## Abstract

The purpose of this paper is a study of computation that can be done locally in a distributed network. By locally we mean within time (or distance) independent of the size of the network. We consider Locally Checkable Labeling (LCL) problems, where the legality of a labeling can be checked locally (e.g., coloring). Our results include the following: .There are non-Trivial LCL problems that have local algorithms. There is a variant of the dining philosophers problem which can be solved locally. Randomization cannot make an LCL problem local; i.e., if a problem has a local randomized algorithm then it has a local deterministic algorithm. It is undecidable, in general, whether a given LCL has a local algorithm. However, it is decidable whether a given LCL has an algorithm that operates in a given time t. Any LCL problem that has a local algorithm has one which is order-invariant (the algorithm depends only on the order of the processor id's).