Cellular Automata Theory

Home , Automata theory, Matthew Cook

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 = 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.

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. 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.

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. Two brilliant minds: Universal Computation and Universal Constructors

Alan M. Turing (1912-1954) John von Neumann (1903-1957) CA as supercomputers (unconventional computing with CA)

Norman Margolus, Tommaso Toffoli, & Gérard Vichniac (1986) Cellular-Automata Supercomputers for Fluid Dynamics Modeling, Physical Review Letters 56(16) 1694-1696. Stephen Wolfram (1988) Cellular Automata Supercomputing, In: High Speed Computing: Scientiﬁc Applications and Algorithm Design, R. B. Wilhelmson (Ed.), University of Illinois Press, 40-48. Toffoli Tommaso (1998) Non-Conventional Computers, In: Encyclopedia of Electrical and Electronics Engineering, J. Webster (Ed.), Wiley & Sons, 455-471. Anthony J. G. Hey (1998) Feynman and computation: exploring the limits of computers, Perseus Books. Andrew Adamatzky (Ed.) (2002) Collision-Based Computing, Springer-London. Tommaso Toffoli & Norman Margolus (1987) Cellular Automata Machine, The MIT Press. John von Neumann (1966) Theory of Self-reproducing Automata (edited and completed by A. W. Burks), University of Illinois Press, Urbana and London. Moshe Sipper (1997) Evolution of Parallel Cellular Machines: The Cellular Programming Approach, Springer. Genaro J. Martínez, Andrew Adamatzky, Kenichi Morita, & Maurice Margenstern (2010) Computation with competing patterns in Life-like automaton, In: Game of Life Cellular Automata, A. Adamatzky (Ed.), Springer, 547-572. 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. Kenichi Morita (2008) Reversible computing and cellular automata--A survey, Theoretical Computer Science 395, 101-131 Symbol super colliders in CA and lattice gas

Tommaso Toffoli 1943 - ?

Tommaso Toffoli & Norman Margolus (1987) Cellular Automata Machine, The MIT Press.

CAM8: a Parallel, Uniform, Scalable Architecture for Cellular Automata Experimentation http://www.ai.mit.edu/projects/im/cam8/ Symbol super colliders in CA and lattice gas To map Toffoli's supercollider onto a one-dimensional CA we use the notion of an idealized particle p ! Z+ (without energy and potential energy). The particle p is represented by a binary string of cell states.

Representation of abstract particles in a one-dimensional CA ring

Figure shows two typical scenarios where particles pf and ps travel in a CA cyclotron. The ﬁrst scenario (Fig. a) shows two particles travelling in opposite directions which then collide. Their collision site is shown by a dark circle in (Fig. a). The second scenario demonstrates a typical beam routing where a fast particle pf eventually catches up with a slow particle ps at a collision site (Fig. b). If the particles collide like solitons, then the faster particle pf simply overtakes the slower particle ps and continues its motion (Fig. c). Typically, we can ﬁnd all types of particles manifest in CA gliders, including positive p+, negative p-, and neutral p0 displacements, and also composite particles assembled from elementary particles.

Tommaso Toffoli (2002) "Symbol Super Colliders," In Collision Based-Computing, A. Adamatzky (Ed.), 1-22. Supercolliders in CA

Transition between two beam routing synchronizing multiple reactions. When the first set of collisions are done a new beam routing is defined with other particles, so that when the second set of collisions is done then one returns to the initial condition of the first beam, constructing a meta-glider or mesh in Rule 110.

In this way, we can design more complex constructions synchronizing multiple collisions with a diversity of speeds and phases on different particles. Figure displays a more sophisticated beam routing design, connecting two of beams and then creating a new beam routing diagram where edges represent a change of particles and collisions contact point on ECA Rule 110. In such a transition, a number of new particles emerge and collide to return to the ﬁrst beam, thus oscillating between two beam routing forever. changing to the set of particles (second beam routing): deﬁning two beam routing connected by a transition of collisions as: Some simulations implemented in Discrete Dynamics Lab (DDLab) A free software created by Andrew Wuensche http://www.ddlab.com/ Algunas simulaciones implementadas en Discrete Dynamics Lab (DDLab - http://www.ddlab.com)

The next collisions form a cycle of reactions,

C1 <- F = F + C1 C1 <- F = E- + C2 C2 <- E- = F + C1

Hence we can coded an initial condition as a regular expression, as follows:

4e-C1(A,f1_1)-e-F(A,f1_1)-4e

or as a binary string, that is:

(11111000100110)^4-111110000-11111000100110-111110001011010-(11111000100110)^4

The cyclic reaction is:

{(C1 <- F) -> (C1 <- F) -> (C2 <- E-)}*

implementado un choque soliton Algunas simulaciones implementadas en Discrete Dynamics Lab (DDLab - http://www.ddlab.com)

implementado un choque soliton Supercolliders in CA

yielding a meta particle by supercolliders

{A(f3_1)-e-A(f1_1)-e-[B-](C,f1_1)-e-2B (f4_1)}*

[e+]-A^4(f3_1)-8e-A(f3_1)-e-A(f1_1)-8e-A^4(f1_1)-7e-A(f1_1)-e-A (f2_1)-2e-[B-](A,f1_1)-e-B(f1_1)-B(f4_1)-E(B,f2_1)-[E-](G,f1_1)-5e- [B-](C,f1_1)-e-B(f1_1)-B(f4_1)-e-E(A,f2_1)-[E-](A,f4_1)-[e+] Algunas simulaciones implementadas en Discrete Dynamics Lab (DDLab - http://www.ddlab.com)

implementado una meta partícula Algunas simulaciones implementadas en Discrete Dynamics Lab (DDLab - http://www.ddlab.com)

implementado una meta partícula Universality and the cyclic tag system in Rule 110

As an advance of our present work, we show the cyclic tag system inside the evolution space of Rule 110. This incredible result is reconstructed using our regular language for Rule 110. You can ﬁnd some differences from A New Kind of Science, because it has mistakes that do not allow a good reconstruction. The mistakes were clariﬁed by Matthew Cook in November 2002 (personal communication).

http://uncomp.uwe.ac.uk/genaro/rule110/ctsRule110.html

Writing the sequence 1110111 on the tape of the cyclic tag system and a leader component at the end with two solitons. Our reconstruction is developed over an evolution space of 56,240 cells in 57,400 generations, i.e., a space of 3,228,176,000 cells with a computer Pentium II to 233 mhz, operating system OpenStep and 256MB of RAM, February 2003. Collaborations of Harold V. McIntosh and Juan C. Seck-Tuoh-Mora.

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. Circular universal machines

We have two important previous results in computer science theory to think in circular computations.

! Circular Turing machines Michael A. Arbib, "Monogenic Normal Systems are Universal," Annual General Meeting of the Australian Mathematical Society, Sydney, August 17, 1962.

! Circular Post machines Manfred Kudlek and Yurii Rogozhin "Small Universal Circular Post Machines," Computer Science Journal of Moldova 9(1), 2001. Universality and the cyclic tag system in Rule 110 Supercolliders in ECA Rule 110 and its CTS

A diagram of a cyclic tag system (CTS) working in Rule 110

Beam routing codiﬁcation representing package of particles which reproduces a CTS in Rule 110 Supercolliders in ECA Rule 110 and its CTS

Beam routing machine transitions to simulate CTS in Rule 110

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. Cyclic tag system in Rule 110 as a supercollider

Project: implement such CTS into a supercollider in software and physically.

Thank you very much! 非常感謝!

International Center of Unconventional Computing (ICUC) http://uncomp.uwe.ac.uk/

Computer Science Laboratory (LCCOMP) http://uncomp.uwe.ac.uk/LCCOMP/