A hierarchy of local decision

We extend the notion of distributed decision in the framework of distributed network computing, inspired by both the polynomial hierarchy for Turing machines and recent results on so-called distributed graph automata. We show that, by using distributed decision mechanisms based on the interaction between a prover and a disprover, the size of the certificates distributed to the nodes for certifying a given network property can be drastically reduced. For instance, we prove that minimum spanning tree (MST) can be certified with O(log⁡n)-bit certificates in n-node graphs, with just one interaction between the prover and the disprover, while it is known that certifying MST requires Ω(log²⁡n)-bit certificates if only the prover can act. The improvement can even be exponential for some simple graph properties. For instance, it is known that certifying the existence of a nontrivial automorphism requires Ω(n²) bits if only the prover can act. We show that there is a protocol with two interactions between the prover and the disprover that certifies nontrivial automorphism with O(log⁡n)-bit certificates. These results are achieved by defining and analyzing a local hierarchy of decision which generalizes the classical notions of proof-labeling schemes and locally checkable proofs.