1,250 research outputs found
Simulation in Algorithmic Self-assembly
Winfree introduced a model of self-assembling systems called the abstract Tile Assembly Model (aTAM) where square tiles with glues on their edges attach spontaneously via matching glues to form complex structures. A generalization of the aTAM called the 2HAM (two-handed aTAM) not only allows for single tiles to bind, but also for supertile assemblies consisting of any number of tiles to attach. We consider a variety of models based on either the aTAM or the 2HAM. The underlying commonality of the work presented here is simulation. We introduce the polyTAM, where a tile system consists of a collection of polyomino tiles, and show that for any polyomino P of size greater than or equal to 3 and any Turing machine M , there exists a temperature-1 polyTAM system containing only shape-P tiles that simulates M . We introduce the RTAM (Reflexive Tile Assembly Model) that works like the aTAM except that tiles can nondeterministically flip prior to binding. We show that the temperature-1 RTAM cannot simulate a Turing machine by showing the much stronger result that the RTAM can only self-assemble periodic patterns. We then define notions of simulation which serve as relations between two tile assembly systems (possibly belonging to different models). Using simulation as a basis of comparison, we first show that cellular automata and the class of all tile assembly systems in the aTAM are equivalent. Next, we introduce the Dupled aTAM (DaTAM) and show that the temperature-2 aTAM and the temperature-1 DaTAM are mutually exclusive by showing that there is an aTAM system that cannot be simulated by any DaTAM system, and vice versa. Third, we consider the restricted glues Tile Assembly Model (rgTAM) and show that there is an aTAM system that cannot be simulated by any rgTAM system. We introduce the Dupled restricted glues Tile Assembly Model (DrgTAM), and show that the DrgTAM is intrinsically universal for the aTAM. Finally, we consider a variation of the Signal-passing Tile Assembly Model (STAM) called the STAM+ and show that the STAM+ is intrinsically universal and that the 3-D 2HAM is intrinsically universal for the STAM+
Noncooperative algorithms in self-assembly
We show the first non-trivial positive algorithmic results (i.e. programs
whose output is larger than their size), in a model of self-assembly that has
so far resisted many attempts of formal analysis or programming: the planar
non-cooperative variant of Winfree's abstract Tile Assembly Model.
This model has been the center of several open problems and conjectures in
the last fifteen years, and the first fully general results on its
computational power were only proven recently (SODA 2014). These results, as
well as ours, exemplify the intricate connections between computation and
geometry that can occur in self-assembly.
In this model, tiles can stick to an existing assembly as soon as one of
their sides matches the existing assembly. This feature contrasts with the
general cooperative model, where it can be required that tiles match on
\emph{several} of their sides in order to bind.
In order to describe our algorithms, we also introduce a generalization of
regular expressions called Baggins expressions. Finally, we compare this model
to other automata-theoretic models.Comment: A few bug fixes and typo correction
Optimal Staged Self-Assembly of General Shapes
We analyze the number of tile types , bins , and stages necessary to
assemble squares and scaled shapes in the staged tile assembly
model. For squares, we prove stages suffice and
are necessary for almost all .
For shapes with Kolmogorov complexity , we prove
stages suffice and are necessary to
assemble a scaled version of , for almost all . We obtain similarly tight
bounds when the more powerful flexible glues are permitted.Comment: Abstract version appeared in ESA 201
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