Homonym Population Protocols

The population protocol model was introduced by Angluin et al. as a model of passively mobile anonymous finite-state agents. This model computes a predicate on the multiset of their inputs via interactions by pairs. The original population protocol model has been proved to compute only semilinear predicates and has been extended in various ways. In the community protocol model by Guerraoui and Ruppert, the n agents have unique identifiers but may only store a finite number of the identifiers they already heard about. The community protocol model is known to provide the power of a non-deterministic Turing machine with an O ( n log n ) space. We consider variants of the two above-mentioned models and we obtain a whole landscape that covers and extends already known results. Namely, by considering the case of homonyms, that is to say the case when several agents may share the same identifier, we provide a hierarchy that goes from the case of no identifier (population protocol model) to the case of unique identifiers (community protocol model). In particular, we obtain that any Turing Machine on space O (log O (1) n ) can be simulated with log r n identifiers, for any r > 0. Our results also extend and revisit the hierarchy provided by Chatzigiannakis et al. on population protocols carrying Turing Machines on limited space, reducing the gap left by this work between per-agent space o (loglog n ) (proved to be equivalent to population protocols) and Ω(log n ) (proved to be equivalent to Turing machines): We prove that per-agent space Θ(loglog n ) corresponds to symmetric predicates computable in polylogarithmic non-deterministic space.