有向连通的单调切换网络严格地比确定知识切换网络更强大
Monotone switching networks for directed connectivity are strictly more powerful than certain-knowledge switching networks
摘要：
L（Logarithmic space）与NL（Non-deterministic logarithmic space）是计算复杂性理论中的一个重要的开放问题。在论文“Bounds on monotone switching networks for directed connectivity”中，我们使用一种称为开关网络模型的模型来分离L和NL的单调类似物。特别地，通过考虑输入仅由一条路径和孤立的顶点组成，我们证明了任何求解$N$顶点上的有向连通性的单调切换网络的大小必须至少为$N^{Ω(\lg(N))}$，并且这个界是紧的。如果我们能证明一般交换网络求解有向连通性的类似结果，那么这将证明$L \neq NL$。然而，证明一般开关网络解决有向连通性的下界需要证明更强的有向连通性的单调开关网络的下界。为了实现这一目标，我们研究了一组不同的输入，我们认为这对单调切换网络来说很难解决，并试图证明类似的下限。相反，我们发现，这组输入实际上很容易为单调切换网络的有向连通性解决，但如果我们限制自己的某些知识切换网络，这是一个简单而直观的子类单调切换网络的有向连通性，那么这些输入确实很难解决。在本文中，我们给出了这一组输入，证明了一个“奇怪的”多项式大小的单调切换网络的有向连通性，解决了这一组输入，并证明了没有多项式大小的确定性知识切换网络可以解决这一组输入，从而证明单调切换网络的有向连通性严格比确定性知识切换网络更强大。
英文摘要：
L (Logarithmic space) versus NL (Non-deterministic logarithmic space) is one of the great open problems in computational complexity theory. In the paper "Bounds on monotone switching networks for directed connectivity", we separated monotone analogues of L and NL using a model called the switching network model. In particular, by considering inputs consisting of just a path and isolated vertices, we proved that any monotone switching network solving directed connectivity on $N$ vertices must have size at least $N^{Ω(\lg(N))}$ and this bound is tight. If we could show a similar result for general switching networks solving directed connectivity, then this would prove that $L \neq NL$. However, proving lower bounds for general switching networks solving directed connectivity requires proving stronger lower bounds on monotone switching networks for directed connectivity. To work towards this goal, we investigated a different set of inputs which we believed to be hard for monotone switching networks to solve and attempted to prove similar lower size bounds. Instead, we found that this set of inputs is actually easy for monotone switching networks for directed connectivity to solve, yet if we restrict ourselves to certain-knowledge switching networks, which are a simple and intuitive subclass of monotone switching networks for directed connectivity, then these inputs are indeed hard to solve. In this paper, we give this set of inputs, demonstrate a "weird" polynomially-sized monotone switching network for directed connectivity which solves this set of inputs, and prove that no polynomially-sized certain-knowledge switching network can solve this set of inputs, thus proving that monotone switching networks for directed connectivity are strictly more powerful than certain-knowledge switching networks.
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。
