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.