reversible cellular automata without memory theofanis raptis computational applications group...

REVERSIBLE CELLULAR AUTOMATA WITHOUT MEMORY

Theofanis Raptis

Computational Applications GroupDivision of Applied Technologies

NCSR Demokritos, Ag. Paraskevi, Attiki, 151 35

A. Cellular History

● First CA introduced by John von Neumann in the 50's as an abstract model of self-replication.● Later used by Edward Fredkin to introduce the idea of “Digital Mechanics” in the 60s.● John Conway's Game of Life at 70s.● Revival after Stephen Wolfram's classic paper at 84 on the properties of elementary 1-D CA.● Several classes of CA proven capable of Universal Computation (equivalence with a Universal Turing Machine) including the Game of Life.●Possibility of a CA computer extensively discussed after Toffoli and Margolus work based on Fredkin ideas.●Japanese company announced the first CA asynchronous computer possible in 5 years based on work by Morita, Matsui and Pepper.

B. Why Reversibility?

● Fredkin' s view on the exact transcription of all physical laws on a computational substrate required reversibility.● Landauer theorem: heat dissipation or entropy production in a logical circuit due to irreversibility of classical logic gates (bit erasure) ●[Bennet 88] “To erase 1 bit of classical information within a computer, 1 bit of entropy must be expelled into the computer's environment (waste heat)”● First classical reversible gates introduced by Fredkin and Toffoli ● Billiard Ball Model of computation (BBM) as a special type of classical CA.● Possibility of “Cold Computing”

C. Elementary CA

Definition : We refer to CA as a tuple <L, S, N, R> where

● L is a n-D lattice of Cell sites● S a set of Cell states with integer values in [0, b-1] (b symbols)● N a neighbourhood of lattice sites S

i Є S of arbitrary topology.

● R a discrete map (Transition Table)

R({Si }

iЄN t) → S

kt+1

Theorem : Every n-D CA can be decomposed in 3 ???? linear mappings.

Proof : ● Perform dimensional reduction by introducing a disconnected neighborhood.

.....Ln-2

Ln-1

Ln L

n+1...

● Let the unfolded one-dimensional representation

correspond to a Ln long configuration vector St containing the values of the lattice sites. ● Let h be a mapping from the initial Configuration Space to a new vector in the Address Space defined by

● C is a Ln x Ln circulant Toeplitz matrix with rows

[ ... 0 1 b2 ... b||N||-1 0 ... ]

● Let g be a mapping from the Address Space to the

Pointer Space of unit vectors of length b||N|| defined by the correspondence

|| ||: [0, 1] [0, 1] :N t th b b Y C S

: :tt t t

ig Y iYY E e

● Let R be a varying kernel mapping from the constant Rule vector back to the Configuration Space

● Dynamics equivalent to the sequence

1: t tR S E r

1... ...t t t t S Y E S

0 1 0 0 0 1 1 0 1 0 1 0 1 1 0 1 1 1 1

... 2 1 0 4 6 3 5 ..........

... 1 0 0 1 1 0 0 ..........

tS

tY

1t S

Continuous generalisation

A “Self-Modulator”

“Rule” signal

Et

C

Yt

Sth

R

● Y(ω) = C(ω)S(ω) Ordinary Filter● S(ω) = E(ω, Y)r(ω) Const. Input Adaptive Filter

D. Inverting the Non-Invertible

● Origin of Irreversibility: Varying Kernel of 3rd map irretrievable

● Alternative explanation:Mapping of const. Rule vector is a contraction from a higher to a lower symbolic alphabet (whole neighborhood mapped to single symbol)

● Correction: Retain the same number of input and output bits (neighborhood to neighborhood mapping)

● Obstacle: non-matching of resulting neighborhoods

●Remedy: 3-step time evolution!

.... Ytn ................

Yt

n+3 ................ Yt

n+6 ....

1st Sublattice

........... Ytn+1

................ Yt

n+4 ................ 2nd Sublattice

................... Ytn+2

................ Yt

n+5 .... .... 3rd Sublattice

Ytn+1

= R([2-1Yt-1n] +4[Yt-1

n+3 ]mod2)

Gn Gn+3 Gn+6 Gn+9

Gn+1 Gn+4 Gn+7

Gn+2 Gn+5 Gn+8

3-step timecorresponds to a Shiftof Logic Gates

Examples of Gate Definition

Reversible-AND Reversible-XOR0 0 0 | 0 0 0 0 0 0 | 0 0 01 0 0 | 1 0 0 1 0 0 | 1 0 10 1 0 | 0 1 0 0 1 0 | 0 1 11 1 0 | 1 1 1 1 1 0 | 1 1 00 0 1 | 0 0 1 0 0 1 | 0 0 11 0 1 | 1 0 1 1 0 1 | 1 0 00 1 1 | 0 1 1 0 1 1 | 0 1 01 1 1 | 1 1 0 1 1 1 | 1 1 1

Equivalent to permutations of the octant alphabet in the Address Space

AND : 0 1 2 7 4 5 6 3

XOR : 0 5 6 3 4 1 2 7

E. WHAT WE EARNED

● Each step totally reversible

● Time evolution of asymmetric patterns

● Enormous number of rules possible even for 1-D CA

Elementary CA Rule space cardinality: bits/Rule #(R)= b||N|| Rules possible b#(R)

(b = number of alphabet symbols, ||N|| = Nearest Neighbours)||N|| = (2r+1)D for a symmetric local Neighborhood

RCA Rule Space cardinality: #(R)!● 1D binary: (23)! = 40320 mappings possible● 2D binary: (29)!● 3D binary: (227)!

●1-D Examples AND – RCA XOR - RCA

Random Permutations

F. Statistical Mechanics of RCA. Is it possible?

● Need for appropriate parametrisation of Rule Space

● Introduce a new parameter k analogous to Langton's λ in ordinary CA

k = 1 – nb - ||N| , k Є [0,1]n = number of invariant addresses (fixed points) under permutations

● Introduce a measure μ of the number of independent cycles per permutation.

● Problem: most RCA have no fixed points. Insufficient information due to the presence of the Right Shift operator.

k

μ

G. Applications

● Possible implementation of the composite mapping h•g•R ● All-optical implementation of h ● Problem with g•R due to varying kernel● All-optical RCA-Machine?

● Problem: Find rules that immitate various logical circuits under various initial conditions● Possible solution by training via genetic algorithms

References● E. F. Codd, “Cellular Automata” (1968), Academic Press, NY.

● S. Wolphram, “ Universality and Complexity in Cellular Automata”, Physica D, 10, 135 (1984).

● A. Adamatzky, “Identification of Cellular Automata ”(1994), Taylor & Francis.

● K. Lindgren, M. Nordahl, “Universal Computation in simple One Dimensional Cellular Automata ”, Complex Systems, 4 (1990), 299

●T. Raptis, D. Whitford, R.T. Kroemer, “Applications of Cellular Automata and Dynamical Systems to the Identification and Reconstruction of Biological Sequences ”, EMBL-EBI Symposium on Gene Prediction, Cambridge, 2000.