Partially Ordered Automata and Piecewise Testability
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Universality is the question whether a system recognizes all words over its alphabet. Complexity of deciding universality provides lower bounds for other problems, including inclusion and equivalence of systems behaviors. We study the complexity of universality for a class of nondeterministic finite automata, models as expressive as boolean combinations of existential first-order sentences. We show that deciding universality is as hard as for general automata if the alphabet may grow with the number of states, but decreases if the alphabet is fixed. The proof requires a novel and nontrivial extension of our recent construction for self-loop-deterministic partially ordered automata. Consequently, we obtain complexity results for several variants of partially ordered automata and problems of inclusion, equivalence, and (k-)piecewise testability, for which we provide a whole complexity picture.
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