  In this work, we explore emergent behaviors by swarms of anonymous,
homogeneous, non-communicating, reactive robots that do not know their global
position and have limited relative sensing. We introduce a novel method that
enables such severely limited robots to autonomously arrange in a desired
pattern and maintain it. The method includes an automatic proof procedure to
check whether a given pattern will be achieved by the swarm from any initial
configuration. An attractive feature of this proof procedure is that it is
local in nature, avoiding as much as possible the computational explosion that
can be expected with increasing robots, states, and action possibilities. Our
approach 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 can
only coexist when the global desired pattern is achieved and that, until this
occurs, there is always a sequence of actions that will lead from the current
pattern to the desired pattern. Furthermore, we show that the agents will never
perform actions that could a) lead to intra-swarm collisions or b) cause the
swarm to separate. After an analysis of the performance of pattern formation in
the discrete domain, we also test the system in continuous time and space
simulations and reproduce the results using asynchronous agents operating in
unbounded space. The agents successfully form the desired patterns while
avoiding collisions and separation.
