The Coarsest Congruence for Timed Automata with Deadlines Contained in Bisimulation

Delaying the synchronisation of actions may reveal some hidden behaviour that would not happen if the synchronisation would meet the specified deadlines. This precise phenomenon makes bisimulation fail to be a congruence for the parallel composition of timed automata with deadlines, a variant of timed automata where time progress is controlled by deadlines imposed on each transition. This problem has been known and unsolved for several years. In this paper we give a characterisation of the coarsest congruence that is included in the bisimulation relation. In addition, a symbolic characterisation of such relation is provided and shown to be decidable. We also discuss the pitfalls of existing parallel compositions in this setting and argue that our definition is both reasonable and sufficiently expressive as to consider the modelling of both soft and hard real-time constraints.