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

Abstract

We investigate the problem of assembling general shapes and patterns in a model in which particles move based on uniform external forces until they encounter an obstacle. In this model, corresponding particles may bond when adjacent with one another. Succinctly, this model considers a 2D grid of "open" and "blocked" spaces, along with a set of slidable polyominoes placed at open locations on the board. The board may be tilted in any of the 4 cardinal directions, causing all slidable polyominoes to move maximally in the specified direction until blocked. By successively applying a sequence of such tilts, along with allowing different polyominoes to stick when adjacent, tilt sequences provide a method to reconfigure an initial board configuration so as to assemble a collection of previous separate polyominoes into a larger shape.While previous work within this model of assembly has focused on designing a specific board configuration for the assembly of a specific given shape, we propose the problem of designing universal configurations that are capable of constructing a large class of shapes and patterns. For these constructions, we present the notions of weak and strong universality which indicate the presence of "excess" polyominoes after the shape is constructed. In particular, for given integers h, w, we show that there exists a strongly universal configuration with O(hw) 1 × 1 slidable particles that can be reconfigured to build any h × w patterned rectangle. We then expand this result to show that there exists a weakly universal configuration that can build any h × w-bounded size connected shape. Following these results, which require an admittedly relaxed assembly definition, we go on to show the existence of a strongly universal configuration (no excess particles) which can assemble any shape within a previously studied "drop" class, while using quadratically less space than previous results.Finally, we include a study of the complexity of motion planning in this model. We consider the problems of deciding if a board location can be occupied by any particle (occupancy problem), deciding if a specific particle may be relocated to another position (relocation problem), and deciding if a given configuration of particles may be transformed into a second given configuration (reconfiguration problem). We show all of these problems to be PSPACE-complete with the allowance of a single 2 × 2 polyomino in addition to 1 × 1 tiles. We further show that relocation and occupancy remain PSPACE-complete even when the board geometry is a simple rectangle if domino polyominos are included.

Hardness of Reconfiguring Robot Swarms with Uniform External Control in Limited Directions

“…Recent work strengthened these results and showed PSPACE-Completeness for all three problems even when only allowing 1 × 1 pieces [2]. Additional work has focused on the application of the tilt model for the assembly of general shapes [8], [10], [11], including universal constructors [1], [2], and sorting polyominoes [9].…”

“…Later work in Ref. [1] relaxed the constraint on tile size and showed the Relocation and Reconfiguration Problems were PSPACE-Complete when only a single 2 × 2 polyomino is allowed. Recent work strengthened these results and showed PSPACE-Completeness for all three problems even when only allowing 1 × 1 pieces [2].…”

“…Figure 1 shows a simple example. We further consider these problems with limited usable directions (e.g., only tilting down and right), as well as considering the effect of limiting the complexity of the geometry of the open spaces 1 The University of Texas Rio Grande Valley, Edinburg, Texas 78539, USA a) amluchsinger@gmail.com of the board. For example, one limited type of geometry is that in which the open spaces form an "x/y-monotone" shape.…”

Hierarchical Shape Construction and Complexity for Slidable Polyominoes under Uniform External Forces

“…Given a shape in their buildable class, a tilt model configuration could be generated to assemble that shape through a sequence of tilts. This drop-shape construction work was extended in [3]. Rather than focusing on designing configurations for specific drop-shapes, the authors create a general drop-shape constructor.…”

“…The authors also showed that a dualrail logic fanout was not possible with only 1×1 tiles, which seemed to serve as evidence that this problem was not PSPACE-complete. Following, in [3] the authors proved that the occupancy problem is PSPACE-complete if a single "large" polyomino is allowed (they used a single 2 × 2 polyomino along with 1 × 1 tiles).…”