Abstract
We investigate properties of locally finite languages introduced by Ressayre (J. Symbolic Logic 53 (4) (1988) 1009–1026). These languages are defined by locally finite sentences and generalize languages recognized by automata or defined by monadic second-order sentences. We give many examples, showing that numerous context free languages are locally finite. Then we study closure properties of the family LOC of locally finite languages, and show that most undecidability results that hold for context free languages may be extended to locally finite languages. In a second part, we consider an extension of these languages to infinite and transfinite length words. We prove that each α-language which is recognized by a Büchi automaton (where α is an ordinal and ω⩽α<ω ω ) is defined by a locally finite sentence. This result, combined with a preceding one of (Finkel and Ressayre (J. Symbolic Logic 61 (2) (1996) 563–585), provides a generalization of Büchi's result about decidability of monadic second-order theory of the structure (α,<).
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