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 $\mathrm{\Omega }(n)$.

我们定义并研究了针对组上扩展有限自动机（EFA）定义的复杂性度量。粗略地讲，EFA是一个有限自动机，其中增加了一个寄存器，用于存储组的元素，最初是标识元素。当执行转换时，不仅状态被更新，而且寄存器内容也被更新。如果在完全读取单词后自动机达到了最终状态，并且寄存器返回了身份元素，则该单词被接受。EFA在组上的组内存复杂度是从$\mathbb{ñ}$ 至 $\mathbb{ñ}$其关联与每个Ñ的值为0，如果没有长的字Ñ由自动机接受，或最小整数Ç使得对于每一个字X长度的Ñ由自动机接受，对一个计算X，使得由该计算中使用的组的非中性元素标记的转换数最多为c。我们证明一种语言是规则的，当且仅当它被具有有限组内存复杂性的EFA接受时。尤其是，某个组中所有其有限生成的子组都是有限的任何EFA都接受常规语言。然后，我们提供了一些接受非常规语言并具有亚线性组内存复杂性的组的EFA示例，即$\mathcal{Ø}（\sqrt{ñ}）$ 要么 $\mathcal{Ø}（\mathrm{日志}ñ）$。存在非常规语言，因此接受该语言的某个组中的任何EFA在以下情况下都具有组内存复杂性：$\mathrm{\Omega }（ñ）$。