自由群上方程的解集是EDT0L语言-- ICALP 2015版
Solution sets for equations over free groups are EDT0L languages -- ICALP 2015 version
摘要：
我们表明，给定一个词方程在一个非线性生成的自由群，所有的解决方案在减少的话形成一个EDT 0 L语言。特别是，它是Aho意义上的索引语言。是否有可能将解集描述为一种索引语言的问题已经公开了好几年，显然没有太多希望得到一个肯定的答案。然而，我们的答案远不止于此：它们是EDT 0 L，这是索引语言的一个适当子类。我们还可以处理自由乘积$\star_{1 \leq i \leq s}F_i$中具有有理约束的方程的存在性理论，其中每个$F_i$是自由或有限群，或者是具有对合的自由幺半群。在所有情况下，结果都是相同的：所有简化字的解的集合是EDT 0 L。这只在Ferté、Marin和Sénizergues（ToCS 2014）的二次词方程中才知道，这是一个非常有限的情况。我们的一般结果成为可能，由于最近的再压缩技术的Jewish。本文利用一种新的方法将线性丢番图方程的解整合到这一过程中，得到了比相关文献（arXiv 1405.5133）更一般的结果。例如，我们将复杂度从（arXiv 1405.5133）中的二次非确定性空间提高到这里的拟线性非确定性空间。这意味着一个改进的复杂性，决定存在的理论，非阿贝尔自由群：NSPACE（$n\log n$）。约束复杂度是NP的，然而，我们相信，我们的结果是最优的空间复杂度，独立于约束NP。
英文摘要：
We show that, given a word equation over a finitely generated free group, the set of all solutions in reduced words forms an EDT0L language. In particular, it is an indexed language in the sense of Aho. The question of whether a description of solution sets in reduced words as an indexed language is possible has been been open for some years, apparently without much hope that a positive answer could hold. Nevertheless, our answer goes far beyond: they are EDT0L, which is a proper subclass of indexed languages. We can additionally handle the existential theory of equations with rational constraints in free products $\star_{1 \leq i \leq s}F_i$, where each $F_i$ is either a free or finite group, or a free monoid with involution. In all cases the result is the same: the set of all solutions in reduced words is EDT0L. This was known only for quadratic word equations by Ferté, Marin and Sénizergues (ToCS 2014), which is a very restricted case. Our general result became possible due to the recent recompression technique of Jeż. In this paper we use a new method to integrate solutions of linear Diophantine equations into the process and obtain more general results than in the related paper (arXiv 1405.5133). For example, we improve the complexity from quadratic nondeterministic space in (arXiv 1405.5133) to quasi-linear nondeterministic space here. This implies an improved complexity for deciding the existential theory of non-abelian free groups: NSPACE($n\log n$). The conjectured complexity is NP, however, we believe that our results are optimal with respect to space complexity, independent of the conjectured NP.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
