Cayley多项式时间可计算群
Cayley Polynomial-Time Computable Groups
摘要：
我们提出了一个新的概括凯莱自动组，不同的时间复杂性计算乘法，语言复杂性的正常形式的代表。我们首先考虑具有$\mathcal C$类中的正规形式语言的群，并且在一定的受限图灵机模型（位置忠实的一带）上乘可在线性时间内计算的生成元。我们发现，许多自动组的算法属性被保存（二次时间字问题），证明各种封闭性，并表明，类是相当大的，例如，它包括所有几乎多环组。然后，我们推广到团体有正规形式的语言在类$\mathcal C$和乘法的发电机可计算的多项式时间（标准）图灵机。特别感兴趣的是当$\mathcal C= \mathrm{REG}$（正则语言的类）。我们证明了$\mathrm{REG}$ -Cayley多项式时间可计算群包含所有的非线性生成的幂零群、圈积$\mathbb Z_2 \wr \mathbb Z^2$和Thompson群$F$ .
英文摘要：
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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