Abstract
The tile assembly model, a formal model of crystal growth, is of special interest to computer scientists and mathematicians because it is universal [1]. Therefore, tile assembly model systems can compute all the functions that computers compute. In this paper, I formally define what it means for a system to nondeterministically decide a set, and present a system that solves an NP-complete problem called SubsetSum. Because of the nature of NP-complete problems, this system can be used to solve all NP problems in polynomial time, with high probability. While the proof that the tile assembly model is universal [2] implies the construction of such systems, those systems are in some sense “large” and “slow.” The system presented here uses 49 = Θ(1) different tiles and computes in time linear in the input size. I also propose how such systems can be leveraged to program large distributed software systems.
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References
Winfree, E.: Simulations of computing by self-assembly of DNA. Technical Report CS-TR:1998:22, California Insitute of Technology, Pasadena, CA, USA (1998)
Winfree, E.: Algorithmic Self-Assembly of DNA. PhD thesis, California Insitute of Technology, Pasadena, CA, USA (June 1998)
Adleman, L.: Molecular computation of solutions to combinatorial problems. Science 266, 1021–1024 (1994)
Braich, R., Johnson, C.R., Rothemund, P.W.K., Hwang, D., Chelyapov, N., Adleman, L.: Solution of a satisfiability problem on a gel-based DNA computer. In: Proceedings of DNA Computing: 6th International Workshop on DNA-Based Computers (DNA 2000), Leiden, The Netherlands, pp. 27–38 (June 2000)
Braich, R., Chelyapov, N., Johnson, C.R., Rothemund, P.W.K., Adleman, L.: Solution of a 20-variable 3-SAT problem on a DNA computer. Science 296(5567), 499–502 (2002)
Winfree, E.: On the computational power of DNA annealing and ligation. DNA Based Computers 199–221 (1996)
Winfree, E., Bekbolatov, R.: Proofreading tile sets: Error correction for algorithmic self-assembly. In: Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science (FOCS 2002), Madison, WI, USA, June 2003, vol. 2943, pp. 126–144. IEEE Computer Society Press, Los Alamitos (2003)
Baryshnikov, Y., Coffman, E.G., Seeman, N., Yimwadsana, T.: Self correcting self assembly: Growth models and the hammersley process. In: Carbone, A., Pierce, N.A. (eds.) DNA Computing. LNCS, vol. 3892, Springer, Heidelberg (2006)
Chen, H.L., Goel, A.: Error free self-assembly with error prone tiles. In: Ferretti, C., Mauri, G., Zandron, C. (eds.) DNA Computing. LNCS, vol. 3384, Springer, Heidelberg (2005)
Reif, J.H., Sahu, S., Yin, P.: Compact error-resilient computational DNA tiling assemblies. In: Ferretti, C., Mauri, G., Zandron, C. (eds.) DNA Computing. LNCS, vol. 3384, Springer, Heidelberg (2005)
Winfree, E.: Self-healing tile sets. Nanotechnology: Science and Computation, 55–78 (2006)
Barish, R., Rothemund, P.W.K., Winfree, E.: Two computational primitives for algorithmic self-assembly: Copying and counting. Nano Letters 5(12), 2586–2592 (2005)
Rothemund, P.W.K., Papadakis, N., Winfree, E.: Algorithmic self-assembly of DNA Sierpinski triangles. PLoS Biology 2(12), 424 (2004)
Adleman, L., Cheng, Q., Goel, A., Huang, M.-D., Wasserman, H.: Linear self-assemblies: Equilibria, entropy, and convergence rates. In: Proceedings of the 6th International Conference on Difference Equations and Applications (ICDEA 2001), Augsburg, Germany (June 2001)
Rothemund, P.W.K., Winfree, E.: The program-size complexity of self-assembled squares. In: Proceedings of the ACM Symposium on Theory of Computing (STOC 2000, Portland, OR, USA, pp. 459–468. ACM Press, New York (2000)
Adleman, L., Cheng, Q., Goel, A., Huang, M.-D., Kempe, D., de Espanes, P.M., Rothemund, P.W.K.: Combinatorial optimization problems in self-assembly. In: Proceedings of the ACM Symposium on Theory of Computing (STOC 2002), Montreal, Quebec, Canada, pp. 23–32. ACM Press, New York (2002)
Adleman, L., Goel, A., Huang, M.-D., de Espanes, P.M.: Running time and program size for self-assembled squares. In: Proceedings of the ACM Symposium on Theory of Computing (STOC 2002), Montreal, Quebec, Canada, pp. 740–748. ACM Press, New York (2001)
de Espanes, P.M.: Computerized exhaustive search for optimal self-assembly counters. In: Proceedings of the 2nd Foundations of Nanoscience: Self-Assembled Architectures and Devices (FNANO 2005), Snowbird, UT, USA, pp. 24–25 (April 2005)
Lagoudakis, M.G., LaBean, T.H.: 2D DNA self-assembly for satisfiability. DIMACS Series in Discrete Mathematics and Theoretical Computer Science 54, 141–154 (1999)
Fu, T.J., Seeman, N.C.: DNA double-crossover molecules. Biochemistry 32(13), 3211–3220 (1993)
Wang, H.: Proving theorems by pattern recognition. II. Bell System Technical Journal 40, 1–42 (1961)
Brun, Y.: Arithmetic computation in the tile assembly model: Addition and multiplication. Theoretical Computer Science 378, 17–31 (2007)
Brun, Y.: Nondeterministic polynomial time factoring in the tile assembly model. Theoretical Computer Science (2007), doi:10.1016/j.tcs.2007.07.051
Brun, Y.: Solving NP-complete problems in the tile assembly model. Theoretical Computer Science (2007), doi:10.1016/j.tcs.2007.07.052
Brun, Y., Medvidovic, N.: An architectural style for solving computationally intensive problems on large networks. In: Proceedings of Software Engineering for Adaptive and Self-Managing Systems (SEAMS 2007), Minneapolis, MN, USA (May 2007)
Brun, Y., Medvidovic, N.: Fault and adversary tolerance as an emergent property of distributed systems’ software architectures. In: Proceedings of the 2nd International Workshop on Engineering Fault Tolerant Systems (EFTS 2007), Dubrovnik, Croatia (September 2007)
Brun, Y.: Discreetly distributing computation via self-assembly. Technical Report USC-CSSE-2007-714, Center for Software Engineering, University of Southern California (2007)
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Brun, Y. (2008). Constant-Size Tileset for Solving an NP-Complete Problem in Nondeterministic Linear Time. In: Garzon, M.H., Yan, H. (eds) DNA Computing. DNA 2007. Lecture Notes in Computer Science, vol 4848. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77962-9_3
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