Relating May and Must Testing Semantics for Discrete Timed Process Algebras Luis Fernando Llana D´ ıaz and David de Frutos Escrig Dept. Sistemas Inform´aticos y Programaci´on. Universidad Complutense de Madrid. {llana,defrutos}@eucmax.sim.ucm.es Abstract. In this paper we prove that for timed algebras may test- ing is much stronger than it could be expected. More exactly, we prove that the may testing semantics is equivalent to the must testing seman- tics for a rather typical discrete timed process algebra when considering divergence-free processes. This is so, because for any adequate test we can define a dual one in such a way that a process passes the original test in the must sense if and only if it does not pass the dual one in the may sense. It is well known that in the untimed case by may testing we can (partially) know the possible behaviors of a process after the instant at which it diverges, which is not possible under must semantics. This is also the case in the timed case. Keywords: process algebra, time, testing semantics, must , may. 1 Introduction and Related Work Testing semantics is introduced in [DH84, Hen88] in order to have an abstract semantics induced by the operational one, which allows us to compare processes in a natural way. Besides, it is a formalization of the classical notion of testing of programs. Tests are applied to processes generating computations that either have success or fail. But as processes are non-deterministic it is possible that the same process sometimes passes a test and sometimes fails to do it. This leads us to two different families of tests associated to each process: those that sometimes are passed, and those that always are passed. Two processes are equivalent if they pass the same tests, but as we have two different ways to pass tests, we obtain two different testing semantics that are called may and must semantics. In this paper we will study testing semantics for timed process algebras, which is also the subject of the Ph.D. Thesis of the first author [Lla96]. As far as we know, there has been not too much previous work on the subject, but [HR95] is an interesting related reference. In the untimed case the may semantics is just trace semantics, while the must semantics is more involved, and so we need acceptance trees [Hen88] in order to characterize it. But when time is introduced the may semantics also becomes more complex, since, as usual, we are assuming the ASAP rule, which allows us to detect by means of tests not only the actions that have been executed, but also those that could have been chosen instead. To be exact, we will prove in this P.S. Thiagarajan, R. Yap (Eds.): ASIAN’99, LNCS 1742, pp. 74–86, 1999. c  Springer-Verlag Berlin Heidelberg 1999