On an Eeective Hierarchy of Communicating Processes: Separation Principle and Testing

In this paper, we will establish precise connections between innnitary languages and test equivalent concurent processes. More precisely, we will extend some negative results on the separability of classes of concurrent processes. Two processes are said separable if they can be separated from each other by means of another test process from the same class i.e., if one can nd a test that is successful for one process while failing on the other. Interpreting the set of innnite visible traces of a process as an !-language, we obtain this language theoretic deenition: given a class C of !-languages, we say that L and L 0 2 C are separable if there exists T 2 C so that L \ T = ; while L 0 \ T 6 = ;. We concentrate on guaranteed processes, i.e processes under a guarantee constraint. Intuitively , a process satisses a guarantee requirement when that requirement holds for any forthcoming point in every computation. Starting from some eeective classes of guarantee properties proven to form a hierarchy. We will rst prove their non-separability. We will then deene classes of processes in such a way that non-separability under testing can be induced from the above language theoretic non-separability result. It nally turns out that some processes, having diierent visible behaviours, are test equivalent. We conclude with logical complexity issues of the testing problem when test criteria and guarantee constraints range over the arithmetical hierarchy.