Post's Correspondence Problem (the PCP) is a classical decision problem in theoretical computer science that asks whether for pairs of free monoid morphisms $g, h\colon\Sigma^*\to\Delta^*$ there exists any non-trivial $x\in\Sigma^*$ such that $g(x)=h(x)$. Post's Correspondence Problem for a group $\Gamma$ takes pairs of group homomorphisms $g, h\colon F(\Sigma)\to \Gamma$ instead, and similarly asks whether there exists an $x$ such that $g(x)=h(x)$ holds for non-elementary reasons. The restrictions imposed on $x$ in order to get non-elementary solutions lead to several interpretations of the problem; we consider the natural restriction asking that $x \notin \ker(g) \cap \ker(h)$ and prove that the resulting interpretation of the PCP is undecidable for arbitrary hyperbolic $\Gamma$, but decidable when $\Gamma$ is virtually nilpotent. We also study this problem for group constructions such as subgroups, direct products and finite extensions. This problem is equivalent to an interpretation due to Myasnikov, Nikolev and Ushakov when one map is injective.
翻译:邮差对应问题(PCP)是一个典型的理论计算机科学决定问题, 询问对于自由单态的一对夫妇来说, 是否存在非三重美元xxxxxxxgmaxxxxxgma=xx美元。 对于一个组的美元=xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxgg, h\colom F(Sigma)\crocolf\gmad\ggmamad = gammam$, 类似地问是否存在美元xxx美元, 美元=xxxxxxxxxx(xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxgxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmlxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx