A self-stabilizing transformer for population protocols with covering

Abstract

a b s t r a c tDeveloping self-stabilizing solutions is considered to be more challenging and complicated than developing classical solutions, where a proper initialization of the variables can be assumed. Hence, to ease the task of the developers, some automatic techniques have been proposed to design self-stabilizing algorithms. In this paper, we propose an automatic transformer for algorithms in an extended population protocol model. Population protocols is a model that was introduced recently for networks with a large number of resource-limited mobile agents. We use a variant of this model. First, we assume agents having characteristics (e.g., moving speed, communication radius) affecting their intercommunication ''speed'', which is reflected by their cover times. Second, we assume the existence of a special agent with an unbounded memory, the base station. The automatic transformer takes as an input an algorithm solving a static problem (and meeting some additional rather natural requirements) and outputs a self-stabilizing algorithm for the same problem. The transformer is built using a re-execution approach (the technique consisting of executing an algorithm repeatedly in order to obtain its self-stabilizing version). We show that in the model we use, a transformer based on such an approach is impossible without the assumption of an unbounded memory agent.

Self-stabilizing Mutual Exclusion and Group Mutual Exclusion for Population Protocols with Covering

“…The self-stabilizing algorithms that estimate cv min and cv max are presented in [9]. They are executed at BS and stabilize in O([cv min ] BS ) events.…”

“…In case (a), if BS does not hold a token, then during the next event of BS with an agent in state out holding a token, token BS becomes true (line 16) and req ctr is decremented (line 20). From this point, BS can dispatch the token to different registered agents until req ctr ≤ 0 (lines [9][10][11][12][13][14][15][16][17][18][19][20]. Then, the end of the phase condition, in line 32, triggers.…”

“…A scheduler schedules the next event according to the cover times of agents. Even though this type of the local fairness condition seems relatively strong, many problems stay impossible to solve in such a model (see, e.g., [6,9]). In this work, we prove a similar impossibility result stating that in the model with only bounded state agents, mutual exclusion has no solution (Sec.…”

“…This result can be easily extended to the group mutual exclusion problem as well. For circumventing the impossibility result, as in [6,9,10], we introduce a special agent, the base station (BS). We assume that BS has a memory size proportional to the number of the other system agents.…”

“…The variables of the algorithm are regularly reset to predetermined values and an instance of a protocol is repeatedly executed. Note that the automatic self-stabilizing transformer of [9], which uses the same technique, cannot be useful here, because it applies only to static problems.…”