Stabilizing token schemes for population protocols

In this paper we address the stabilizing token circulation and leader election problems in population protocols model augmented with oracles. Population protocols is a recent model of computation that captures the interactions of biological systems. In this model emergent global behavior is observed while anonymous finite-state agents(nodes) perform local peer interactions. Uniform self-stabilizing leader election or token circulation are impossible in such systems without additional assumptions. Therefore, the classical model has been augmented with the eventual leader detector, $\Omega?$, that eventually detects the presence or the absence of a leader. In this work we propose some impossibility results related to self-stabilizing implementation of leader election and token circulation in this model. Then we propose deterministic and probabilistic self-stabilizing solutions for token circulation and leader election for various topologies (chains, trees and arbitrary networks). Additionally, we prove the necessity of the eventual leader or token detector even in environments helped by randomization. All the proposed algorithms are memory optimal -- they need only one memory bit per agent.