On regions and zones for event-clock automata

Event clock automata (ECA) are a model for timed languages that has been introduced by Alur, Fix and Henzinger as an alternative to timed automata, with better theoretical properties (for instance, ECA are determinizable while timed automata are not). In this paper, we revisit and extend the theory of ECA. We first prove that no finite time abstract language equivalence exists for ECA, thereby disproving a claim in the original work on ECA. This means in particular that regions do not form a time abstract bisimulation. Nevertheless, we show that regions can still be used to build a finite automaton recognizing the untimed language of an ECA. Then, we extend the classical notions of zones and DBMs to let them handle event clocks instead of plain clocks (as in timed automata) by introducing event zones and Event DBMs (EDBMs). We discuss algorithms to handle event zones represented as EDBMs, as well as (semi-) algorithms based on EDBMs to decide language emptiness of ECA. This work has been supported by the projects: (i) QUASIMODO (FP7ICT-STREP-214755), Quasimodo: “Quantitative System Properties in Model-Driven-Design of Embedded”, http://www.quasimodo. aau.dk/http://www.quasimodo.aau.dk/, (ii) GASICS (ESF-EUROCORES LogiCCC), Gasics: “Games for Analysis and Synthesis of Interactive Computational Systems”, http://www.ulb.ac.be/di/ gasics/http://www.ulb.ac.be/di/gasics/, (iii) Moves: “Fundamental Issues in Modelling, Verification and Evolution of Software”, http://moves.vub.ac.behttp://moves.vub.ac.be, a PAI program funded by the Federal Belgian Government and (iv) The European Union Seventh Framework Programme under Grant Agreement 601148 (Cassting), http://www.cassting-project.euhttp://www.cassting-project.eu. Gilles Geeraerts has been supported by a ‘Crédit aux chercheurs’ from the Belgian FRS/F.N.R.S. Gilles Geeraerts Université libre de Bruxelles, Faculté des Sciences, Département d’Informatique, Brussels, Belgium E-mail: [email protected] Jean-François Raskin Université libre de Bruxelles, Faculté des Sciences, Département d’Informatique, Brussels, Belgium E-mail: [email protected] Nathalie Sznajder Sorbonne Universités, Université Pierre et Marie Curie UMR CNRS 7606, LIP6, Paris, France E-mail: [email protected] 2 Gilles Geeraerts et al.