Eeective Control of Logical Discrete Event Systems in a Trace Theory Setting Using the Reeection Operator ?

Logical discrete event systems can be modelled using trace theory. In this paper we present an eeective algorithm to nd a controller using an operator (the reeection) that leads to systems that go beyond our scope. We deene a discrete event system (DES) to be a triple, see Sme93b, Sme93c]: P = haP; bP; tPi with aP the alphabet (set of events), bP (aP) the behaviour set, and tP (aP) the task set. aP is a nite set of symbols, bP and tP are possibly innnite sets of strings of symbols (traces). For any x 2 tP we assume that, after x, P may stop without performing another event, while after x 2 bP ntP the system will eventually perform another event or it deadlocks. We call a DES realistic, if the behaviour is preex-closed: bP = pref(bP) and each completed task is a behaviour: tP bP. An unrealistic DES goes beyond our scope of a discrete event system. Nevertheless, it will play a crucial role in the remainder of this paper. The restriction of a trace x to some alphabet A, xdA, is deened by dA = , and xadA = xdA, if a 6 2 A, and xadA = (xdA)a, otherwise. Here denotes the empty string. Alphabet restriction can easily be extended to work on trace sets: TdA = fxdA j x 2 Tg, and on DESs: PdA = haP \ A; bPdA; tPdAi. For DESs P and R we deene the interaction, see Sme93b, Sme93c], by: P kR = haP aR; fx j xdaP 2 bP ^ xdaR 2 bRg; fx j xdaP 2 tP ^ xdaR 2 tRgi For interactions of systems the common events can be seen as internal events. Such events need no longer be visible outside the interaction. Therefore, we introduce the external interaction operator that deletes the common events: P ed R = (P kR)d(aP aR)