Abstract
This paper considers extended finite automata over monoids, in the sense of Dassow and Mitrana. We show that the family of languages accepted by extended finite automata over a monoid K is controlled by the word problem of K in a precisely stated manner. We also point out a critical error in the proof of the main result in the paper by Dassow and Mitrana. However as one consequence of our approach, by analyzing a certain word problem, we obtain a complete proof of this result, namely that the family of languages accepted by extended finite automata over the free group of rank two is exactly the family of context-free languages. We further deduce that along with the free group of rank two, the only finitely generated groups with this property are precisely the groups that have a nonabelian free subgroup of finite index.
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