Homonym Population Protocols, or Providing a Small Space of Computation Using a Few Identifiers

TL;DR

This paper explores how varying the number of identifiers in population protocols affects computational power, establishing a hierarchy that bridges models from no identifiers to unique identifiers, and simulating Turing machines with limited space.

Contribution

It introduces a hierarchy of population protocols with homonyms, showing how the number of shared identifiers influences computational capabilities, and fills gaps in previous models.

Findings

Any Turing machine with O(1) space can be simulated with a logarithmic number of identifiers.

The hierarchy extends from no identifiers to unique identifiers, covering and surpassing known models.

It resolves the gap between population protocols and community protocols regarding space limitations.

Abstract

Population protocols have been introduced by Angluin et al. as a model in which n passively mobile anonymous finite-state agents stably compute a predicate on the multiset of their inputs via interactions by pairs. The model has been extended by Guerraoui and Ruppert to yield the community protocol models where agents have unique identifiers but may only store a finite number of the identifiers they already heard about. The population protocol models can only compute semi-linear predicates, whereas in the community protocol model the whole community of agents provides collectively the power of a Turing machine with a O(n log n) space. We consider variations on the above models and we obtain a whole landscape that covers and extends already known results: 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 (i.e. a single one for all, i.e. the population protocol model) to the case of unique identifiers (i.e. community protocol model). We obtain in particular that any Turing Machine on space O(logO(1) n) can be simulated with at least O(logO(1) n) identifiers, a result filling a gap left open in all previous studies. Our results also extend and revisit in particular the hierarchy provided by Chatzigiannakis et al. on population protocols carrying Turing Machines on limited space, solving the problem of the gap left by this work between per-agent space o(log log n) (proved to be equivalent to population protocols) and O(log n) (proved to be equivalent to Turing machines).