We propose a new generalization 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, prove various closure properties, and show that the class is quite large. We then generalize 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 $\mathcal{C}=\mathsf{REG}$ (the class of regular languages). We prove that $\mathsf{REG}$–Cayley polynomial–time computable groups include all finitely generated nilpotent groups, the wreath product ${\mathbb{Z}}_2\wr {\mathbb{Z}}^2$, and Thompson's group F.

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和范式代表的语言复杂度。我们首先考虑班级中具有正常形式语言的组$\mathcal{C}$以及在某个受限图灵机模型（位置-忠实单带）上可在线性时间内计算的生成器的乘法。我们展示了自动组的许多算法属性被保留，证明了各种闭包属性，并表明该类相当大。然后我们推广到类中具有正常形式语言的组$\mathcal{C}$并通过在（标准）图灵机上以多项式时间计算的生成器进行乘法。特别感兴趣的是当$\mathcal{C}=\mathsf{注册}$（常规语言类）。我们证明$\mathsf{注册}$–Cayley 多项式时间可计算群包括所有有限生成的幂零群，即花环乘积${\mathbb{Z}}_2\wr {\mathbb{Z}}^2$，和汤普森的组F。