Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Full Tilt: Universal Constructors for General Shapes with Uniform
  External Forces   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 $\mathcal{O}(hw)$
$1 \times 1$ slidable particles that can be reconfigured to build any $h \times
w$ patterned rectangle. We then expand this result to show that there exists a
weakly universal configuration that can build any $h \times w$-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 $2\times 2$ polyomino in addition to $1\times 1$
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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The categories of this paper are: nlin
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The categories of this paper are: solv-int
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-ex
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The categories of this paper are: q-bio
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The categories of this paper are: econ
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The categories of this paper are: supr-con
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The categories of this paper are: stat
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The categories of this paper are: ao-sci
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The categories of this paper are: nucl-th
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The categories of this paper are: gr-qc
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The categories of this paper are: q-alg
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The categories of this paper are: funct-an
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The categories of this paper are: nucl-ex
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The categories of this paper are: astro-ph
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The categories of this paper are: q-fin