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Cellular Automata Theory Cellular Automaton Supercollider: an abstract model Genaro Juárez Martínez [email protected] http://uncomp.uwe.ac.uk/genaro/ International Center of Unconventional Computing, University of the West of England, United Kingdom http://uncomp.uwe.ac.uk/ Laboratorio de Ciencias de la Computación, Universidad Nacional Autónoma de México, México http://uncomp.uwe.ac.uk/LCCOMP/ Centre for Chaos and Complex Networks, City University of Hong Kong, Hong Kong, P. R. China http://www.ee.cityu.edu.hk/~cccs/ Foundation of Computer Science Laboratory, Hiroshima University, Hiroshima, Japan http://www.iec.hiroshima-u.ac.jp/ Laboratoire de Recherche Scientifique, Paris, France http://labores.eu/ Department of Electronic Engineering, City University of Hong Kong Hong Kong SAR, P. R. China, April 2, 2012 School of Science, Hangzhou Dianzi University Hanhzhou, Zhejiang, P. R. China, April 11, 2012 Cellular Automata Theory Cellular automata (CA) are very simple mathematical functions that evolve massively in parallel on 1, 2, 3, 4, ..., n, dimensions in a regular lattice. Invented by John von Neumann in 1956 and popularized amply by John Horton Conway and his famous additive-binary 2d CA The Game of Life in 1970. So, in 1986 Stephen Wolfram has been introduced the 1D CA and its famous “classes” where CA may fall. class I. Homogenous class II. Periodic class III. Chaos class IV. Complexity This classification is extended to any dynamical systems in general. the unpredictable ... Simplicity in CA are famous representing “patterns”, evolving as: complex dynamics, chaotic systems, and trivial behaviour. All they captured from random initial conditions generally. In this way, kaleidoscope is a funny example related of this problem. ! How many configurations there are? ! How long shall be the evolution to repeat the configuration? ! Is here impossible to predict the behaviour in each evolution Could you write an algorithm to calculate the set of configurations or determines the evolution? Sir David Brewster (Scot. Brit.), the physicist who invented de kaleidoscope in 1816 Hulton-Deutsch Collection/Corbis Encyclopaedia Britannica A kaleidoscope patterns Mark Gibson-Corbis Encyclopaedia Britannica Cellular Automata Cellular automata (CA) are discrete dynamical systems evolving on an infinite regular lattice. A CA is a 4-tuple A = <!, µ, ", c0> evolving in d-dimensional lattice, where d ! Z+. Such that: • ! represents the finite alphabet • µ is the local connection, where, µ = {x0,1,...,n-1:d | x ! !}, therefore, µ is a neighbourhood • " is the local function, such that, " : !µ " ! Z • c0 is the initial condition, such that, c0 ! ! Also, the local function induces a global transition between configurations: Z Z #" : ! " ! . CA dynamics in one dimension Elemental CA (ECA) is defined as follows: • ! = {0,1} • µ = (x+1,x0,x-1) such that x ! ! • " : !3 " ! • µ = {c0 | x ! !} the initial condition is the first ring with t = 0 History in ECA Rule 110 Stephen Wolfram, 1959-? Stephen Wolfram (1994) Cellular Automata and Complexity: collected papers, Addison-Wesley Publishing Company. Stephen Wolfram (2002) A New Kind of Science, Wolfram Media, Inc., Champaign, Illinois. History (literature) in Rule 110 In November 1998 at the Santa Fe Institute Matthew Cook had demonstrates that Rule 110 is Universal! Simulating a novel cyclic tag system. Wentian Li & Mats G. Nordahl (1992) "Transient behavior of cellular automaton rule 110," Physics Letters A 166, 335-339. Stephen Wolfram (1994) Cellular Automata and Complexity: collected papers, Addison-Wesley Publishing Company. Matthew Cook (1999) “Introduction to the activity of rule 110,” (copyright 1994-1998 Matthew Cook), http:// w3.datanet.hu/~cook/Workshop/CellAut/Elementary/Rule110/110pics.html Harold V. McIntosh (1999) “Rule 110 as it relates to the presence of gliders,” http://delta.cs.cinvestav.mx/~mcintosh/oldweb/pautomata.html Genaro J. Martínez & Harold V. McIntosh (2001) "ATLAS: Collisions of gliders like phases of ether in Rule 110," http://uncomp.uwe.ac.uk/genaro/Papers/Papers_on_CA.html Stephen Wolfram (2002) A New Kind of Science, Wolfram Media, Inc., Champaign, Illinois. Harold V. McIntosh (2002) “Rule 110 Is Universal!” http://delta.cs.cinvestav.mx/~mcintosh/oldweb/pautomata.html Genaro J. Martínez, Harold V. McIntosh & Juan C. Seck Tuoh Mora (2003) "Production of gliders by collisions in Rule 110," Lecture Notes in Computer Science 2801, 175-182. Matthew Cook (2004) “Universality in Elementary Cellular Automata,” Complex Systems 15(1), 1-40. History (literature) in Rule 110 Turlough Neary & Damien Woods (2006) "P-completeness of cellular automaton Rule 110," Lecture Notes in Computer Science 4051, 132-143. Genaro J. Martínez, Harold V. McIntosh & Juan C. Seck Tuoh Mora (2006) "Gliders in Rule 110," International Journal of Unconventional Computing 2(1), 1-49. Genaro J. Martínez, Harold V. McIntosh, Juan C. Seck Tuoh Mora, & Sergio V. Chapa Vergara (2007) "Rule 110 objects and other constructions based-collisions," Journal of Cellular Automata 2(3), 219-242. Matthew Cook (2008) "A Concrete View of Rule 110 Computation," In The Complexity of Simple Programs, T. Neary, D. Woods, A. K. Seda, & N. Murphy (Eds.), 31-55. Genaro J. Martínez, Harold V. McIntosh, Juan C. Seck Tuoh Mora, & Sergio V. Chapa Vergara (2008) "Determining a regular language by glider-based structures called phases fi_1 in Rule 110," Journal of Cellular Automata 3(3), 231-270. José M. Sausedo-Solorio (2010) Conservation features in binary collisions for rule 110 cellular automaton, International Journal of Modern Physics C 21(7), 931-942. Genaro J. Martínez, Harold V. McIntosh, Juan C. Seck-Tuoh-Mora, & Sergio V. Chapa-Vergara (2011) "Reproducing the cyclic tag system developed by Matthew Cook with Rule 110 using the phases f1_1," Journal of Cellular Automata 6(2-3), 121-161. Genaro J. Martínez, Andrew Adamatzky, Christopher R. Stephens, Alejandro F. Hoeflich (2011) "Cellular automaton supercolliders," International Journal of Modern Physics C 22(4), 419-439. Fangyue Chen, Weifeng Jin, Guangrong Chen, & Lin Chen (2012) "Chaos emerged on the “edge of chaos”," International Journal of Computer Mathematics, by publish. "Rule 110 repository" http://uncomp.uwe.ac.uk/genaro/Rule110.html Dynamics in Rule 110 Rule 110 is an elemental cellular automaton. The local function determining the behaviour is: Random evolution in Rule 110. Initial density 50% with a ring of 539 cells for 568 generations. Set of particles in Rule 110 Formal languages and particles as strings The formal languages theory provides a way to study sets of chains from a finite alphabet. The languages can be seen as inputs for some classes of machines or as the final result from a typesetter substitution system i.e., a generative grammar into the Chomsky's classification. Lyman P. Hurd (1987) "Formal Language Characterizations of Cellular Automaton Limit Sets," Complex Systems 1, 69-80. Stephen Wolfram (1984) "Computation Theory on Cellular Automata," Communication in Mathematical Physics 96, 15-57. John E. Hopcroft & Jeffrey D. Ullman (1987) Introduction to Automata Theory Languajes, and Computation, Addison-Wesley Publishing Company. Collaborative Research Center SFB 676, "Particles, Strings, and the Early Feynman diagram Universe", Universität Hamburg , http://wwwiexp.desy.de/sfb676/ Particles as strings in CA: the de Bruijn diagram For an one-dimensional cellular automaton of order (k,r), the de Bruijn diagram is defined as a directed graph with k2r vertices and k2r+1 edges. The vertices are labeled with the elements of the alphabet of length 2r. An edge is directed from vertex i to vertex j, if and only if, the 2r-1 final symbols of i are the same that the 2r-1 initial ones in j forming a neighbourhood of 2r+1 states represented by i # j. In this case, the edge connecting i to j is labeled with "(i # j). The connection matrix M corresponding with the de Bruijn diagram is as follows: Harold V. McIntosh (1991) "Linear cellular automata via de Bruijn diagrams," http://delta.cs.cinvestav.mx/~mcintosh/ oldweb/pautomata.html Harold V. McIntosh (2009) One Dimensional Cellular Automata, Luniver Press. Burton H. Voorhees (1996) Computational analysis of one-dimensional cellular automata, World Scientific Series on Nonlinear Science, Series A, Vol. 15. Particles as strings in CA: the de Bruijn diagram Paths in the de Bruijn diagram may represent chains, configurations or classes of configurations in the evolution space. Now we must discuss another variant where the de Bruijn diagram can be extended to determine greater sequences by the period and the shift of their cells in the evolution space in Rule 110. A problem is that the calculation of extended de Bruijn diagrams grows exponentially with order k2rn $ n ! Z+. de Bruijn diagram for Rule 110 Particles as strings in CA: attractors class IV: very long transients, very long periodic attractors low in-degree, low leaf density (complex dynamics). Particles as strings in Rule 110 Regular language in Rule 110 (2001-2004) particle A [111110] = A(f1_1), 6 cells, 1l-0r [11111000111000100110] = A(f2_1), 20 cells, 2l-3r [11111000100110100110] = A(f3_1), 20 cells, 3l-2r A(f4_1) = A(f1_1) particle B [11111010] = B(f1_1), 8 cells, 1l-1r [11111000] = B(f2_1), 8 cells, 2l-0r [1111100010011000100110] = B(f3_1), 22 cells, 3l-3r [11100110] = B(f4_1), 8 cells, 0l-2r ... Doug Lind, “Structures in rule 110,” In Cellular Automata and Complexity: Collected Papers, Stephen Wolfram, Table of Properties, page 577, http://www.stephenwolfram.com/publications/articles/ca/86-caappendix/16/text.html Genaro J. Martínez, Harold V. McIntosh, Juan C. Seck Tuoh Mora, & Sergio V. Chapa Vergara (2008) "Determining a regular language by glider-based structures called phases fi_1 in Rule 110," Journal of Cellular Automata 3(3), 231-270. "Regular language in Rule 110" (2004) http://uncomp.uwe.ac.uk/genaro/rule110/listPhasesR110.txt Collisions in Rule 110 producing a producing a producing a particle generator gap of 28 cells meta particle Genaro J. Martínez & Harold V. McIntosh (2001) "ATLAS: Collisions of gliders like phases of ether in Rule 110," http:// uncomp.uwe.ac.uk/genaro/Papers/Papers_on_CA.html Genaro J.
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