9666 10.1007/s11047-017-9666-6 15 Particle computation: complexity, algorithms, and logic 181 201 2017 12 2 2017 12 8 National Science Foundation IIS-1553063 IIS-1619278 Springer Science+Business Media B.V., part of Springer Nature 2017 11047 18 18 1 1 Aaron T. Becker atbecker@uh.edu Erik D. Demaine edemaine@mit.edu Sándor P. Fekete s.fekete@tu-bs.de Jarrett Lonsford JLLonsford@uh.edu Rose Morris-Wright rmorriswright@brandeis.edu 0000 0004 1569 9707 grid.266436.3 Department of Electrical and Computer Engineering University of Houston Houston TX USA 0000 0001 2341 2786 grid.116068.8 Computer Science and Artificial Intelligence Laboratory MIT Cambridge MA USA 0000 0001 1090 0254 grid.6738.a Department of Computer Science TU Braunschweig Brunswick Germany 0000 0004 1936 9473 grid.253264.4 Department of Mathematics Brandeis University Waltham MA USA Abstract We investigate algorithmic control of a large swarm of mobile particles (such as robots, sensors, or building material) that move in a 2D workspace using a global input signal (such as gravity or a magnetic field). Upon activation of the field, each particle moves maximally in the same direction until forward progress is blocked by a stationary obstacle or another stationary particle. In an open workspace, this system model is of limited use because it has only two controllable degrees of freedom—all particles receive the same inputs and move uniformly. We show that adding a maze of obstacles to the environment can make the system drastically more complex but also more useful. We provide a wide range of results for a wide range of questions. These can be subdivided into external algorithmic problems, in which particle configurations serve as input for computations that are performed elsewhere, and internal logic problems, in which the particle configurations themselves are used for carrying out computations. For external algorithms, we give both negative and positive results. If we are given a set of stationary obstacles, we prove that it is NP-hard to decide whether a given initial configuration of unit-sized particles can be transformed into a desired target configuration. Moreover, we show that finding a control sequence of minimum length is PSPACE-complete. We also work on the inverse problem, providing constructive algorithms to design workspaces that efficiently implement arbitrary permutations between different configurations. For internal logic, we investigate how arbitrary computations can be implemented. We demonstrate how to encode dual-rail logic to build a universal logic gate that concurrently evaluates and, nand, nor, and or operations. Using many of these gates and appropriate interconnects, we can evaluate any logical expression. However, we establish that simulating the full range of complex interactions present in arbitrary digital circuits encounters a fundamental difficulty: a fan-out gate cannot be generated. We resolve this missing component with the help of 2 × 1 particles, which can create fan-out gates that produce multiple copies of the inputs. Using these gates we provide rules for replicating arbitrary digital circuits. Keywords Programmable matter Robot swarms Nano-particles Uniform inputs Parallel motion planning Complexity Array permutations NP-completeness PSPACE-completeness Efficient algorithms Logic gates Universal computation This paper provides full details for and combines results of a number of different extended abstracts that have appeared in the International Symposium on Algorithms and Experiments for Sensor Systems, Wireless Networks and Distributed Robotics (ALGOSENSORS 2013) (Becker et al. 2014a), IEEE International Conference on Robotics and Automation (ICRA 2014) (Becker et al. 2014b) and ICRA 2015 (Shad et al. 2015). See video from the 31st International Symposium on Computational Geometry (SoCG’15) (Becker et al. 2015) for illustrations and animations.