Full Tilt: Universal Constructors for General Shapes with Uniform External Forces
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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. 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. In particular, for given integers h,w, we show that there exists a strongly universal configuration (no excess particles) 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, we go on to show the existence of a strongly universal configuration 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 polyominoes are included.
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