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