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In this work, we explore emergent behaviors by swarms of anonymous,homogeneous, non-communicating, reactive robots that do not know their globalposition and have limited relative sensing. We introduce a novel method thatenables such severely limited robots to autonomously arrange in a desiredpattern and maintain it. The method includes an automatic proof procedure tocheck whether a given pattern will be achieved by the swarm from any initialconfiguration. An attractive feature of this proof procedure is that it islocal in nature, avoiding as much as possible the computational explosion thatcan be expected with increasing robots, states, and action possibilities. Ourapproach is based on extracting the local states that constitute a global goal(in this case, a pattern). We then formally show that these local states canonly coexist when the global desired pattern is achieved and that, until thisoccurs, there is always a sequence of actions that will lead from the currentpattern to the desired pattern. Furthermore, we show that the agents will neverperform actions that could a) lead to intra-swarm collisions or b) cause theswarm to separate. After an analysis of the performance of pattern formation inthe discrete domain, we also test the system in continuous time and spacesimulations and reproduce the results using asynchronous agents operating inunbounded space. The agents successfully form the desired patterns whileavoiding collisions and separation.

Provable Emergent Pattern Formation by a Swarm of Anonymous,  Homogeneous, Non-Communicating, Reactive Robots with Limited Relative Sensing  and no Global Knowledge or Positioning