Coefficient Synthesis for Threshold Automata

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.