A Meta-Theorem for Distributed Certification

Abstract

Distributed certification, whether it be proof-labeling schemes, locally checkable proofs, etc., deals with the issue of certifying the legality of a distributed system with respect to a given boolean predicate. A certificate is assigned to each process in the system by a non-trustable oracle, and the processes are in charge of verifying these certificates, so that two properties are satisfied: completeness, i.e., for every legal instance, there is a certificate assignment leading all processes to accept, and soundness, i.e., for every illegal instance, and for every certificate assignment, at least one process rejects. The verification of the certificates must be fast, and the certificates themselves must be small. A large quantity of results have been produced in this framework, each aiming at designing a distributed certification mechanism for specific boolean predicates. This paper presents a “meta-theorem”, applying to many boolean predicates at once. Specifically, we prove that, for every boolean predicate on graphs definable in the monadic second-order (MSO) logic of graphs, there exists a distributed certification mechanism using certificates on $O(\log ^2n)$ bits in n-node graphs of bounded treewidth, with a verification protocol involving a single round of communication between neighbors.

This work was partially done during the visit of the second and third authors to IRIF at Université de Paris, and LIFO at Université d’Orléans, partially supported by ANR project DUCAT and FONDECYT 1220142.

P. Fraigniaud—Additional support for ANR projects QuData and DUCAT.

P. Montealegre—This work was supported by Centro de Modelamiento Matemático (CMM), ACE210010 and FB210005, BASAL funds for centers of excellence from ANID-Chile, FONDECYT 11190482, and PAI 77170068.