Cellular Automaton Supercollider: An abstract model

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

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

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?

A kaleidoscope patterns

Mark Gibson-Corbis Encyclopaedia Britannica

Sir David Brewster (Scot. Brit.), the physicist who invented de kaleidoscope in 1816

Cellular Automata

Cellular automata

(CA) are discrete dynamical systems evolving on an

infinite regular lattice.

A

CA

is a 4-tuple

A

= <

!

, µ,

"

,

c

> evolving in

d

-dimensional lattice,

where

d

!

Z

. Such that:

•

!

represents the finite

alphabet

•

µ is the

local connection

, where, µ = {

x

|

x

!

!

}, therefore, µ is a

neighbourhood

•

"

is the

local function

, such that,

"

:

!

"

!

•

c

is the

initial condition

, such that,

c

!

!

Also, the local function induces a

global transition

between

configurations:

CA dynamics in one dimension

Elemental CA (ECA) is defined as follows:

•

!

= {0,1}

•

µ = (

x

x

x

) such that

x

!

!

•

"

:

!

"

!

•

µ = {

c

|

x

!

!

} the initial condition is the

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

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.

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:

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.

Formal languages and particles as strings

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

For an one-dimensional cellular automaton of order (

k

,

r

), the

de Bruijn diagram

is defined as a

directed graph with

k

vertices and

k

edges. The vertices are labeled with the elements of

the alphabet of length 2

r

. An edge is directed from vertex

i

to vertex

j

, if and only if, the 2

r

-1

final symbols of

i

are the same that the 2

r

-1 initial ones in

j

forming a neighbourhood of 2

r

+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.

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

k

$

n

!

Z

.

Particles as strings in CA: the de Bruijn diagram

class IV: very long transients, very long periodic attractors low in-degree, low leaf density (complex dynamics).

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

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.

Collisions in Rule 110

producing a particle generator

producing a gap of 28 cells

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

Symbol super colliders in CA and lattice gas

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

p

and

p

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

p

eventually catches up with a slow particle

p

at a

collision site (Fig. b). If the particles collide like solitons, then the faster particle

p

simply

overtakes the slower particle

p

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

p

displacements, and also composite particles assembled from elementary particles.

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

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

implementado un choque soliton

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:

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

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

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

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

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

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

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.

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.

Beam routing codification representing package of particles which reproduces a CTS in Rule 110 A diagram of a cyclic tag system (CTS) working in Rule 110

Beam routing machine transitions to simulate CTS in Rule 110

Supercolliders in ECA Rule 110 and its CTS

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.

Cyclic tag system in Rule 110 as a supercollider

Project:

implement such CTS into a supercollider in

The End

Thank you very much!

非常感謝

!

International Center of Unconventional Computing (ICUC)

http://uncomp.uwe.ac.uk/