  A central theme in distributed network algorithms concerns understanding and
coping with the issue of locality. Inspired by sequential complexity theory, we
focus on a complexity theory for distributed decision problems. In the context
of locality, solving a decision problem requires the processors to
independently inspect their local neighborhoods and then collectively decide
whether a given global input instance belongs to some specified language. This
paper introduces several classes of distributed decision problems, proves
separation among them and presents some complete problems. More specifically,
we consider the standard LOCAL model of computation and define LD (for local
decision) as the class of decision problems that can be solved in constant
number of communication rounds. We first study the intriguing question of
whether randomization helps in local distributed computing, and to what extent.
Specifically, we define the corresponding randomized class BPLD, and ask
whether LD=BPLD. We provide a partial answer to this question by showing that
in many cases, randomization does not help for deciding hereditary languages.
In addition, we define the notion of local many-one reductions, and introduce
the (nondeterministic) class NLD of decision problems for which there exists a
certificate that can be verified in constant number of communication rounds. We
prove that there exists an NLD-complete problem. We also show that there exist
problems not in NLD. On the other hand, we prove that the class NLD#n, which is
NLD assuming that each processor can access an oracle that provides the number
of nodes in the network, contains all (decidable) languages. For this class we
provide a natural complete problem as well.
