Abstract
We explore a method of designing algorithms using two types of DNA strands, namely rule strands (rules) and input strands. Rules are fixed in advance, and their task is to bind with the input strands in order to produce an output. We present algorithms for divisibility and primality testing as well as for square root computation. We measure the complexity of our algorithms in terms of the necessary rule strands. Our three algorithms utilize a super-constant amount of complex rules.
Can one solve interesting problems using only few—or at least simple—rule strands? Our main result proves that restricting oneself to a constant number of rule strands is equivalent to deciding regular languages. More precisely, we show that an algorithm (possibly using infinitely many rule strands of arbitrary length) can merely decide regular languages if the structure of the rules themselves is simple, i.e., if the rule strands constitute a regular language.
All algorithms and proofs are presented in the full version of this paper, available at http://disco.ethz.ch/publications/ISAAC2015-dna.pdf.
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Notes
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A tree is ordered if the children of every node are ordered, e.g., from left to right.
References
Lakowicz, J.R.: DNA technology. In: Lakowicz, J.R. (ed.) Principles of Fluorescence Spectroscopy, pp. 705–740. Springer, New York (2006)
Adleman, L.M.: Molecular computation of solutions to combinatorial problems. Science 266(5187), 1021–1024 (1994)
Angluin, D., Aspnes, J., Eisenstat, D.: A simple population protocol for fast robust approximate majority. Distrib. Comput. 21(2), 87–102 (2008)
Bottoni, P., Labella, A., Manca, V., Mitrana, V.: Superposition based on Watson-Crick-like complementarity. Theory Comput. Syst. 39(4), 503–524 (2006)
Breaker, R.R.: Engineered allosteric ribozymes as biosensor components. Curr. Opin. Biotechnol. 13(1), 31–39 (2002)
Cardelli, L., Csikász-Nagy, A.: The cell cycle switch computes approximate majority. Scientific reports 2, September 2012
Kari, L., Păun, G., Rozenberg, G., Salomaa, A., Yu, S.: DNA computing, sticker systems, and universality. Acta Informatica 35(5), 401–420 (1998)
Kobayashi, S., Mitrana, V., Păun, G., Rozenberg, G.: Formal properties of PA-matching. Theoret. Comput. Sci. 262(1–2), 117–131 (2001)
Lakin, M.R., Phillips, A.: Modelling, simulating and verifying turing-powerful strand displacement systems. In: Cardelli, L., Shih, W. (eds.) DNA 17 2011. LNCS, vol. 6937, pp. 130–144. Springer, Heidelberg (2011)
Lipton, R.J.: DNA solution of hard computational problems. Science 268(5210), 542–545 (1995)
Manea, F., Martín-Vide, C., Mitrana, V.: Hairpin lengthening: language theoretic and algorithmic results. J. Logic Comput. 25(4), 987–1009 (2015). doi:10.1093/logcom/exs076. http://logcom.oxfordjournals.org/content/25/4/987.abstract
Patitz, M.: An introduction to tile-based self-assembly and a survey of recent results. Nat. Comput. 13(2), 195–224 (2014)
Păun, G., Rozenberg, G., Yokomori, T.: Hairpin languages. Int. J. Found. Comput. Sci. 12(06), 837–847 (2001)
Qian, L., Winfree, E.: Scaling up digital circuit computation with DNA strand displacement cascades. Science 332(6034), 1196–1201 (2011)
Seelig, G., Soloveichik, D., Zhang, D.Y., Winfree, E.: Enzyme-free nucleic acid logic circuits. Science 314(5805), 1585–1588 (2006)
Seeman, N.C.: An overview of structural DNA nanotechnology. Mol. Biotechnol. 37(3), 246–257 (2007)
Sipser, M.: Introduction to the Theory of Computation. International Thomson Publishing, Boston (1996)
Winfree, E., Yang, X., Seeman, N.C.: Universal computation via self-assembly of DNA: some theory and experiments. In: DNA Based Computers II. DIMACS, vol. 44, pp. 191–213. American Mathematical Society (1996)
Zhang, D.Y., Turberfield, A.J., Yurke, B., Winfree, E.: Engineering entropy-driven reactions and networks catalyzed by DNA. Science 318(5853), 1121–1125 (2007)
Zhang, D.Y., Winfree, E.: Control of DNA strand displacement kinetics using toehold exchange. J. Am. Chem. Soc. 131(47), 17303–17314 (2009)
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Brandt, S., Mattia, N., Seidel, J., Wattenhofer, R. (2015). Toehold DNA Languages are Regular (Extended Abstract). In: Elbassioni, K., Makino, K. (eds) Algorithms and Computation. ISAAC 2015. Lecture Notes in Computer Science(), vol 9472. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48971-0_65
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DOI: https://doi.org/10.1007/978-3-662-48971-0_65
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