REVERSIBLE CELLULAR AUTOMATA WITHOUT MEMORY

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REVERSIBLE CELLULAR AUTOMATA WITHOUT
MEMORY Theofanis Raptis Computational
Applications Group Division of Applied
Technologies NCSR Demokritos, Ag. Paraskevi,
Attiki, 151 35
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• 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.

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• 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

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• C. Elementary CA
• Definition We refer to CA as a tuple ltL, S, N,
  Rgt 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 Si ? S of
  arbitrary topology.
• R a discrete map (Transition Table)

R(Si i?N t) ? Skt1
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• 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 Ln1...
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• 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 ... bN-1 0 ...
• Let g be a mapping from the Address Space to
  the Pointer Space of unit vectors of length
  bN defined by the correspondence

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• Let R be a varying kernel mapping from the
  constant Rule vector back to the Configuration
  Space
• Dynamics equivalent to the sequence

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 ..........
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Continuous generalisation
A Self-Modulator
R
Yt
h
St

• Y(?) C(?)S(?) Ordinary Filter
• S(?) E(?, Y)r(?) Const. Input Adaptive
  Filter

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• 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!

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.... Ytn ................ Ytn3
................ Ytn6 .... 1st Sublattice
........... Ytn1 ................ Ytn4
................ 2nd Sublattice
.................. Ytn2 ................ Ytn5
.... .... 3rd Sublattice
.
Ytn1 R(2-1Yt-1n 4Yt-1n3 mod2)
3-step time corresponds to a Shift of Logic Gates
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Examples of Gate Definition Reversible-AND Rever
sible-XOR 0 0 0 0 0 0 0 0 0 0 0 0 1
0 0 1 0 0 1 0 0 1 0 1 0 1 0 0 1 0
0 1 0 0 1 1 1 1 0 1 1 1 1 1 0 1 1
0 0 0 1 0 0 1 0 0 1 0 0 1 1 0 1 1 0 1
1 0 1 1 0 0 0 1 1 0 1 1 0 1 1 0 1
0 1 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
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• 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) bN
• Rules possible b(R)
• (b number of alphabet symbols, N Nearest
  Neighbours)
• N (2r1)D for a symmetric local
  Neighborhood
• RCA Rule Space cardinality (R)!
• 1D binary (23)! 40320 mappings possible
• 2D binary (29)!
• 3D binary (227)!

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• 1-D Examples
• AND RCA XOR - RCA

Random Permutations
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• F. Statistical Mechanics of RCA. Is it possible?
• Need for appropriate parametrisation of Rule
  Space
• Introduce a new parameter k analogous to
  Langton's ? in ordinary CA
• k 1 nb - N , k ? 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.

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k
µ
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• G. Applications
• Possible implementation of the composite mapping
  hgR
• All-optical implementation of h
• Problem with gR 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

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• 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.