  Threshold automata are a formalism for modeling and analyzing fault-tolerant
distributed algorithms, recently introduced by Konnov, Veith, and Widder,
describing protocols executed by a fixed but arbitrary number of processes. We
conduct the first systematic study of the complexity of verification and
synthesis problems for threshold automata. We prove that the coverability,
reachability, safety, and liveness problems are NP-complete, and that the
bounded synthesis problem is $\Sigma_p^2$ complete. A key to our results is a
novel characterization of the reachability relation of a threshold automaton as
an existential Presburger formula. The characterization also leads to novel
verification and synthesis algorithms. We report on an implementation, and
provide experimental results.
