Lower bounds on separation automata for Parity Games
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Several recently developed quasi-polynomial time algorithms for Parity Games rely on construction of an automaton with quasi-polynomial number of states that separates specific languages of infinite words. This motivates a question of whether a matching quasi-polynomial lower bound on the number of states can be proved. This would mean impossibility for a separation approach to give a polynomial-time algorithm for Parity Games. In this paper we study a restricted version of the problem. We bound the number of moves to be read by a separation automaton before it makes a decision. In the general problem there is no restriction of this sort. But we show that lower bounds for unrestricted version of the problem are related to lower bounds for a restricted one with some explicit bound on the number of moves. We apply communication complexity techniques to get an exponential lower bound on the number of states of a separation automata which makes a decision after reading linear number of moves. We also prove an exponential lower bound on the number of states of any deterministic automaton which outputs any repetition in an infinite string after reading some finite prefix of it. The motivation of this result in context of separation automata is discussed in a paper. Finally, we show an upper bound on a communication problem that we call Inevitable Intersection problem. Using this upper bound we conclude that a certain class of reductions from communication complexity are unable to show super-polynomial lower bounds on separation automata.
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