Abstract
The process where simple entities form more complex structures acting autonomously is called self-assembly; it lies at the centre of many physical, chemical and biological phenomena. Massively parallel formation of nanostructures or DNA computation are just two examples of possible applications of self-assembly once it is technologically harnessed. Various mathematical models have been proposed for self-assembly, including the well-known Winfree’s Tile Assembly Model based on Wang tiles on a two-dimensional plane. In the present paper we propose a model based on directed figures with partial catenation. Directed figures are defined as labelled polyominoes with designated start and end points, and catenation is defined for non-overlapping figures. This is one of possible extensions generalizing words and variable-length codes to planar structures, and a flexible model, allowing for a natural expression of self-assembling entities as well as e.g. image representation or “pictorial barcoding.” We prove several undecidability results related to filling the plane with a given set of figures and formation of infinite and semi-infinite zippers, demonstrating a unifying approach that could be useful for the study of self-assembly.
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References
Adleman, L.: Molecular computation of solutions to combinatorial problems. Science 266(5187), 1021–1024 (1994)
Adleman, L., Kari, J., Kari, L., Reishus, D., Sosik, P.: The undecidability of the infinite ribbon problem: Implications for computing by self-assembly. SIAM Journal on Computing 38(6), 2356–2381 (2009)
Berger, R.: The undecidability of the domino problem. Memoirs of the American Mathematical Society 66, 1–72 (1966)
Chen, H.L., Goel, A.: Error free self-assembly using error prone tiles. In: Ferretti, C., Mauri, G., Zandron, C. (eds.) DNA 2004. LNCS, vol. 3384, pp. 62–75. Springer, Heidelberg (2005)
Costagliola, G., Ferrucci, F., Gravino, C.: Adding symbolic information to picture models: definitions and properties. Theoretical Computer Science 337, 51–104 (2005)
Kolarz, M.: The code problem for directed figures. Theoretical Informatics and Applications RAIRO 44(4), 489–506 (2010)
Kolarz, M., Moczurad, W.: Directed figure codes are decidable. Discrete Mathematics and Theoretical Computer Science 11(2), 1–14 (2009)
Moczurad, W.: Directed figure codes with weak equality. In: Fyfe, C., Tino, P., Charles, D., Garcia-Osorio, C., Yin, H. (eds.) IDEAL 2010. LNCS, vol. 6283, pp. 242–250. Springer, Heidelberg (2010)
Winfree, E.: Algorithmic Self-Assembly of DNA. Ph.D. thesis, California Institute of Technology (1998)
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Moczurad, W. (2011). Plane-Filling Properties of Directed Figures. In: Atallah, M., Li, XY., Zhu, B. (eds) Frontiers in Algorithmics and Algorithmic Aspects in Information and Management. Lecture Notes in Computer Science, vol 6681. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21204-8_28
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DOI: https://doi.org/10.1007/978-3-642-21204-8_28
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