This dissertation is about local certification, a central topic in distributed decision, a subfield of distributed computing. The distributed decision mechanism consists, for the nodes of a network, in deciding in a distributed manner whether the network is in a proper configuration or not, with respect to some fixed predicate. This decision is said to be local because the nodes of the network can communicate only with their neighbours. After communication, every node outputs a decision, stating whether the network is locally correct, that is, correct given the partial information gathered so far by this node. The network is declared to be globally correct, if and only if, it is declared to be locally correct by every node.

Most predicates cannot be verified by this type of computation, due to the locality constraint. Local certification is a mechanism that enables to circumvent this difficulty, and to check any property. It consists in providing the nodes of the network with labels, called certificates, that can be verified locally by a distributed algorithm. A local certification scheme is correct if only the networks that satisfy the predicate can be certified. In addition to its theoretical appeal, as a form of distributed non-determinism, the concept of local certification is especially relevant in the study of fault-tolerant distributed algorithms, where a key step consists in checking the status of the network, based on information stored at the nodes.

This dissertation deals with four aspects of local certification: error-sensitivity, uniformity, redundancy, and interactivity. The study of these four topics is motivated by the same essential question: How to reduce the resources needed for certification, and/or ensure a better fault-tolerance? In order to tackle this question we have to understand fundamental properties of certification. In particular, in this dissertation we answer questions such as: How redundant the certificates need to be for a proper certification? Are the classic certification protocols robust to a strengthening of the acceptance condition? and, How does introducing interactivity in the process changes the complexity of certification?

Keywords: Distributed network computing, distributed decision, local certification, proof-labelling scheme, fault-tolerance.