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1 LaBRI - Laboratoire Bordelais de Recherche en Informatique

Abstract : The controller synthesis problem as formalized by Ramadge and Wonham consists of finding a finite controller that when synchronized with a given plant results in a system satisfying a required property. In this setting, both a plant and a controller are deterministic fi- nite automata, while synchronization is modelled by a synchronous product. Originally, the framework was developed only for safety and some deadlock properties. More recently, Arnold et. al. have extended the setting to all mu-calculus expressible properties and proposed a reduction of the synthesis problem to the satisfiability problem of the mu-calculus. They have also presented some results on decidability of distributed synthesis problem where one requires to find several controllers that control the plant at the same time. The additional difficulty in this case is that each controller is aware of a different part of the whole system. In this paper, an extension of the setting to nondeterministic pro- cesses is studied. In other words, the case when both a system and a controller can be presented by a nondeterministic automaton. This extension is motivated by examples in control where a continuous quantity is measured and digitized thereby introducing imprecision and uncertainty. It is shown that nondeterminism of the plant can be handled at no extra cost, both for centralized and decentralized con- trol. Centralized synthesis remains decidable even for nondetermin- istic controllers. In contrast, very few cases of decentralized control are decidable when controllers are allowed to be nondeterministic. A classification of decidable-undecidable variants of this problem is given.

Author: André Arnold - Igor Walukiewicz -



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