Undecidable problems of decentralized observation and control on regular languages
Information Processing Letters
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We introduce problems of decentralized control with delayed communication, where delays are either unbounded or bounded by a given constant k. In the k-bounded-delay model, between the transmission of a message and its reception, the plant can execute at most k events. In the unbounded-delay model, the plant can execute any number of events between transmission and reception. We show that our framework yields an infinite hierarchy of control problems,CC = DCC_0\supset DCC_1\supset DCC_2\supset\cdots\supset DCUC \supset DCwhere CC is the set of control problems solvable with a single controller (centralized case) and DCCk (resp. DCUC, DC) is the set of problems solvable with two controllers in a k-bounded-delay network (resp. two controllers in an unbounded-delay network, two controllers without communication). The above containments are strict. We prove undecidability of checking the existence of controllers in the unbounded-delay case, or in the case without any communication. Finally, we prove that a decentralized observation problem with bounded-delay communication is decidable.