Decidability problems in automaton semigroups
نویسنده
چکیده
We consider decidability problems in self-similar semigroups, and in particular in semigroups of automatic transformations of X. We describe algorithms answering the word problem, and bound its complexity under some additional assumptions. We give a partial algorithm that decides in a group generated by an automaton, given x, y, whether an Engel identity (r ̈ ̈ ̈ rrx, ys, ys, . . . , ys “ 1 for a long enough commutator sequence) is satisfied. This algorithm succeeds, importantly, in proving that Grigorchuk’s 2-group is not Engel. We consider next the problem of recognizing Engel elements, namely elements y such that the map x ÞÑ rx, ys attracts to t1u. Although this problem seems intractable in general, we prove that it is decidable for Grigorchuk’s group: Engel elements are precisely those of order at most 2. We include, in the text, a large number of open problems. Our computations were implemented using the package Fr within the computer algebra system Gap.
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ورودعنوان ژورنال:
- CoRR
دوره abs/1705.04598 شماره
صفحات -
تاریخ انتشار 2017