关于L-可达性的复杂性
On the Complexity of L-reachability
摘要：
本文从复杂性理论的角度出发，对基于语言的图可达性问题（L-REACH）进行了研究。给定一个图，其边用语言L的字母符号和两个特殊顶点s和t标记，测试图中是否存在从s到t的路径P，使得路径P中从s到t的符号的连接形成语言L的字符串。我们研究这个问题的变种，不同的图类和不同的语言类，并获得所有这些复杂性理论的刻画。我们的主要结果如下：1.用形式语言理论对语言进行限制，我们证明了L-REACH的复杂性随着形式语言类的能力而增加。我们证明了存在一种正则语言，它的L-REACH是NL-完全的，即使是无向图。在线性语言的情况下，L-REACH的复杂性不会超过L本身的复杂性。此外，存在确定性上下文无关语言L，对于该语言，L-DJACH是LogCFL-完全的。2.我们使用L-REACH作为研究结构复杂性的透镜。在这个方向上，我们表明，有一个语言A在TC 0的A-DALACH是NP-完全的。利用这一点，我们证明了P vs NP问题等价于P vs DALUACH-1（P）问题。这导致了一种有趣的可能性，即通过证明DALACH-1（P）包含在P的某个子类中，我们可以证明复杂性类的分离的向上平移。请注意，我们不知道向上转换复杂性类的分离的方法。
英文摘要：
We initiate a complexity theoretic study of the language based graph reachability problem (L-REACH) : Fix a language L. Given a graph whose edges are labeled with alphabet symbols of the language L and two special vertices s and t, test if there is path P from s to t in the graph such that the concatenation of the symbols seen from s to t in the path P forms a string in the language L. We study variants of this problem with different graph classes and different language classes and obtain complexity theoretic characterizations for all of them. Our main results are the following: 1.Restricting the language using formal language theory we show that the complexity of L-REACH increases with the power of the formal language class. We show that there is a regular language for which the L-REACH is NL-complete even for undirected graphs. In the case of linear languages, the complexity of L-REACH does not go beyond the complexity of L itself. Further, there is a deterministic context-free language L for which L-DAGREACH is LogCFL-complete. 2.We use L-REACH as a lens to study structural complexity. In this direction we show that there is a language A in TC0 for which A-DAGREACH is NP-complete. Using this we show that P vs NP question is equivalent to P vs DAGREACH-1(P) question. This leads to the intriguing possibility that by proving DAGREACH-1(P) is contained in some subclass of P, we can prove an upward translation of separation of complexity classes. Note that we do not know a way to upward translate the separation of complexity classes.
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。
