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Monotone switching networks for directed connectivity are strictly more powerful than certain-knowledge switching networks

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.