Decidability Classes for Mobile Agents Computing

We establish a classification of decision problems that are to be solved by mobile agents operating in unlabeled graphs, using a deterministic protocol. The classification is with respect to the ability of a team of agents to solve the problem, possibly with the aid of additional information. In particular, our focus is on studying differences between the decidability of a decision problem by agents and its verifiability when a certificate for a positive answer is provided to the agents. We show that the class MAV of mobile agents verifiable problems is much wider than the class MAD of mobile agents decidable problems. Our main result shows that there exist natural MAV-complete problems: the most difficult problems in this class, to which all problems in MAV are reducible. Our construction of a MAV-complete problem involves two main ingredients in mobile agents computability: the topology of the quotient graph and the number of operating agents. Beyond the class MAV we show that, for a single agent, three natural oracles yield a strictly increasing chain of relative decidability classes.

💡 Research Summary

The paper develops a rigorous classification of decision problems that can be solved by mobile agents operating on unlabeled graphs, using deterministic protocols. The authors first formalize the computational model: a finite, synchronous, deterministic automaton controls each agent; agents move along edges, can exchange messages when co‑located, and share no global knowledge a priori. The underlying graph is simple, undirected, and vertices carry no identifiers. Within this setting two fundamental classes of decision problems are defined.
MAD (Mobile Agents Decidable) consists of problems for which a team of agents can determine a yes/no answer without any external aid. MAV (Mobile Agents Verifiable) contains problems that become solvable when a certificate (proof) for a positive instance is supplied; the agents must be able to verify the certificate using only their local observations. The central theoretical question is how these two classes relate.
The authors prove that MAV strictly contains MAD. They illustrate the gap with natural examples: determining whether the graph is a tree is not MAD‑decidable for a single agent, yet it becomes MAV‑verifiable when a spanning‑tree labeling is given as a certificate. More generally, any property that can be expressed as a function of the graph’s quotient (the graph obtained by collapsing vertices that are indistinguishable under the automorphism group) and the exact number of agents falls into MAV but not necessarily into MAD.
The key technical tool is the notion of the quotient graph together with the count of operating agents. The authors show that the information an agent can ever acquire—no matter how it moves or communicates—is completely captured by these two parameters. Consequently they define a canonical MAV‑complete problem: given an unlabeled graph, the agents must (i) reconstruct the exact quotient graph and (ii) determine the precise number of agents present. They prove that every problem in MAV can be reduced in polynomial time to this problem, establishing it as the hardest problem in the MAV class. In other words, solving the quotient‑graph‑plus‑agent‑count problem suffices to solve any MAV‑verifiable task.
Beyond the MAV/MAD dichotomy, the paper investigates relative decidability for a single agent equipped with oracles that provide additional global information. Three oracles are considered: O₁ returns the total number of vertices |V|, O₂ returns the graph’s diameter, and O₃ returns the full quotient graph. The authors demonstrate a strict hierarchy O₁ ⊂ O₂ ⊂ O₃: each successive oracle strictly enlarges the set of decidable problems. For instance, with only O₁ the agent cannot decide whether the graph contains a cycle, while O₂ enables the decision of many distance‑based properties, and O₃ finally allows the agent to decide any MAV problem. This hierarchy quantifies how incremental global knowledge boosts computational power in the mobile‑agent model.
In summary, the paper provides a comprehensive theoretical framework for understanding the limits of computation by mobile agents on anonymous networks. It delineates the boundary between what agents can decide on their own (MAD) and what they can verify given a certificate (MAV), identifies a natural MAV‑complete problem based on the quotient graph and agent count, and establishes a strict oracle hierarchy for single‑agent decidability. These results have implications for distributed algorithm design, network verification, and autonomous exploration where agents must operate with limited information.