Real-Time systems (RTS for short) are those systems whose behavior is time dependent. Researchers and scientists always point out that reliability and safety are of paramount importance in designing and building RTS because a failure of a RTS put the public and/or the environment at risk. For the purpose of effective error reporting and testing, this paper considers the trace inclusion problem for RTS: given a path p (resp. p') of length n of a timed automaton A (resp. B), find whether the set of timed traces of p of length n are included in the set of timed traces of p' of length n such that p is known (the different constraints and clock updates of p are given) and p' is unknown (only the set of traces is given and the different constraints and clock updates of p' are unknown). The solution proposed to this problem is based on the identification of the timed diagnostics of bounds. These latter give a finite representation of the trace space corresponding to a path. Their number varies between 1 and 2x(n+1). The the problem of trace inclusion is then reduced to the inclusion of timed diagnostics of bounds. By the way, the paper shows how to use these results to minimize the number of test cases considered when testing RTS.
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