Synthesis of Over-Approximating Inference-based Decentralized Supervisors for Discrete Event Systems

In our past work [1], we presented a framework for the decentralized control of discrete event systems involving inferencing over ambiguities, about the system state, of various local decision-makers. Using the knowledge of the self- ambiguities and that of the others, each local control decision is tagged with a certain ambiguity level (level zero being the minimum and representing no ambiguity). A global control decision is taken to be a "winning" local control decision, i.e., one with a minimum ambiguity level. For the existence of a decentralized supervisor, so that for each controllable event the ambiguity levels of all winning disablement or enablement decisions are bounded by some number N (such a supervisor is termed N-inferring), the notion of N-inference-observability was introduced. When the given specification fails to satisfy the N-inference-observability property, an N-inferring supervisor achieving the entire specification does not exist, and a technique for synthesizing a decentralized supervisor that achieves an N-inference-observable sublanguage of the specification was presented in [7]. The present paper studies the dual problem, namely, the synthesis of a decentralized supervisor for achieving an N-inference-observable superlanguage. This requires a key modification in the technique proposed for the sublanguage case. The superlanguage achieved equals the specification language when the specification is strongly N-inference-observable, a new property introduced in this paper. A formula for the synthesized superlanguage is also presented. The synthesized superlanguage is parameterized by N (the parameter bounding the ambiguity level), and as N is increased, the superlanguage becomes smaller. Further, regardless of the choice of N, the synthesized superlanguage is smaller than the infimal closed, controllable, and C&P-coobservable superlanguage.