  In this paper, we investigate shape-assembling power of a tile-based model of
self-assembly called the Signal-Passing Tile Assembly Model (STAM). In this
model, the glues that bind tiles together can be turned on and off by the
binding actions of other glues via "signals". Specifically, the problem we
investigate is "shape replication" wherein, given a set of input assemblies of
arbitrary shape, a system must construct an arbitrary number of assemblies with
the same shapes and, with the exception of size-bounded junk assemblies that
result from the process, no others. We provide the first fully universal shape
replication result, namely a single tile set capable of performing shape
replication on arbitrary sets of any 3-dimensional shapes without requiring any
scaling or pre-encoded information in the input assemblies. Our result requires
the input assemblies to be composed of signal-passing tiles whose glues can be
deactivated to allow deconstruction of those assemblies, which we also prove is
necessary by showing that there are shapes whose geometry cannot be replicated
without deconstruction. Additionally, we modularize our construction to create
systems capable of creating binary encodings of arbitrary shapes, and building
arbitrary shapes from their encodings. Because the STAM is capable of universal
computation, this then allows for arbitrary programs to be run within an STAM
system, using the shape encodings as input, so that any computable
transformation can be performed on the shapes.

Formal Languages and Automata Theory