Myhill-Nerode Theorem for Sequential Transducers over Unique GCD-Monoids
We generalize the classical Myhill-Nerode theorem for finite automata to the setting of sequential transducers over unique GCD-monoids, which are cancellative monoids in which every two non-zero elements admit a unique greatest common (left) divisor. We prove that a given formal power series is sequential, if and only if it is directed and our Myhill-Nerode equivalence relation has finite index. As in the classical case, our Myhill-Nerode equivalence relation also admits the construction of a minimal (with respect to the number of states) sequential transducer recognizing the given formal power series.
KeywordsPower Series Classical Case Formal Power Series Finite Index Minimization Algorithm
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