We extend some of the classical connections between automata and logic due to Büchi [B¨60] and McNaughton and Papert [MP71], to languages of finitely varying functions or “signals”. In particular we introduce a natural class of automata for generating finitely varying functions called ST-NFA’s, and show that it coincides in terms of language-definability with a natural monadic second-order logic interpreted over finitely varying functions [Rab02]. We also identify a “counter-free” subclass of ST-NFA’s which characterizes the first-order definable languages of finitely varying functions. Our proofs mainly factor through the classical results for word languages. These results have applications in automata characterisations for continuously interpreted real-time logics like Metric Temporal Logic (MTL) [CDP06, CDP07]
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