RAIRO-Theor. Inf. Appl. 44 (2010) 79–111 Available online at: DOI: 10.1051/ita/2010006 www.rairo-ita.org FORMAL LANGUAGE PROPERTIES OF HYBRID SYSTEMS WITH STRONG RESETS ∗ Thomas Brihaye 1 , V´ eronique Bruy` ere 2 and Elaine Render 3 Abstract. We study hybrid systems with strong resets from the per- spective of formal language theory. We define a notion of hybrid reg- ular expression and prove a Kleene-like theorem for hybrid systems. We also prove the closure of these systems under determinisation and complementation. Finally, we prove that the reachability problem is undecidable for synchronized products of hybrid systems. Mathematics Subject Classification. 68Q68, 68Q45. 1. Introduction Nowadays more and more real-life systems are (automatically) controlled by computer programs. It is of capital importance to know whether the programs governing these systems are correct; these questions gave rise to the theory of ver- ification. In order to model real-life systems, various extensions of finite automata, such as timed automata [2] and hybrid systems [29], have been studied. Together with these models, various (temporal) logics, such as LTL [43] and CTL [24,44], have been considered, which capture properties of the systems in which we are in- terested. For instance, an important question is to know whether the system can reach some prohibited states. This question is known as the reachability problem. Keywords and phrases. Hybrid systems with strong resets, formal language theory. * The research of the third author was supported by a Modnet grant. This work has been partially supported by a grant from the National Bank of Belgium, and by the grant 2.4530.02 of the Belgian Fund for Scientific Research (F.R.S.-FNRS). 1 Institute of Mathematics, University of Mons 20, Place du Parc, 7000 Mons, Belgium; thomas.brihaye@umons.ac.be 2 Institute of Computer Science, University of Mons 20, Place du Parc, 7000 Mons, Belgium; veronique.bruyere@umh.ac.be 3 School of Mathematics, University of Manchester, Oxford Road, Manchester, U.K.; e.render@maths.man.ac.uk Article published by EDP Sciences c  EDP Sciences 2010