Complexity of Verification and Synthesis of Threshold Automata

TL;DR

This paper investigates the computational complexity of verification and synthesis tasks for threshold automata, revealing NP-completeness for several problems and introducing new algorithms based on Presburger formulas.

Contribution

It provides the first systematic complexity analysis for threshold automata verification and synthesis, including novel algorithms and an implementation.

Findings

Coverability, reachability, safety, and liveness are NP-complete.

Bounded synthesis is $\

$ ext{Sigma}_p^2$-complete.

Abstract

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.