In this work we study local checkability of network properties like s-t reachability, or whether the network is acyclic or contains a cycle. A structural property S of a graph G is locally checkable, if there is a prover-and-verifier pair (P, V) as follows. The prover P assigns a label to each node in graphs satisfying S. The verifier V is a constant time distributed algorithm that returns Yes at all nodes if G satisfies S and was labeled by P, and No for at least one node if G does not satisfy S, regardless of the node labels. The quality of (P, V) is measured in terms of the label size. We obtain (asymptotically) tight bounds for the bit complexity of the latter two problems for undirected as well as directed networks, where in the directed case we consider one-way and two-way communication, i.e., we distinguish whether communication is possible only in the edge direction or not. For the one-way case we obtain a new asymptotically tight lower bound for the bit complexity of s-t reachability. For the two-way case we devise an emulation technique that allows us to transfer a previously known s-t reachability upper bound without asymptotic loss in the bit complexity.

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