🤖 AI Summary
This work systematically characterizes the computational complexity of verification and synthesis for threshold automata—formal models of fault-tolerant distributed protocols—with respect to fundamental properties including coverability, reachability, safety, and liveness.
Method: We introduce a novel characterization of the reachability relation via existential Presburger formulas—the first such formulation—and establish tight complexity bounds: coverability is NP-complete, while bounded synthesis is Σ₂^p-complete. Leveraging these results, we design new symbolic verification and synthesis algorithms and implement a prototype tool.
Results: Experiments demonstrate that multiple verification tasks are solvable within PSPACE, significantly outperforming brute-force enumeration; the NP-completeness results provide theoretically tight complexity bounds for optimization of tool implementations. This work establishes foundational complexity benchmarks and delivers practical algorithmic support for formal analysis of distributed protocols.
📝 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.
