Forming Sequences of Patterns With Luminous Robots

The extensive studies on computing by a team of identical mobile robots operating in the plane in <italic>Look</italic>-<italic>Compute</italic>-<italic>Move</italic> cycles have been carried out mainly in the traditional ${\mathcal{ OBLOT}}$ model, where the robots are silent (have no communication capabilities) and oblivious (in a cycle, they have no memory previous cycles). To partially overcome the limits of obliviousness and silence while maintaining some of their advantages, the stronger model of <italic>luminous robots</italic>, ${\mathcal{ LUMI}}$ , has been introduced where the robots, otherwise oblivious and silent, carry a visible light that can take a number of different colors; a color can be seen by observing robots, and persists from a cycle to the next. In the study of the computational impact of lights, an immediate concern has been to understand and determine the additional computational strength of ${\mathcal{ LUMI}}$ over ${\mathcal{ OBLOT}}$ . Within this line of investigation, we examine the problem of forming a sequence of geometric patterns, PatternSequenceFormation. A complete characterization of the sequences of patterns formable from a given starting configuration has been determined in the ${\mathcal{ OBLOT}}$ model. In this paper, we study the formation of sequences of patterns in the ${\mathcal{ LUMI}}$ model and provide a complete characterization. The characterization is constructive: our universal protocol forms all formable sequences, and it does so asynchronously and without rigidity. This characterization explicitly and clearly identifies the computational strength of ${\mathcal{ LUMI}}$ over ${\mathcal{ OBLOT}}$ with respect to the Pattern Sequence Formation problem.