The monadic second-order logic of graphs. I. recognizable sets of finite graphs
Information and Computation
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Theoretical Computer Science
Finite automata, formal logic, and circuit complexity
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We consider regular languages of labeled trees. We give an effective characterization of the regular languages over such trees that are definable in first-order logic in the language of labeled graphs. These languages are the analog on trees of the “locally threshold testable” languages on strings. We show that this characterization yields a decision procedure for determining whether a regular tree language is first-order definable: The procedure is polynomial time in the minimal automaton presenting the regular language. We also provide an algorithm for deciding whether a regular language is definable in first-order logic supplemented with modular quantifiers.