It is well known that the problem solving equations in virtually free groups can be reduced to the problem of solving twisted word equations with regular constraints over free monoids with involution. In this paper, we prove that the set of all solutions of a twisted word equation is an EDT0L language whose specification can be computed in PSPACE . Within the same complexity bound we can decide whether the solution set is empty, finite, or infinite. In the second part of the paper we apply the results for twisted equations to obtain in PSPACE an EDT0L description of the solution set of equations with rational constraints for finitely generated virtually free groups in standard normal forms with respect to a natural set of generators. If the rational constraints are given by a homomorphism into a fixed (or “small enough”) finite monoid, then our algorithms can be implemented in [Formula: see text], that is, in quasi-quadratic nondeterministic space. Our results generalize the work by Lohrey and Sénizergues (ICALP 2006) and Dahmani and Guirardel (J. of Topology 2010) with respect to both complexity and expressive power. Neither paper gave any concrete complexity bound and the results in these papers are stated for subsets of solutions only, whereas our results concern all solutions.

中文翻译：

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几乎自由群中扭曲词方程和方程的解

众所周知，实际上自由群中的问题求解方程可以简化为求解具有对合自由幺半群的规则约束的扭曲字方程的问题。在本文中，我们证明了扭曲词方程的所有解的集合是一种 EDT0L 语言，其规范可以在 PSPACE 中计算。在相同的复杂度范围内，我们可以决定解集是空的、有限的还是无限的。在论文的第二部分中，我们应用扭曲方程的结果在 PSPACE 中获得了方程组的解集的 EDT0L 描述，该方程组具有关于自然生成元集的标准范式中有限生成的虚拟自由群的有理约束。如果有理约束由同态给出为固定（或“足够小”）有限幺半群，那么我们的算法可以在[公式：见正文]中实现，即在准二次非确定性空间中。我们的结果概括了 Lohrey 和 Sénizergues (ICALP 2006) 以及 Dahmani 和 Guirardel (J. of Topology 2010) 在复杂性和表达能力方面的工作。这两篇论文都没有给出任何具体的复杂性限制，并且这些论文中的结果仅针对解决方案的子集进行说明，而我们的结果涉及所有解决方案。