On January 19, 2023, Pedro Montealegre from Universidad Adolfo Ibáñez gave an online talk at the Virtual Discrete Math Colloquium on a proof labeling scheme certifying that a graph has tree-width at most k and satisfies a monadic second-order property with a certificate of small size at each vertex. The title of his talk was “A Meta-Theorem for Distributed Certification“.

## Pedro Montealegre, A Meta-Theorem for Distributed Certification

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. In this talk, I will present a “meta-theorem”, applying to many boolean predicates at once. Specifically, I will show 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 $O(log^2 n)$ bits in n-node graphs of bounded treewidth, with a verification protocol involving a single round of communication between neighbors.