We propose a new generalisation of Cayley automatic groups, varying the time complexity of computing multiplication, and language complexity of the normal form representatives. We first consider groups which have normal form language in the class $\mathcal C$ and multiplication by generators computable in linear time on a certain restricted Turing machine model (position-faithful one-tape). We show that many of the algorithmic properties of automatic groups are preserved (quadratic time word problem), prove various closure properties, and show that the class is quite large; for example it includes all virtually polycyclic groups. We then generalise to groups which have normal form language in the class $\mathcal C$ and multiplication by generators computable in polynomial time on a (standard) Turing machine. Of particular interest is when $\mathcal C= \mathrm{REG}$ (the class of regular languages). We prove that $\mathrm{REG}$-Cayley polynomial-time computable groups includes all finitely generated nilpotent groups, the wreath product $\mathbb Z_2 \wr \mathbb Z^2$, and Thompson's group $F$.

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Cayley 多项式时间可计算群

我们提出了 Cayley 自动群的新泛化，改变了计算乘法的时间复杂度，以及范式代表的语言复杂度。我们首先考虑在类 $\mathcal C$ 中具有范式语言的组，以及在特定受限图灵机模型（位置忠实单带）上可在线性时间内计算的生成器的乘法。我们证明了自动组的许多算法性质都被保留了（二次时间词问题），证明了各种闭包性质，并表明该类相当大；例如，它包括所有实际上的多环基团。然后，我们推广到在类 $\mathcal C$ 中具有标准形式语言的组，并且可以在（标准）图灵机上以多项式时间计算生成器的乘法。特别感兴趣的是 $\mathcal C= \mathrm{REG}$（常规语言的类）。我们证明 $\mathrm{REG}$-Cayley 多项式时间可计算群包括所有有限生成的幂零群、花环积 $\mathbb Z_2 \wr \mathbb Z^2$ 和 Thompson 群 $F$。