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In our past work, 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-ambiguity and those 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 iV-inference-observability property, an iV-inferring supervisor achieving the entire specification does not exist. We first show that the class of iV-inference-observable sublanguages is not closed under union implying that the supremal N- inference-observable sublanguage need not exist. We next provide a technique for synthesizing an N -inferring decentralized supervisor that achieves an N -inference-observable sublanguage of the specification. The sublanguage achieved equals the specification language when the specification itself is iV-inference-observable. A formula for the synthesized sublanguage is also presented. For the special cases of N = 0 and N = 1, the proposed supervisor achieves the same language as those reported in ,  (for N = 0) and  (for N = 1). The synthesized supervisor is parameterized by N (the parameter bounding the ambiguity level), and as N is increased, the supervisor becomes strictly more permissive in general. Thus, a user can choose N based on the degree of permissiveness and the degree of computational complexity desired.