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The standard population protocol model assumes that when two agents interact, each observes the entire state of the other agent. We initiate the study of the message complexity for population protocols, where the state of an agent is divided into an externally-visible message and an internal component, where only the message can be observed by the other agent in an interaction. We consider the case of O(1) message complexity. When time is unrestricted, we obtain an exact characterization of the stably computable predicates based on the number of internal states s(n): If s(n) = o(n) then the protocol computes a semilinear predicate (unlike the original model, which can compute non-semilinear predicates with s(n) = O(log n)), and otherwise it computes a predicate decidable by a nondeterministic O(n log s(n))-space-bounded Turing machine. We then consider time complexity, introducing novel O(polylog(n)) expected time protocols for junta/leader election and general purpose broadcast correct with high probability, and approximate and exact population size counting correct with probability 1. Finally, we show that the main constraint on the power of bounded-message-size protocols is the size of the internal states: with unbounded internal states, any computable function can be computed with probability 1 in the limit by a protocol that uses only one-bit messages. 

Third-state dynamics (Angluin et al. 2008; Perron et al. 2009) is a well-known process for quickly and robustly computing approximate majority through interactions between randomly-chosen pairs of agents. In this paper, we consider this process in a new model with persistent-state catalytic inputs, as well as in the presence of transient leak faults. Based on models considered in recent protocols for populations with persistent-state agents (Dudek et al. 2017; Alistarh et al. 2017; Alistarh et al. 2020), we formalize a Catalytic Input (CI) model comprising n input agents and m worker agents. For m = Θ(n), we show that computing the parity of the input population with high probability requires at least Ω(n2) total interactions, demonstrating a strong separation between the CI and standard population protocol models. On the other hand, we show that the third-state dynamics can be naturally adapted to this new model to solve approximate majority in O(n log n) total steps with high probability when the input margin is Ω(√(n log n)), which preserves the time and space efficiency of the corresponding protocol in the original model. We then show the robustness of third-state dynamics protocols to the transient leak faults considered by (Alistarh et al. 2017; Alistarh et al 2020). In both the original and CI models, these protocols successfully compute approximate majority with high probability in the presence of leaks occurring at each time step with probability β ≤ O(√(n log n}/n). The resilience of these dynamics to adversarial leaks exhibits a subtle connection to previous results involving Byzantine agents.