Expressive Power of Oblivious Consensus Protocols

Population protocols are a formal model of computation by identical, anonymous mobile agents interacting in pairs. It has been shown that their computational power is rather limited: They can only compute the predicates expressible in Presburger arithmetic. Population protocols are oblivious, in the sense that their behavior only depends on the number of agents in each state of the current configuration, and nothing else. Obliviousness has advantages for applications where agents want to reveal as little as possible about their trajectories in a computation. We investigate the computational power of oblivious protocols. We first show that, under a weak assumption, oblivious protocols can only compute number predicates $\varphi : \mathbb{N}^m \rightarrow \{0, 1\}$ in NSPACE(n) (with the input written, as usual, in binary), while all predicates computed by population protocols are in DSPACE(log n), thus proving an exponential gap. Then we introduce broadcast consensus protocols, in which agents can also broadcast signals to all other agents. We prove that they compute all predicates in NSPACE(n), reaching the theoretical limit for oblivious protocols. Finally, we conduct the first systematic comparison of different models introduced in the literature (population protocols, broadcast protocols, community protocols, and mediated protocols) with respect to their computational power and their privacy guarantees.