Path finding in the tile assembly model

Swarm robotics, active self-assembly, and amorphous computing are fields that focus on designing systems of large numbers of small, simple components that can cooperate to complete complex tasks. Many of these systems are inspired by biological systems, and all attempt to use the simplest components and environments possible, while still being capable of achieving their goals. The canonical problems for such biologically-inspired systems are shape assembly and path finding. In this paper, we demonstrate path finding in the well-studied tile assembly model, a model of molecular self-assembly that is strictly simpler than other biologically-inspired models. As in related work, our systems function in the presence of obstacles and can be made fault-tolerant. The path-finding systems use @Q(1) distinct components and find minimal-length paths in time linear in the length of the path.

[1]  E. Klavins,et al.  Programmable Self-Assembly , 2007, IEEE Control Systems.

[2]  E. Winfree,et al.  Algorithmic Self-Assembly of DNA Sierpinski Triangles , 2004, PLoS biology.

[3]  Erik Winfree,et al.  The program-size complexity of self-assembled squares (extended abstract) , 2000, STOC '00.

[4]  Radhika Nagpal,et al.  Programming Methodology for Biologically-Inspired Self-Assembling Systems , 2003 .

[5]  V. Michael Bove,et al.  Programming a paintable computer , 2002 .

[6]  N. Seeman,et al.  DNA double-crossover molecules. , 1993, Biochemistry.

[7]  Aristides A. G. Requicha,et al.  Self-repairing Self-assembled Structures , 2006 .

[8]  Yuriy Brun Nondeterministic polynomial time factoring in the tile assembly model , 2008, Theor. Comput. Sci..

[9]  L. Adleman,et al.  Self-assembly of DNA double-double crossover complexes into high-density, doubly connected, planar structures. , 2005, Journal of the American Chemical Society.

[10]  Yuriy Brun Solving NP-complete problems in the tile assembly model , 2008, Theor. Comput. Sci..

[11]  E. Demaine,et al.  Folding and unfolding , 2002 .

[12]  Sudheer Sahu,et al.  Compact Error-Resilient Computational DNA Tiling Assemblies , 2004, DNA.

[13]  Yuriy Brun Discreetly Distributing Computation via Self-Assembly , 2007 .

[14]  Radhika Nagpal,et al.  Distributed construction by mobile robots with enhanced building blocks , 2006, Proceedings 2006 IEEE International Conference on Robotics and Automation, 2006. ICRA 2006..

[15]  Hao Wang Proving theorems by pattern recognition — II , 1961 .

[16]  Matthew Cook,et al.  Combining self-healing and proofreading in self-assembly , 2008, Natural Computing.

[17]  Hao Wang,et al.  Proving theorems by pattern recognition I , 1960, Commun. ACM.

[18]  Aristides A. G. Requicha,et al.  Shape restoration by active self-assembly , 2005 .

[19]  Ashish Goel,et al.  Error Free Self-assembly Using Error Prone Tiles , 2004, DNA.

[20]  James McLurkin,et al.  Speaking Swarmish: Human-Robot Interface Design for Large Swarms of Autonomous Mobile Robots , 2006, AAAI Spring Symposium: To Boldly Go Where No Human-Robot Team Has Gone Before.

[21]  Yuriy Brun,et al.  An Architectural Style for Solving Computationally Intensive Problems on Large Networks , 2007, International Workshop on Software Engineering for Adaptive and Self-Managing Systems (SEAMS '07).

[22]  Robert L. Berger The undecidability of the domino problem , 1966 .

[23]  E. Winfree Simulations of Computing by Self-Assembly , 1998 .

[24]  Paul W. K. Rothemund,et al.  Scaffolded DNA Origami: from Generalized Multicrossovers to Polygonal Networks , 2006, Nanotechnology: Science and Computation.

[25]  Radhika Nagpal,et al.  Organizing a Global Coordinate System from Local Information on an Ad Hoc Sensor Network , 2003, IPSN.

[26]  L M Adleman,et al.  Molecular computation of solutions to combinatorial problems. , 1994, Science.

[27]  Dustin Reishus Design of a Self-Assembled Memory Circuit , 2008 .

[28]  Erik Winfree,et al.  Proofreading Tile Sets: Error Correction for Algorithmic Self-Assembly , 2003, DNA.

[29]  Yuriy Brun Arithmetic computation in the tile assembly model: Addition and multiplication , 2007, Theor. Comput. Sci..

[30]  Ming-Yang Kao,et al.  Complexities for generalized models of self-assembly , 2004, SODA '04.

[31]  Erik Winfree,et al.  Two computational primitives for algorithmic self-assembly: copying and counting. , 2005, Nano letters.

[32]  Erik Winfree,et al.  On the computational power of DNA annealing and ligation , 1995, DNA Based Computers.

[33]  Jarkko Kari,et al.  On the decidability of self-assembly of infinite ribbons , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[34]  David Harel,et al.  On the Solvability of Domino Snake Problems , 1994, Theor. Comput. Sci..

[35]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[36]  Edward G. Coffman,et al.  Self-correcting Self-assembly: Growth Models and the Hammersley Process , 2005, DNA.

[37]  Heinz-Dieter Ebbinghaus Undecidability Of Some Domino Connectability Problems , 1982, Math. Log. Q..

[38]  Radhika Nagpal Programmable self-assembly: constructing global shape using biologically-inspired local interactions and origami mathematics , 2001 .

[39]  Matthew Cook,et al.  Self-Assembled Circuit Patterns , 2003, DNA.

[40]  Ming-Yang Kao,et al.  Reducing tile complexity for self-assembly through temperature programming , 2006, SODA '06.

[41]  Clifford R. Johnson,et al.  Solution of a 20-Variable 3-SAT Problem on a DNA Computer , 2002, Science.

[42]  Aristides A. G. Requicha,et al.  Active self-assembly , 2004, IEEE International Conference on Robotics and Automation, 2004. Proceedings. ICRA '04. 2004.

[43]  Paul W. K. Rothemund,et al.  Design of DNA origami , 2005, ICCAD-2005. IEEE/ACM International Conference on Computer-Aided Design, 2005..

[44]  M. Sahani,et al.  Algorithmic Self-Assembly of DNA , 2006 .

[45]  Ashish Goel,et al.  Combinatorial optimization problems in self-assembly , 2002, STOC '02.

[46]  Leonard M. Adleman,et al.  Solution of a Satisfiability Problem on a Gel-Based DNA Computer , 2000, DNA Computing.

[47]  Yuriy Brun,et al.  Fault and adversary tolerance as an emergent property of distributed systems' software architectures , 2007, EFTS '07.

[48]  Wei-Min Shen,et al.  Multimode locomotion via SuperBot reconfigurable robots , 2006, Auton. Robots.

[49]  Yuriy Brun,et al.  DNA triangles and self-assembled hexagonal tilings. , 2004, Journal of the American Chemical Society.

[50]  Yan Liu,et al.  DNA-Templated Self-Assembly of Protein Arrays and Highly Conductive Nanowires , 2003, Science.

[51]  Michail G. Lagoudakis,et al.  2D DNA self-assembly for satisfiability , 1999, DNA Based Computers.

[52]  Ashish Goel,et al.  Running time and program size for self-assembled squares , 2001, STOC '01.

[53]  Heinz-Dieter Ebbinghaus,et al.  Domino Threads and Complexity , 1987, Computation Theory and Logic.

[54]  Qi Cheng,et al.  Linear Self-Assemblies: Equilibria, Entropy and Convergence Rates , 2003 .

[55]  P. Rothemund Folding DNA to create nanoscale shapes and patterns , 2006, Nature.

[56]  R. Robinson Undecidability and nonperiodicity for tilings of the plane , 1971 .

[57]  Erik Winfree,et al.  Self-healing Tile Sets , 2006, Nanotechnology: Science and Computation.

[58]  Chris Hanson,et al.  Amorphous computing , 2000, Commun. ACM.

[59]  Erik Winfree,et al.  Complexity of Self-assembled Shapes , 2004, DNA.