FEDER-020486.