Third-state dynamics [AAE08, PVV09] 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 persistentstate 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 [DK18, ADK + 17, ATU20] 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(n 2 ) 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 [ADK + 17, ATU20]. 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.

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