  Threshold automata are a formalism for modeling fault-tolerant distributed
algorithms. The main feature of threshold automata is the notion of a threshold
guard, which allows us to compare the number of received messages with the
total number of different types of processes. In this paper, we consider the
coefficient synthesis problem for threshold automata, in which we are given a
sketch of a threshold automaton (with the constants in the threshold guards
left unspecified) and a specification and we want to synthesize a set of
constants which when plugged into the sketch, gives a threshold automaton
satisfying the specification. Our main result is that this problem is
undecidable, even when the specification is a coverability specification and
the underlying sketch is acyclic.
