论文标题
图形产品中的电源单词问题
The Power Word Problem in Graph Products
论文作者
论文摘要
组$ g $的电源单词问题询问表达式$ u_1^{x_1} \ cdots u_n^{x_n} $，其中$ u_i $是$ g $的有限生成器和$ x_i $ x $ biary nocded Integers的单词，等于$ g $的标识。这是对压缩单词问题的限制，其中输入单词由直线程序表示（即，$ g $以上的代数电路）。我们首先显示有关电源单词问题的一些简单结果。特别是，组$ g $的电源单词问题是$ nc^1 $ -MANY-一个可降低为$ g $的有限索引子组的电源单词问题。 对于我们的主要结果，我们考虑了没有订单级别元素的组的图形产品。我们表明，固定的图形产品中的电源单词问题为$ ac^0 $ -Turing-turing-turn-turn-turn-turn-turn-turn-turn-turn-turn-truct-turn-turn-turn-turn-turn-turn-turn-truck y-turia-tribuly $ f_2 $的单词问题和基本组的电源单词问题。此外，我们研究了图产品中统一的功率单词问题，其中依赖图和基数是输入的一部分。给定一类有限生成的组$ \ MATHCAL {C} $没有订单两个元素，图中均匀的电源单词问题可以在$ \ Mathsf {ac^0（c_ = l^{upowwp（\ m noupcal {c}}）} $中，$ upowWp（commath for for Mathcal for for） $ \ Mathcal {C} $。由于我们的结果，右角ARTIN组中的统一背包问题是NP完整的。 本文是两篇会议论文的结合。在[Stoberw22]和本文的先前迭代中，我们的图形产品的结果是错误地陈述的，而没有其他假设，即基本组没有二的元素。在目前的工作中，我们纠正了这个错误。尽管我们强烈猜测，如前所述的结果是正确的，但我们的证明依赖于这一额外的假设。
The power word problem for a group $G$ asks whether an expression $u_1^{x_1} \cdots u_n^{x_n}$, where the $u_i$ are words over a finite set of generators of $G$ and the $x_i$ binary encoded integers, is equal to the identity of $G$. It is a restriction of the compressed word problem, where the input word is represented by a straight-line program (i.e., an algebraic circuit over $G$). We start by showing some easy results concerning the power word problem. In particular, the power word problem for a group $G$ is $NC^1$-many-one reducible to the power word problem for a finite-index subgroup of $G$. For our main result, we consider graph products of groups that do not have elements of order two. We show that the power word problem in a fixed such graph product is $AC^0$-Turing-reducible to the word problem for the free group $F_2$ and the power word problems of the base groups. Furthermore, we look into the uniform power word problem in a graph product, where the dependence graph and the base groups are part of the input. Given a class of finitely generated groups $\mathcal{C}$ without order two elements, the uniform power word problem in a graph product can be solved in $\mathsf{AC^0(C_=L^{UPowWP(\mathcal{C})})}$, where $UPowWP(\mathcal{C})$ denotes the uniform power word problem for groups from the class $\mathcal{C}$. As a consequence of our results, the uniform knapsack problem in right-angled Artin groups is NP-complete. The present paper is a combination of the two conference papers. In [StoberW22] and previous iterations of this paper our results on graph products were wrongly stated without the additional assumption that the base groups do not have elements of order two. In the present work we correct this mistake. While we strongly conjecture that the result as stated previously is true, our proof relies on this additional assumption.
