On the Complexity of Projections of Discrete - Event

Natural projections play an important role in decentralized control and partial observation of discrete-event systems (DES), where they are used to constructìocal' models of a given`global' system. However, in general a local model obtained by natural projection may be more complex (i.e., have more states and transitions) than the original system. This raises the question: Under what conditions will a projected DES be simpler than the original system? In this paper we show that a property called observer, introduced in an investigation of nonblocking hierarchical control of DES, provides an answer. If the natural projection on a given system possesses the observer property, then the (minimal) projected model will always be simpler than the original system. Also, it can be shown that in general the time complexity of computing a projected model is at worst exponential. It turns out that the observer condition ensures that the computation of a projected model is at worst polynomial.