On the Convergence of a Population Protocol When Population Goes to Infinity

Population protocols have been introduced as a model of sensor networks consisting of very limited mobile agents with no control over their own movement. A population protocol corresponds to a collection of anonymous agents, modeled by finite automata, that interact with one another to carry out computations, by updating their states, using some rules. Their computational power has been investigated under several hypotheses but always when restricted to finite size populations. In particular, predicates stably computable in the original model have been characterized as those definable in Presburger arithmetic. In this paper, we study mathematically a particular population protocol that we show to compute in some natural sense some algebraic irrational number, whenever the population goes to infinity. Hence we show that these protocols seem to have a rather different computational power when considered as computing functions, and when a huge population hypothesis is considered.