Ability to Count Messages Is Worth Θ(Δ) Rounds in Distributed Computing.

Hella et al. (PODC 2012, Distributed Computing 2015) identified seven different message-passing models of distributed computing---one of which is the port-numbering model---and provided a complete classification of their computational power relative to each other. However, their method for simulating the ability to count incoming messages causes an additive overhead of 2Δ -- 2 communication rounds, and it was not clear if this is actually optimal. In this paper we give a positive answer, by using bisimulation as our main tool: there is a matching linear-in-Δ lower bound. This closes the final gap in our understanding of the models, with respect to the number of communication rounds. By a previously identified connection to modal logic, our result has implications to the relationship between multimodal logic and graded multimodal logic.