Abstract
Fault-tolerance is a critical issue for biochemical computation. Recent theoretical work on algorithmic self-assembly has shown that error correcting tile sets are possible, and that they can achieve exponential decrease in error rates with a small increase in the number of tile types and the scale of the construction [24, 4]. Following [17], we consider the issue of applying similar schemes to achieve error correction without any increase in the scale of the assembled pattern. Using a new proofreading transformation, we show that compact proofreading can be performed for some patterns with a modest increase in the number of tile types. Other patterns appear to require an exponential number of tile types. A simple property of existing proofreading schemes – a strong kind of redundancy – is the culprit, suggesting that if general purpose compact proofreading schemes are to be found, this type of redundancy must be avoided.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Adleman, L.M., Cheng, Q., Goel, A., Huang, M.-D.A.: Running time and program size for self-assembled squares. In: ACM Symposium on Theory of Computing (STOC), pp. 740–748 (2001)
Aggarwal, G., Goldwasser, M., Kao, M., Schweller, R.T.: Complexities for generalized models of self-assembly. In: Symposium on Discrete Algorithms (SODA), pp. 880–889 (2004)
Barish, R.D., Rothemund, P.W.K., Winfree, E.: Two computational primitives for algorithmic self-assembly: Copying and counting. NanoLetters (to appear)
Chen, H.-L., Goel, A.: Error free self-assembly using error prone tiles. In: Ferretti, et al. (eds.) [7], pp. 62–75
Chen, J., Reif, J.H. (eds.): DAN 2003. LNCS, vol. 2943. Springer, Heidelberg (2004)
Cook, M., Rothemund, P.W.K., Winfree, E.: Self-assembled circuit patterns. In: Chen, Reif (eds.) [5], pp. 91–107
Ferretti, C., Mauri, G., Zandron, C. (eds.): DNA 2004. LNCS, vol. 3384. Springer, Heidelberg (2005)
Rothemund, P.W.K.: Theory and Experiments in Algorithmic Self-Assembly. PhD thesis, University of Southern California, Los Angeles (2001)
Rothemund, P.W.K., Papakakis, N., Winfree, E.: Algorithmic self-assembly of DNA Sierpinski triangles. PLoS Biology 2, e424 (2004)
Rothemund, P.W.K., Winfree, E.: The program-size complexity of self-assembled squares. In: ACM Symposium on Theory of Computing (STOC), pp. 459–468 (2000)
Hurd, L., Kari, J., Culik, K.: The topological entropy of cellular automata is uncomputable. Ergodic Theory Dynamical Systems 12, 255–265 (1992)
LaBean, T.H., Yan, H., Kopatsch, J., Liu, F., Winfree, E., Reif, J.H., Seeman, N.C.: Construction, analysis, ligation, and self-assembly of DNA triple crossover complexes. Journal of the Americal Chemical Society 122, 1848–1860 (2000)
Lagoudakis, M.G., LaBean, T.H.: 2-D DNA self-assembly for satisfiability. In: Istrail, S., Waterman, M.S., Clark, A. (eds.) DIMACS/RECOMB Satellite Workshop 2002. LNCS (LNBI), vol. 2983, pp. 141–154. Springer, Heidelberg (2004)
Mao, C., LaBean, T.H., Reif, J.H., Seeman, N.C.: Logical computation using algorithmic self-assembly of DNA triple-crossover molecules. Nature 407, 493–496 (2000)
Mao, C., Sun, W., Seeman, N.C.: Designed two-dimensional DNA holliday junction arrays visualized by atomic force microscopy. Journal of the Americal Chemical Society 121, 5437–5443 (1999)
Reif, J.: Local parallel biomolecular computing. In: DNA Based Computers III. DIMACS, vol. 48, pp. 217–254. American Mathematical Society, Providence (1999)
Reif, J.H., Sahu, S., Yin, P.: Compact error-resilient computational DNA tiling assemblies. In: Ferretti, et al. (eds.) [7], pp. 293–307
Schulman, R., Winfree, E.: Programmable control of nucleation for algorithmic self-assembly. In: Ferretti, et al. (eds.) [7], pp. 319–328
Schulman, R., Winfree, E.: Self-replication and evolution of DNA crystals. In: Capcarrère, M.S., Freitas, A.A., Bentley, P.J., Johnson, C.G., Timmis, J. (eds.) ECAL 2005. LNCS (LNAI), vol. 3630, pp. 734–743. Springer, Heidelberg (2005)
Soloveichik, D., Winfree, E.: Complexity of self-assembled shapes (2005) (extended abstract); Preprint of the full paper is cs.CC/0412096 on arXiv.org
Winfree, E.: On the computational power of DNA annealing and ligation. In: Lipton, R.J., Baum, E.B. (eds.) DNA Based Computers. DIMACS, vol. 27, pp. 199–221. American Mathematical Society, Providence (1996)
Winfree, E.: Algorithmic Self-Assembly of DNA. PhD thesis, California Institute of Technology, Pasadena (1998)
Winfree, E.: Simulations of computing by self-assembly. Technical Report CS-TR:1998.22, Caltech (1998)
Winfree, E., Bekbolatov, R.: Proofreading tile sets: Error-correction for algorithmic self-assembly. In: Chen, Reif (eds.) [5], pp. 126–144
Winfree, E., Liu, F., Wenzler, L.A., Seeman, N.C.: Design and self-assembly of two dimensional DNA crystals. Nature 394, 539–544 (1998)
Winfree, E., Yang, X., Seeman, N.C.: Universal Computation via Self-assembly of DNA: Some Theory and Experiments. In: Landweber, L.F., Baum, E.B. (eds.) DNA Based Computers II. DIMACS, vol. 44, pp. 191–213. American Mathematical Society, Providence (1998)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Soloveichik, D., Winfree, E. (2006). Complexity of Compact Proofreading for Self-assembled Patterns. In: Carbone, A., Pierce, N.A. (eds) DNA Computing. DNA 2005. Lecture Notes in Computer Science, vol 3892. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11753681_24
Download citation
DOI: https://doi.org/10.1007/11753681_24
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-34161-1
Online ISBN: 978-3-540-34165-9
eBook Packages: Computer ScienceComputer Science (R0)