Scaled tree fractals do not strictly self-assemble

In this paper, we show that any scaled-up version of any discrete self-similar {\it tree} fractal does not strictly self-assemble, at any temperature, in Winfree's abstract Tile Assembly Model.Comment: 13 pages, 3 figures, Appeared in the Proceedings of UCNC-2014, pp 27-39; Unconventional Computation and Natural Computation - 13th International Conference, UCNC 2014, London, ON, Canada, July 14-18, 2014, Springer Lecture Notes in Computer Science ISBN 978-3-319-08122-

Self-Assembly of Infinite Structures

We review some recent results related to the self-assembly of infinite structures in the Tile Assembly Model. These results include impossibility results, as well as novel tile assembly systems in which shapes and patterns that represent various notions of computation self-assemble. Several open questions are also presented and motivated

Self-Assembly of 4-sided Fractals in the Two-handed Tile Assembly Model

We consider the self-assembly of fractals in one of the most well-studied models of tile based self-assembling systems known as the Two-handed Tile Assembly Model (2HAM). In particular, we focus our attention on a class of fractals called discrete self-similar fractals (a class of fractals that includes the discrete Sierpi\'nski carpet). We present a 2HAM system that finitely self-assembles the discrete Sierpi\'nski carpet with scale factor 1. Moreover, the 2HAM system that we give lends itself to being generalized and we describe how this system can be modified to obtain a 2HAM system that finitely self-assembles one of any fractal from an infinite set of fractals which we call 4-sided fractals. The 2HAM systems we give in this paper are the first examples of systems that finitely self-assemble discrete self-similar fractals at scale factor 1 in a purely growth model of self-assembly. Finally, we show that there exists a 3-sided fractal (which is not a tree fractal) that cannot be finitely self-assembled by any 2HAM system

Fractals, Randomization, Optimal Constructions, and Replication in Algorithmic Self-Assembly

The problem of the strict self-assembly of infinite fractals within tile self-assembly is considered. In particular, tile assembly algorithms are provided for the assembly of the discrete Sierpinski triangle and the discrete Sierpinski carpet. The robust random number generation problem in the abstract tile assembly model is introduced. First, it is shown this is possible for a robust fair coin flip within the aTAM, and that such systems guarantee a worst case O(1) space usage. This primary construction is accompanied with variants that show trade-offs in space complexity, initial seed size, temperature, tile complexity, bias, and extensibility. This work analyzes the number of tile types t, bins b, and stages necessary and sufficient to assemble n × n squares and scaled shapes in the staged tile assembly model. Further, this work shows how to design a universal shape replicator in a 2-HAM self-assembly system with both attractive and repulsive forces

The Need for Seed (in the abstract Tile Assembly Model)

In the abstract Tile Assembly Model (aTAM) square tiles self-assemble, autonomously binding via glues on their edges, to form structures. Algorithmic aTAM systems can be designed in which the patterns of tile attachments are forced to follow the execution of targeted algorithms. Such systems have been proven to be computationally universal as well as intrinsically universal (IU), a notion borrowed and adapted from cellular automata showing that a single tile set exists which is capable of simulating all aTAM systems (FOCS 2012). The input to an algorithmic aTAM system can be provided in a variety of ways, with a common method being via the "seed" assembly, which is a pre-formed assembly from which all growth propagates. In this paper we present a series of results which investigate the the trade-offs of using seeds consisting of a single tile, versus those containing multiple tiles. We show that arbitrary systems with multi-tile seeds cannot be converted to functionally equivalent systems with single-tile seeds without using a scale factor > 1. We prove tight bounds on the scale factor required, and also present a construction which uses a large scale factor but an optimal number of unique tile types. That construction is then used to develop a construction that performs simultaneous simulation of all aTAM systems in parallel, as well as to display a connection to other tile-based self-assembly models via the notion of intrinsic universality.Comment: To appear in the SODA 2023 proceeding

Self-Replication via Tile Self-Assembly (Extended Abstract)

In this paper we present a model containing modifications to the Signal-passing Tile Assembly Model (STAM), a tile-based self-assembly model whose tiles are capable of activating and deactivating glues based on the binding of other glues. These modifications consist of an extension to 3D, the ability of tiles to form "flexible" bonds that allow bound tiles to rotate relative to each other, and allowing tiles of multiple shapes within the same system. We call this new model the STAM*, and we present a series of constructions within it that are capable of self-replicating behavior. Namely, the input seed assemblies to our STAM* systems can encode either "genomes" specifying the instructions for building a target shape, or can be copies of the target shape with instructions built in. A universal tile set exists for any target shape (at scale factor 2), and from a genome assembly creates infinite copies of the genome as well as the target shape. An input target structure, on the other hand, can be "deconstructed" by the universal tile set to form a genome encoding it, which will then replicate and also initiate the growth of copies of assemblies of the target shape. Since the lengths of the genomes for these constructions are proportional to the number of points in the target shape, we also present a replicator which utilizes hierarchical self-assembly to greatly reduce the size of the genomes required. The main goals of this work are to examine minimal requirements of self-assembling systems capable of self-replicating behavior, with the aim of better understanding self-replication in nature as well as understanding the complexity of mimicking it