{"title":"图形产品中的幂字问题","authors":"Markus Lohrey, Florian Stober, Armin Weiß","doi":"10.1007/s00224-024-10173-z","DOIUrl":null,"url":null,"abstract":"<p>The power word problem for a group <span>\\(\\varvec{G}\\)</span> asks whether an expression <span>\\(\\varvec{u_1^{x_1} \\cdots u_n^{x_n}}\\)</span>, where the <span>\\(\\varvec{u_i}\\)</span> are words over a finite set of generators of <span>\\(\\varvec{G}\\)</span> and the <span>\\(\\varvec{x_i}\\)</span> binary encoded integers, is equal to the identity of <span>\\(\\varvec{G}\\)</span>. 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 <span>\\(\\varvec{G}\\)</span>). We start by showing some easy results concerning the power word problem. In particular, the power word problem for a group <span>\\(\\varvec{G}\\)</span> is <span>\\(\\varvec{\\textsf{uNC}^{1}}\\)</span>-many-one reducible to the power word problem for a finite-index subgroup of <span>\\(\\varvec{G}\\)</span>. 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 <span>\\(\\varvec{\\textsf{AC} ^0}\\)</span>-Turing-reducible to the word problem for the free group <span>\\(\\varvec{F_2}\\)</span> 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 <span>\\(\\varvec{\\mathcal {C}}\\)</span> without order two elements, the uniform power word problem in a graph product can be solved in <span>\\(\\varvec{\\textsf{AC} ^0[\\textsf{C}_=\\textsf{L} ^{{{\\,\\textrm{UPowWP}\\,}}(\\mathcal {C})}]}\\)</span>, where <span>\\(\\varvec{{{\\,\\textrm{UPowWP}\\,}}(\\mathcal {C})}\\)</span> denotes the uniform power word problem for groups from the class <span>\\(\\varvec{\\mathcal {C}}\\)</span>. As a consequence of our results, the uniform knapsack problem in right-angled Artin groups is <span>\\(\\varvec{\\textsf{NP}}\\)</span>-complete. The present paper is a combination of the two conference papers (Lohrey and Weiß 2019b, Stober and Weiß 2022a). In Stober and Weiß (2022a) 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 in Stober and Weiß (2022a) is true, our proof relies on this additional assumption.</p>","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The Power Word Problem in Graph Products\",\"authors\":\"Markus Lohrey, Florian Stober, Armin Weiß\",\"doi\":\"10.1007/s00224-024-10173-z\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The power word problem for a group <span>\\\\(\\\\varvec{G}\\\\)</span> asks whether an expression <span>\\\\(\\\\varvec{u_1^{x_1} \\\\cdots u_n^{x_n}}\\\\)</span>, where the <span>\\\\(\\\\varvec{u_i}\\\\)</span> are words over a finite set of generators of <span>\\\\(\\\\varvec{G}\\\\)</span> and the <span>\\\\(\\\\varvec{x_i}\\\\)</span> binary encoded integers, is equal to the identity of <span>\\\\(\\\\varvec{G}\\\\)</span>. 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 <span>\\\\(\\\\varvec{G}\\\\)</span>). We start by showing some easy results concerning the power word problem. In particular, the power word problem for a group <span>\\\\(\\\\varvec{G}\\\\)</span> is <span>\\\\(\\\\varvec{\\\\textsf{uNC}^{1}}\\\\)</span>-many-one reducible to the power word problem for a finite-index subgroup of <span>\\\\(\\\\varvec{G}\\\\)</span>. 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 <span>\\\\(\\\\varvec{\\\\textsf{AC} ^0}\\\\)</span>-Turing-reducible to the word problem for the free group <span>\\\\(\\\\varvec{F_2}\\\\)</span> 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 <span>\\\\(\\\\varvec{\\\\mathcal {C}}\\\\)</span> without order two elements, the uniform power word problem in a graph product can be solved in <span>\\\\(\\\\varvec{\\\\textsf{AC} ^0[\\\\textsf{C}_=\\\\textsf{L} ^{{{\\\\,\\\\textrm{UPowWP}\\\\,}}(\\\\mathcal {C})}]}\\\\)</span>, where <span>\\\\(\\\\varvec{{{\\\\,\\\\textrm{UPowWP}\\\\,}}(\\\\mathcal {C})}\\\\)</span> denotes the uniform power word problem for groups from the class <span>\\\\(\\\\varvec{\\\\mathcal {C}}\\\\)</span>. As a consequence of our results, the uniform knapsack problem in right-angled Artin groups is <span>\\\\(\\\\varvec{\\\\textsf{NP}}\\\\)</span>-complete. The present paper is a combination of the two conference papers (Lohrey and Weiß 2019b, Stober and Weiß 2022a). In Stober and Weiß (2022a) 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 in Stober and Weiß (2022a) is true, our proof relies on this additional assumption.</p>\",\"PeriodicalId\":22832,\"journal\":{\"name\":\"Theory of Computing Systems\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2024-05-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Theory of Computing Systems\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://doi.org/10.1007/s00224-024-10173-z\",\"RegionNum\":4,\"RegionCategory\":\"计算机科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Theory of Computing Systems","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.1007/s00224-024-10173-z","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
摘要
群(\varvec{G}\)的幂词问题问的是表达式 \(\varvec{u_1^{x_1} \cdots u_n^{x_n}}\) 是否是幂词、其中,\(\varvec{u_i}\)是有限的\(\varvec{G}\)生成器集合上的词;\(\varvec{x_i}\)是二进制编码的整数,等于\(\varvec{G}\)的标识。它是压缩字问题的一个限制条件,在压缩字问题中,输入字由直线程序(即 \(\varvec{G}\) 上的代数电路)表示。我们首先展示一些关于幂词问题的简单结果。特别是,群 \(\varvec{G}\) 的幂级数问题是 \(\varvec{textsf{uNC}^{1}}\)-many-one reducible to the power word problem for a finite-index subgroup of \(\varvec{G}\).对于我们的主要结果,我们考虑的是没有二阶元素的群的图积。我们证明了在一个固定的这样的图积中,幂词问题是 \(\varvec{textsf{AC} ^0}\)-Turing-reducible 到自由群 \(\varvec{F_2}\) 的词问题和基群的幂词问题的。此外,我们还研究了图积中的均匀幂词问题,其中隶属图和基群是输入的一部分。给定一类无二阶元素的有限生成群((\varvec{textsf{AC}\)),图积中的均匀幂词问题可以在(\varvec{\textsf{AC}\)中求解。^0[\textsf{C}_=\textsf{L}^{{\textrm{UPowWP}\,}}}(\mathcal {C})}]}\), where \(\varvec{{\,\textrm{UPowWP}\、(\mathcal{C})})表示来自类 \(\varvec\{mathcal {C}}\) 的群的均匀幂词问题。)由于我们的结果,直角阿汀群中的均匀knapsack问题是 \(\varvec{textsf{NP}}\)-完全的。本文是两篇会议论文(Lohrey and Weiß 2019b, Stober and Weiß 2022a)的合并。在 Stober and Weiß (2022a)中,我们关于图积的结果被错误地表述为没有额外假设基群没有二阶元素。在本论文中,我们纠正了这一错误。虽然我们强烈推测 Stober 和 Weiß (2022a) 中的结果是正确的,但我们的证明依赖于这个额外的假设。
The power word problem for a group \(\varvec{G}\) asks whether an expression \(\varvec{u_1^{x_1} \cdots u_n^{x_n}}\), where the \(\varvec{u_i}\) are words over a finite set of generators of \(\varvec{G}\) and the \(\varvec{x_i}\) binary encoded integers, is equal to the identity of \(\varvec{G}\). 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 \(\varvec{G}\)). We start by showing some easy results concerning the power word problem. In particular, the power word problem for a group \(\varvec{G}\) is \(\varvec{\textsf{uNC}^{1}}\)-many-one reducible to the power word problem for a finite-index subgroup of \(\varvec{G}\). 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 \(\varvec{\textsf{AC} ^0}\)-Turing-reducible to the word problem for the free group \(\varvec{F_2}\) 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 \(\varvec{\mathcal {C}}\) without order two elements, the uniform power word problem in a graph product can be solved in \(\varvec{\textsf{AC} ^0[\textsf{C}_=\textsf{L} ^{{{\,\textrm{UPowWP}\,}}(\mathcal {C})}]}\), where \(\varvec{{{\,\textrm{UPowWP}\,}}(\mathcal {C})}\) denotes the uniform power word problem for groups from the class \(\varvec{\mathcal {C}}\). As a consequence of our results, the uniform knapsack problem in right-angled Artin groups is \(\varvec{\textsf{NP}}\)-complete. The present paper is a combination of the two conference papers (Lohrey and Weiß 2019b, Stober and Weiß 2022a). In Stober and Weiß (2022a) 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 in Stober and Weiß (2022a) is true, our proof relies on this additional assumption.
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