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Particle computation: complexity, algorithms, and logic

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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.

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Acknowledgements

We thank an anonymous reviewer for carefully going through our work and making numerous constructive suggestions that helped to improve the presentation of our paper. We thank Hamed Mohtasham Shad for building and testing the first experimental tilt tables that brought these algorithms to life. We acknowledge the helpful discussion and motivating experimental efforts with T. pyriformis cells by Yan Ou and Agung Julius at RPI and Paul Kim and MinJun Kim at Drexel University. Preliminary versions of Sects. 4 and 5 are main topics of our paper Becker et al. (2014a) with an extra result proving the system to give rise to pspace-completeness in Sect. 5.3 from paper Becker et al. (2014b). The particle logic in Sects. 6 and 7 was introduced in Becker et al. (2014b) and completed in paper Shad et al. (2015). This work has been partially supported by the National Science Foundation (Grant Nos. [IIS-1553063] and [IIS-1619278]).

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Authors and Affiliations

  1. Department of Electrical and Computer Engineering, University of Houston, Houston, TX, USA

    Aaron T. Becker & Jarrett Lonsford

  2. Computer Science and Artificial Intelligence Laboratory, MIT, Cambridge, MA, USA

    Erik D. Demaine

  3. Department of Computer Science, TU Braunschweig, Brunswick, Germany

    Sándor P. Fekete

  4. Department of Mathematics, Brandeis University, Waltham, MA, USA

    Rose Morris-Wright

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  1. Aaron T. Becker
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  2. Erik D. Demaine
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  3. Sándor P. Fekete
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Correspondence to Sándor P. Fekete.

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Becker, A.T., Demaine, E.D., Fekete, S.P. et al. Particle computation: complexity, algorithms, and logic. Nat Comput 18, 181–201 (2019). https://doi.org/10.1007/s11047-017-9666-6

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