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 investigates decision problems that mobile agents must solve when they operate in unlabeled, undirected connected graphs. The authors introduce two complexity classes tailored to this setting. The first, MAD (Mobile Agents Decidable), contains problems for which there exists a deterministic protocol that guarantees all agents eventually output “yes” when the instance belongs to the problem, and at least one agent outputs “no” otherwise, without any additional information. The second, MAV (Mobile Agents Verifiable), relaxes this requirement by allowing a certificate (or proof) to be supplied when the correct answer is “yes”. In MAV, a verification protocol must accept some certificate for every yes‑instance and reject all certificates for every no‑instance. This mirrors the classic NP versus P distinction.

The model assumes synchronous rounds, ports numbered locally at each node, and agents that have unique identifiers and unlimited memory but no prior knowledge of the graph beyond their own identifier and input. An initial configuration is a quadruple (G, S, Id, w) where G is the graph, S the multiset of starting nodes, Id the injective mapping of agents to positive integers, and w the input strings. Decision problems are sets of such configurations closed under graph automorphisms preserving ports, identifiers, and inputs.

The authors first show that MAD is a strict subset of MAV. Simple global properties such as “the graph contains an odd‑degree node” belong to MAD, whereas natural problems like determining whether the graph is a path (#nodes) or whether it is a tree are not in MAD because a single agent cannot distinguish a long path from a cycle without extra information. However, both become MAV‑solvable when a certificate encoding the exact size of the graph (or a leaf‑to‑root path) is supplied. Moreover, the family of problems degree k (does the graph contain a node of degree k?) forms an infinite antichain under reducibility, illustrating that MAV \ MAD is large.

The central contribution is the identification of a natural MAV‑complete problem. This problem is the conjunction of two orthogonal sub‑problems: (1) teamsize – given an integer k, decide whether the number of agents exceeds k; and (2) quotient – given a graph H, decide whether the quotient graph of the current configuration (the graph obtained by identifying nodes that have identical infinite “views”) differs from H. The authors prove that any MAV problem can be reduced in polynomial time to this combined problem, establishing its completeness. The reduction proceeds by having agents explore their local neighborhoods to reconstruct the quotient structure and by counting agents through coordinated token‑passing, both achievable without certificates.

Beyond MAV, the paper examines relative decidability for a single agent equipped with oracles. Three natural oracles are considered: (i) an oracle that returns the exact number of nodes in the graph, (ii) an oracle that returns the maximum degree, and (iii) an oracle that decides whether the graph admits a consistent port labeling. By constructing decision problems that become solvable only when a particular oracle is available, the authors demonstrate a strict hierarchy: problems solvable with the first oracle form a proper subset of those solvable with the second, which in turn are a proper subset of those solvable with the third. This hierarchy underscores how incremental informational power expands the set of decidable tasks.

In summary, the paper provides a rigorous classification of decision problems for mobile agents in unlabeled graphs, delineates the gap between pure decidability (MAD) and verifiability with certificates (MAV), introduces a natural MAV‑complete problem based on team size and quotient graph structure, and establishes a strict oracle hierarchy for single‑agent computation. These results deepen our understanding of the computational limits of distributed mobile agents and open avenues for further exploration of certificate‑based and oracle‑enhanced protocols in this domain.