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On the group memory complexity of extended finite automata over groups. (English) Zbl 1462.68099
Summary: We define and investigate a complexity measure defined for extended finite automata over groups (EFA). Roughly, an EFA is a finite automaton augmented with a register storing an element of a group, initially the identity element. When a transition is performed, not only the state, but the register contents are updated. A word is accepted if, after reading completely the word, the automaton reached a final state, and the register returned to the identity element. The group memory complexity of an EFA over a group is a function from $$\mathbb{N}$$ to $$\mathbb{N}$$ which associates with each $$n$$ the value 0, if there is no word of length $$n$$ accepted by the automaton, or the minimal integer $$c$$ such that for every word $$x$$ of length $$n$$ accepted by the automaton, there is a computation on $$x$$ such that the number of transitions labeled by non-neutral element of the group used in that computation is at most $$c$$. We prove that a language is regular if and only if it is accepted by an EFA with a finite group memory complexity. In particular, any EFA over a group such that all its finitely generated subgroups are finite accepts a regular language. We then provide examples of EFA over some groups that accept non-regular languages and have a sublinear group memory complexity, namely a function in $$\mathcal{O}(\sqrt{n})$$ or $$\mathcal{O}(\log n)$$. There are non-regular languages such that any EFA over some group that accepts that language has a group memory complexity in $$\Omega(n)$$.
##### MSC:
 68Q45 Formal languages and automata 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
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