Particle computation: complexity, algorithms, and logic

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 unt...

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Bibliographic Details
Published in:Natural computing Vol. 18; no. 1; pp. 181 - 201
Main Authors: Becker, Aaron T., Demaine, Erik D., Fekete, Sándor P., Lonsford, Jarrett, Morris-Wright, Rose
Format: Journal Article
Language:English
Published: Dordrecht Springer Netherlands 01.03.2019
Springer Nature B.V
Subjects:
Algorithms
Artificial Intelligence
Complex Systems
Complexity
Computer Science
Computer simulation
Configurations
Digital electronics
Evolutionary Biology
Gates
Gravitation
Inverse problems
Logic
Logic circuits
Permutations
Processor Architectures
Replicating
Stability
Theory of Computation
Robot swarms
Uniform inputs
Nano-particles
Programmable matter
PSPACE-completeness
Parallel motion planning
NP-completeness
Logic gates
Array permutations
Universal computation
Efficient algorithms
Complexity
ISSN:1567-7818, 1572-9796
Online Access:Get full text
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Summary: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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ISSN:1567-7818
1572-9796
DOI:10.1007/s11047-017-9666-6