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The ___domain of an automatic sequence can be extended from the natural numbers to the integers via ''two-sided'' automatic sequences. This stems from the fact that, given ''k'' ≥ 2, every integer can be represented uniquely in the form <math> \sum_{0 \leq i \leq r} a_{i}(-k)^{i} , </math> where <math>a_{i} \in \{0, \dots, k-1\}</math>. Then a two-sided infinite sequence ''a''(''n'')<sub>''n'' <math>\in \mathbb{Z}</math></sub> is (−''k'')-automatic if and only if its subsequences ''a''(''n'')<sub>n ≥ 0</sub> and ''a''(−''n'')<sub>n ≥ 0</sub> are ''k''-automatic.<ref name=AS162>Allouche & Shallit (2003) p. 162</ref>
The alphabet of a ''k''-automatic sequence can be extended from finite size to infinite size via [[k-regular sequence|''k''-regular sequences]].<ref>{{cite journal | last1 = Allouche | first1 = J.-P. | last2 = Shallit | first2 = J. | title = The ring of ''k''-regular sequences | journal = Theoret. Comput. Sci. | volume = 98 | issue = 2 | year = 1992 | pages = 163–197 | doi=10.1016/0304-3975(92)90001-v| doi-access = free }}</ref> The ''k''-regular sequences can be characterized as those sequences whose ''k''-kernel is finitely-generated. Every bounded ''k''-regular sequence is automatic.<ref>{{cite web |last1=Shallit |first1=Jeffrey |title=The Logical Approach to Automatic Sequences, Part 1: Automatic Sequences and ''k''-Regular Sequences |url=https://cs.uwaterloo.ca/~shallit/Talks/linz1a.pdf |accessdate=April 1, 2020}}</ref>
==Logical approach==
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