REVERSIBILITY AND CRITICALITY IN A DETERMINISTIC “SELF- EXTRACTING” BAK-SNEPPEN AUTOMATON Theofanis Raptis Computational Applications Group Division of Applied Technologies NCSR Demokritos, Ag. Paraskevi, Attiki,

A. The Bak-Sneppen Model of Evolution ● First introduced by Per Bak and Kim Sneppen to explain Punctuated Equilibrium and Extinction Events in biological Evolution [1],[2]. ● Model dynamics repeatedly eliminates the least adapted species and randomly mutates it and its neighbors to recreate the interaction between species. ● Characterised by a limiting distribution where all fitness minima are bounded from below by the same critical value. ● Distribution of Avalanche Sizes leads to a characteristic exponent which defines a Universality Class.

B. Reversible Evolution? ● Cosmic Egg Scenario: Assuming an initial device full of nano-bots arrives on Marsh, opens and spreads several thousands nanomechanisms that start reproducing and spreading according to a prespecified protocol, also capable of controllable mutations. Under what conditions would it be possible for the initial device to send a stop-signal, invert the evolution and gather all nanos back to the source in their initial state, ready to go for another planet? ● In order to reconcile reversibility and criticality we seek after a deterministic, fully reversible automaton upon which we emulate the original Bak-Sneppen algorithm. ● In [4] we presented a new type of reversible automaton, based on permutation gates originally invented for cryptographic applications.

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

● RCA dynamics is decomposable in two consecutive invertible mappings acting on all binary triplets of a 1D Lattice. where Pk is an element of the Lexicographic Permutation Group and SL,SR Left and Right Shifts by 1 bit respectively

● D. “Self – Extracting” Automata (SEA) 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 of radius r. RCA Rule Space cardinality: #(R)! ● 1D binary: ( 2 3 )! = mappings possible ● 2D binary: (2 9 )! ● 3D binary: (2 27 )! ● Possibility of separate control of different Rules in different areas of the Lattice (Self Modifying Finite Automata) [5],[6],[7]. ● Assigning control of local Rules to a special function (Interpreter) of the Lattice cells leads to a Self-Extracting or “self-referential” Automaton.

Abstract Definitions ● Ordinary Turing Automaton U( p, x ) -> y where integers p and x stand for the “program” and “input” respectively. ● SEA: U( ] x [, x ) -> y where ] x [ = f(|x|) is the “Interpretation” of current |x| as a program. ● Reducibility of finite SEA: Let x E S and f:S -> S', with S and S' finite. Then, there always exists an isomorphic Turing Automaton with a SuperRule R:SΧS' -> SΧS':U( R, x ) -> y

E. Modified Bak-Sneppen Algorithm ● Let C be a lattice with N cells. Each cell encodes a binary triplet in the octal alphabet. ● Let N = 4k, where k is a number of registers {Ri}i=1,k each containing 4 cells. Initial distribution of values is assigned to the registers in the interval [1,4096]. ● Pick up min{Ri} (alternatively max{Ri}) and update the three consecutive registers {Ri-1,Ri,Ri+1} according to where Pj is a permutation indexed in the lexicographic order applied at all cells c forming register Ri at time t. ● A “Self-Extracting” automaton corresponds to j(i) = f(|Ri|) at every time step t.

● We run the algorithm for the simple choice j(i) = |Ri| where we observe the formation of avalanches shown in figure 1. (Lattice length = 200, time = 10000) Time Sites Fig 1

F. Reversibility ● In order to achieve a 1-1 mapping between curent register values and next ones, we isolated a special subgroup of permutations such that all pairs are unique. The subset of 4096 permutations is shown graphically in figure 2 and the composite map in figure 3. Chaotic nature of this map is the equivalent of the random number generator used in the original algorithm of Bak and Sneppen. ● Inversion algorithm differs. First pick Min or Max of current configuration. Then pick Min/Max of all preimages and replace if lesser than present Min/Max else choose nearest preimage.

Fig 2 Fig 3

● Avalanches still form but with a different limiting distribution. Characteristic exponents are shown in figures 3 and 4 (-4.3 and -0.8 respectively) from the Avalanche Size Distribution in log-log scale. Fig 3 Fig 4

Evolution of the initial distribution for both forward and backward evolution are shown in figures 5 and 6 Fig 5 Fig 6

G. Conclusions ● Deterministic analogues of self-organised criticality do exist ● Evolution in certain cases can be run “backwards” even though exact reproduction of avalanches may not be meaningful. Distribution can be made to “spread” back to a more uniform one. ● In principle a system could be build capable of altering its internal states towards a more or less organized configuration according to external signals that would trigger an appropriate “switch” function. ● Possible application in “Extremal Optimization” for escaping local minima.

References [1] M. Paczuski, S. Maslov, “Avalanche dynamics in evolution, growth and depinning models”, P. Bak, Phys. Rev. E 53 (1996). [2] S. Boetcher, M. Paczuski, “Exact results for Spatiotemporal Correlations in a Self-Organized Model of Punctuated Equilibrium”, Phys. Rev. Lett. 76 (19). [3] S. Boetcher, A. Percus, “Nature's way of optimizing”, Artificial Intelligence, V. 119, I. 1-2 (2000) [4] T. Raptis, “Reversible Cellular Automata without memory”, 19 th Summer School on Nonlinear Science and Complexity, [5] J. Shutt, R. Rubinstein, “Self-Modifying Finite Automata”, Information Processing Letters 56, N. 4 (1995)