NDLI: On the decidability of self-assembly of infinite ribbons

Content Provider	IEEE Xplore Digital Library
Author	Adleman, L. Kari, J. Kari, L. Reishus, D.
Copyright Year	2002
Description	Author affiliation: Lab. for Molecular Sci., Univ. of Southern California, VA, USA (Adleman, L.)
Abstract	Self-assembly, the process by which objects autonomously come together to form complex structures, is omnipresent in the physical world. A systematic study of self-assembly as a mathematical process has been initiated. The individual components are modelled as square tiles on the infinite two-dimensional plane. Each side of a tile is covered by a specific "glue", and two adjacent tiles will stick if they have matching glues on their abutting edges. Tiles that stick to each other may form various two-dimensional "structures" such as squares, rectangles, or may cover the entire plane. In this paper we focus on a special type of structure, called ribbon: a non-self-crossing sequence of tiles on the plane, in which successive tiles are adjacent along an edge, and abutting edges of consecutive tiles have matching glues. We prove that it is undecidable whether an arbitrary finite set of tiles with glues (infinite supply of each tile type available) can be used to assemble an infinite ribbon. The proof is based on a construction, due to Robinson (1971), of a special set of tiles that allow only aperiodic tilings of the plane. This construction is used to create a special set of directed tiles (tiles with arrows painted on the top) with the "strong plane filling property" - a variation of the "plane filling property" previously defined by Kari (1990, 1994). A construction of "sandwich" tiles is then used in conjunction with this special tile set, to reduce the well-known undecidable tiling problem to the problem of the existence of an infinite directed zipper (a special kind of ribbon). A "motif" construction is then introduced that allows one tile system to simulate another by using geometry to represent glues. Using motifs, the infinite directed zipper problem is reduced to the infinite ribbon problem, proving the latter undecidable. The result settles an open problem formerly known as the "unlimited infinite snake problem". Moreover, an immediate consequence is the undecidability of the existence of arbitrarily large structures self-assembled using tiles from a given tile set.
Sponsorship	IEEE Comput. Soc. Tech. Committee on Maht. Found. Comput. (TCMF) ACM/SIGACT
Starting Page	530
Ending Page	537
File Size	313718
Page Count	8
File Format	PDF
ISBN	0769518222
ISSN	02725428
DOI	10.1109/SFCS.2002.1181977
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2002-11-19
Publisher Place	Canada
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Self-assembly Tiles DNA computing Computer science Laboratories Assembly Biology computing Amorphous materials Plastics Solid modeling
Content Type	Text
Resource Type	Article

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