Undecidability of the Emptiness Problem for Weak Models of Distributed Computing

2025-04-09

AI Summary

This paper investigates the emptiness problem for weak asynchronous distributed automata—i.e., networked finite-state machines: given such an automaton, does there exist any input graph it accepts? Within the Esparza–Reiter framework, this problem encompasses seven equivalence classes of weakly distributed models. Using computability-theoretic analysis and distributed automata theory, the authors establish, for the first time, that emptiness is undecidable for six of these seven classes, exposing a fundamental limitation in the verification of weak distributed systems; for the seventh class, emptiness is trivially decidable. This result fully characterizes the decidability boundary of emptiness checking within the Esparza–Reiter framework, providing a foundational theoretical benchmark for formal verification of distributed systems.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Multiagent Systems: Distributed Problem Solving

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Esparza and Reiter have recently conducted a systematic comparative study of weak asynchronous models of distributed computing, in which a network of identical finite-state machines acts cooperatively to decide properties of the network's graph. They introduced a distributed automata framework encompassing many different models, and proved that w.r.t. their expressive power (the graph properties they can decide) distributed automata collapse into seven equivalence classes. In this contribution, we turn our attention to the formal verification problem: Given a distributed automaton, does it decide a given graph property? We consider a fundamental instance of this question - the emptiness problem: Given a distributed automaton, does it accept any graph at all? Our main result is negative: the emptiness problem is undecidable for six of the seven equivalence classes, and trivially decidable for the remaining class.

Problem

• Study undecidability of emptiness in weak distributed models
• Analyze distributed automata's graph property verification problem
• Determine emptiness problem decidability across seven model classes

Innovation

• Distributed automata framework for weak models
• Emptiness problem undecidable for six classes
• Comparative study of asynchronous distributed computing