Abstract
In a distributed locally-checkable proof, we are interested in checking the legality of a given network configuration with respect to some Boolean predicate. To do so, the network enlists the help of a prover—a computationally-unbounded oracle that aims at convincing the network that its state is legal, by providing the nodes with certificates that form a distributed proof of legality. The nodes then verify the proof by examining their certificate, their local neighborhood and the certificates of their neighbors.
In this paper we examine the power of a randomized form of locally-checkable proof, called distributed Merlin-Arthur protocols, or \({\textsf {dMA}}\) for short. In a \({\textsf {dMA}}\) protocol, the prover assigns each node a short certificate, and the nodes then exchange random messages with their neighbors. We show that while there exist problems for which \({\textsf {dMA}}\) protocols are more efficient than protocols that do not use randomness, for several natural problems, including Leader Election, Diameter, Symmetry, and Counting Distinct Elements, \({\textsf {dMA}}\) protocols are no more efficient than standard nondeterministic protocols. This is in contrast with Arthur-Merlin (\({\textsf {dAM}}\)) protocols and Randomized Proof Labeling Schemes (RPLS), which are known to provide improvements in certificate size, at least for some of the aforementioned properties.
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Acknowledgements
Partially supported by CONICYT PIA/Apoyo a Centros Científicos y Tecnológicos de Excelencia AFB 170001 (P.M. and I.R.), Fondecyt 1170021 (I.R.) and CONICYT via PAI + Convocatoria Nacional Subvención a la Incorporación en la Academia Año 2017 + PAI77170068 (P.M.). Rotem Oshman is supported by ISF i-core Center for Excellence, No. 4/11.
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Fraigniaud, P., Montealegre, P., Oshman, R., Rapaport, I., Todinca, I. (2019). On Distributed Merlin-Arthur Decision Protocols. In: Censor-Hillel, K., Flammini, M. (eds) Structural Information and Communication Complexity. SIROCCO 2019. Lecture Notes in Computer Science(), vol 11639. Springer, Cham. https://doi.org/10.1007/978-3-030-24922-9_16
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