Particle computation: complexity, algorithms, and logic

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.

Full Tilt: Universal Constructors for General Shapes with Uniform External Forces

“…This section is a continuation of the work done by [5][6][7]. In their papers, they discuss the occupancy problem in a tilt-based system which asks whether a given location can be occupied by any tile on the board.…”

“…Our proofs rely on a reduction from a 2-tunnel gadget network game with different gadget types that was recently proven to be PSPACE-complete by Demaine, Grosof, Lynch, and Rudoy [11]. Previous work on occupancy has shown the problem to be NP-hard [6] even when restricted to 1 × 1 pieces that do not stick. A closely related problem of computing the minimum number of tilts needed to reconfigure between two board configurations has been shown to PSPACE-complete for 1 × 1 non-sticking pieces [6].…”

“…Previous work on occupancy has shown the problem to be NP-hard [6] even when restricted to 1 × 1 pieces that do not stick. A closely related problem of computing the minimum number of tilts needed to reconfigure between two board configurations has been shown to PSPACE-complete for 1 × 1 non-sticking pieces [6]. A summary of our complexity results, along with closely related prior work, is provided in Table 2.…”

“…The occupancy problem was shown to be NP-Hard by [5][6][7] when only using 1 × 1 tiles. Their reduction also works to show NP-hardness for the relocation problem when only using 1 × 1 tiles.…”

“…A related problem, which asks if a given location can be occupied by any polyomino, has also been considered. This problem was shown to be NP-hard [6] in the restricted case of 1 × 1 polyominoes that never stick together (but is not known to be in NP). We also extend this line of work by showing this problem is PSPACE-Complete if polyominoes larger than than 1 × 1 tiles are allowed.…”