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

2. 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.

3. 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”

4. C. Elementary CA Definition : We refer to CA as a tuple 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({S i } iЄN t ) → S k t+1

5. Theorem : Every n-D CA can be decomposed in 3 ???? linear mappings. Proof : ● Perform dimensional reduction by introducing a disconnected neighborhood......L n-2 L n-1 L n L n+1...

6. ● Let the unfolded one-dimensional representation correspond to a L n long configuration vector S t 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 L n x L n circulant Toeplitz matrix with rows [... 0 1 b 2... 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

7. ● 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..........

8. Continuous generalisation A “Self-Modulator” “Rule” signal E t C YtYt StSt h R ● Y(ω) = C(ω)S(ω)Ordinary Filter ● S(ω) = E(ω, Y)r(ω) Const. Input Adaptive Filter

9. D. Inverting the Non-Invertible ● Origin of Irreversibility: Varying Kernel of 3 rd 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!

10. .... Y t n................ Y t n+3................ Y t n+6.... 1 st Sublattice........... Y t n+1................ Y t n+4................ 2 nd Sublattice................... Y t n+2................ Y t n+5........ 3 rd Sublattice Y t n+1 = R([2 -1 Y t-1 n ] +4[Y t-1 n+3 ]mod2) GnGn+3Gn+6Gn+9 Gn+1Gn+4Gn+7 Gn+2Gn+5Gn+8 3-step time corresponds to a Shift of Logic Gates

11. Examples of Gate Definition Reversible-ANDReversible-XOR 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 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

12. 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: ( 2 3 )! = 40320 mappings possible ● 2D binary: (2 9 )! ● 3D binary: (2 27 )!

13. ● 1-D Examples AND – RCA XOR - RCA Random Permutations

14. 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 – n b - ||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.

15. k μ

16. 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

17. 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.