@MISC{Kozen_onthe, author = {Dexter Kozen}, title = {On the Myhill-Nerode Theorem for Trees}, year = {} }

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Abstract

The Myhill-Nerode Theorem as stated in [6] says that for a set R of strings over a finite alphabet, the following statements are equivalent: (i) R is regular (ii) R is a union of classes of a right-invariant equivalence relation of index finite (iii) the relation R is of finite index, where x R y i 8z 2 xz 2 R $ yz 2 R. This result generalizes in a straightforward way to automata on finite trees. I rediscovered this generalization in connection with work on finitely presented algebras, and stated it without proof or attribution in [7, 8], being at that time under the impression that it was folklore and completely elementary. Itwas again rediscovered independently by Z. Fülöp and S. Vagvolgyi and reported in a recent contribution to this Bulletin [5]. In that paper they attribute the result to me. In fact, the result goes back at least ten years earlier to the late 1960s. It is difficult to attribute it to any one paper, since it seems to have been in the air at a time when the theory of finite automata on trees was undergoing intense development. In a sense, it is an inevitable consequence Myhill and Nerode's work [9, 10], since "conventional finite automata theory goes through for the generalization|and it goes through quite neatly " [11]. The first explicit