Self-replication at the edge of chaos

Eleonora Bilotta a Antonio Lafusa a Pietro Pantano a

aCentro Interdipartimentale della Comunicazione, Universita della CalabriaCubo 17, 87036 Arcavacata di Rende (CS) Italy

Abstract

Our objective, in this work, is to verify the ”edge of chaos” hypothesis in the space ofcellular automata, and in particular for systems in which self-replicating structuresemerge from a random initial condition.

In order to find the critical region, we develop a parameterization of the cellularautomata rule-space. The results clearly identify an abrupt transition between orderand chaos, in the proximity of critical parameters values, in which complex systemsare likely to be found.

Key words: Cellular automata, edge of chaos, self-replicating structures,complexity

1 Introduction

According to the ”edge of chaos” hypothesis formulated by Langton (2; 3), inthe space of dynamical systems of a given type there generically exist regions ofordered behavior and regions of chaotic behavior. Near the boundaries betweenorder and chaos complex systems, neither simple nor chaotic, are likely to befound.

Our objective, in this work, is to verify the edge of chaos hypothesis inthe space of cellular automata, and in particular for systems in which self-replicating structures emerge from a random initial condition (primordialsoup).

In detail, in the space of all possible rules (i.e. the rule-space), we will verifythat :

a) There exist a boundary between two distinct regions: the region of orderedrules and the region of chaotic rules. This separatrix corresponds to asharp transition from order to chaos (or vice versa).

Preprint submitted to Elsevier Science 10 October 2003

b) Complex systems are likely to be found in this transition. As suggested byLangton, in analogy with phase transitions, complex systems correspondto a critical transition (or second-order transition), that can be identifiedby means of critical parameters.

c) Systems with spontaneous emergence of self-replicating structures arelikely to be found in the critical region corresponding to the edge ofchaos.

From a technical point of view, considering multi-value cellular automata, wewill introduce a parameterization of the rule-space, taking two measures:

I) an order-chaos index;II) a complexity index.

The results of the first measure, in the parameter-space, will clearly identifythe ordered and chaotic regions, and their boundary. The second measure willconfirm that this boundary is also characterized by a sharp peak of complexity.

In our previous works, we proved that in the same rule-spaces, many com-plex systems allow spontaneous formation of self-replicating structures (1).Finally, the results obtained for self-replication phenomena, combined withthe evidence of the edge of chaos outlined in this work, let us conclude thatLangton’s hypothesis also holds for these kind of systems.

2 Cellular Automata and k-totalistic rules

Let us now introduce the cellular automata (CA) rule-space, the statisticalmeasures and the parameters we used.

Cellular Automata are dynamical systems in which space and time are discrete.A CA consists of a regular lattice of cells, in which each cell can be in one ofk states, changing its state according to a local rule.

The environment considered is a two-dimensional CA.

The rule can be thought as a function that maps from the states of the neigh-bors of a cell to the new state of the cell. The neighborhood of a cell is identifiedby means of a local radius r, and consists of I = (2r + 1)2 cells (including thecell itself).

An exhaustive rule considers all k(2r+1)2 possible cases. With the increase of k

and r, the exhaustive rules become non-treatable and it is necessary to definea new simpler form.

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Here we define a new form for the rule - called hereafter k-totalistic - thatdoesn’t distinguish the position of neighbors in the surrounding area, but justconsiders how many cells are in a given state.

Let hs(t) be the number of cells of the neighborhood that are in the state s attime t. We denote with V the set of all possible configurations of the neighbor-hood, whose elements can be represented by a numerical string (h0h1...hk−1).hi are positive integers and must satisfy the following constraint:

h0 + h1 + ... + hk−1 = (2r + 1)2 (1)

Let Σ be the set of states, from 0 to k − 1. By definition a k-totalistic rule T

is an application that associates to any configuration v ∈ V a Σ element:

T : V −→ Σ (2)

To estimate the number of all possible configurations, we consider the problemas being the same as placing I undistinguishable objects in k containers. Fromelements of combinatorial analysis, such a number is given by:

L(k, I) =

k + I − 1

I

=(k + I − 1)!

I!(k − 1)!(3)

The application (2) can be explicitly represented as a table that associates tothe first k − 1 totals (for the constraint (1), the last of which depends on theothers), a number included between 0 and k − 1. Alternatively, by fixing anorder in the set V, a k-totalistic rule can be represented as a numeric string,of length L = L(k, I):

l = (s1, s2, ..., sL) (4)

with si ∈ S. To make a configuration (h0h1...hk−1) ∈ V correspond to a si

value of the string (4), we fix the convention that si corresponds to the ith

configuration in lexicographical order.

3 Ordered, chaotic and complex dynamics: Automatic classifying

cellular automata with input-entropy

The input-entropy is an useful measure that we applied to the k-totalistic rules(in (1)), following the same method devised by Wuensche (6) for exhaustive

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(a)

(b)

(c)

Fig. 1. Input-entropy plot starting from a random initial state, for 400 time-steps,considering a lattice of 60x60 cells. In (a) an ordered rule, in (b) a chaotic rule andin (c) a complex rule. It is possible to note that in (c) the input-entropy remarkablyvaries over time.

rules. For a step of simulation, the input-entropy St is defined as:

St = −

L∑

i=1

(

Qti

nlog

(

Qti

n

))

(5)

where n is the number of cells in the CA lattice and Qi is the look-up frequencyof the ith neighborhood configuration. We verified that the input-entropy onk-totalistic rules works in a similar way as found by Wuensche for exhaustiverules. In ordered systems, it goes immediately to a minimum constant value; inchaotic systems, it stays constantly on high values, with very little variations.On the contrary, in complex systems, (5) remarkably varies over time (Fig. 1).

The mean value of the input-entropy can be considered as a mean order-chaos level, that is near 0 for ordered rules and near the maximum valuefor completely chaotic rules. Therefore the mean value (that we used as anorder-chaos index) is suitable for identify ordered and chaotic rules, but not

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for complex rules, that are identified by means of erratic variations of input-entropy over time.

From these properties of input-entropy, we define an index of complexity thatput in evidence the typical variations of the function over time. In order tocalculate this index, starting from a random initial state, we take trace of thefollowing values of S(t):- the absolute minimum Smin:

Smin = min St

- the absolute maximum Smax, calculated from the minimum point tmin:

Smax = max St; t ≥ tmin

- the absolute minimum S ′

min, calculated from tmax:

S ′

min = min St; t ≥ tmax

We define the index I1 as the sum of the width of two jumps, from tmin totmax and from tmax to t′min

I1 = (Smax − Smin) + (Smax − S ′

min) (6)

The higher the input-entropy variations are, the greater I1 will be.

In order to calculate the input-entropy function, each rule is simulated startingfrom a random initial state (in which each cell has the same probability tobe in any state), for 350 time-steps, using a lattice of 40x40 or more cells,and ignoring the first 50 time-steps to allow the CA to adhere to its typicalbehavior. Moreover, in order to smooth the measure, we do not estimate theentropy for each time step, but we take a mean of the Qi frequencies in a timewindow of 5 steps.

4 Evolving cellular automata toward increasing complexity and

consequential discovery of self-replicating structures

In (1) we used an evolutionary process driven by this complexity index, withthe consequential discovery of complex CA. In many systems discovered withthis method, self replicating structures emerge from a chaotic (i.e. completelyrandom) initial condition.

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Fig. 2. A self-replicating structure (t=0), using an evolved rule with k = 4 and r = 2.After 8 time-steps the first structure moves down and generates another structureon the left. After 16 time-steps, two identical copies of the original structure arepresent. At time t=40, the generation of a colony of self-replicating structures andother secondary structures.

Fig. 3. On the left, a self-replicating structure (k = 5, r = 1); on the right, fouridentical structures are generated after 48 time-steps.

From the theory of the edge-of-chaos we expect these systems, and complexsystems in general, to be concentrated near a critical region of the rule-spacein which there is an abrupt transition between order and chaos.

In the next paragraphs we define several mathematical tools for studying thewhole structure of the rule-space. The empirical results will show that complexsystems are likely to be found in the boundary between two distinct regions:the region of chaotic rules and the region of ordered rules.

In figures 3 and 2, examples of self-replicating structures in evolved rules.

5 Parameterization of the rule-space

We will now define a parameterized survey of the k-totalistic rule-space, similarto the mean field theory parameterization (5; 4), but in a simpler form, to dealwith multi-value CA.

Let q ∈ S be an arbitrary quiescent state, and T a k-totalistic rule. Without

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loss of generality, we choose q = 0.

For the k-totalistic rule T , we define the parameter aj as the number of con-figurations (h0h1...hk−1) ∈ V which map to a non-quiescent state and containsj cells at a non-quiescent state (i.e. hq = h0 = I − j). In this definition, theconstraint is 0 ≤ j ≤ I and therefore there are I + 1 parameters.

The maximum value for the jth parameter is given by the number of configura-tions containing j cells at a non-quiescent state (and I−j cells at the quiescentstate), which is L(k − 1, j). Worth noticing, this number is equal to the num-ber of configurations of a k-totalistic rule with k − 1 states and j neighbors,because we have to allocate j neighbors in the remaining non-quiescent k − 1states. The minimum value for each parameter is 0 and therefore:

0 ≤ aj ≤ L(k − 1, j) (7)

If we consider the string (4) that represents the rule T , the parameter aj canbe calculated counting the number of non-quiescent states from the positionp1(j) to the position p2(j) (we fix the position of the first element of the stringto 0), where :

p1(j) =I∑

x=j+1

L(k − 1, x) (8)

and

p2(j) = p1(j) + L(k − 1, j) − 1 (9)

The formulas for p1 and p2 are easily obtained looking at the string represen-tation of the k-totalistic rule. We can divide the string in contiguous blocks :the first block corresponds to the configurations for which h0 = 0 (relative tothe parameter aI); for the second block h0 = 1 (relative to aI−1); and so on.

In analogy with Gutowitz and Langton (4), each setting of parameter valuescorresponds to a subset of the rule-space. Small perturbations to parametervalues generally lead to small changes of the dynamical regime, but in thevicinity of critical values small changes lead to an abrupt change in behavior,as showed in the next section.

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6 Verifying the edge of chaos hypothesis with experiments

We considered k=4 and r=1 (I=9) rule-space, varying the a3, a4 and a5 pa-rameters, and setting the other parameters to 0. In particular, we generatedsamples of rules varying (a3, a4), (a3, a5) and (a4, a5), by keeping each timethe third parameter to the maximum value.

A sample of 100 rules were generated for each parameters setting, takingthe average order-chaos degree and the average complexity index (i.e. I1). Toevaluate the order-chaos degree for a given rule, we took the mean value ofthe input-entropy over a simulation.

The results in Fig. 4 show that there is an abrupt transition from the or-dered region (mean input-entropy near 0) and the chaotic region (mean input-entropy near the maximum value).

Fig. 5 shows that a complexity peak occurs in the same critical region betweenorder and chaos.

We can identify the critical parameters values with an (ideal) plane that in-tersects the three-dimensional parameter-space (a3, a4, a5). In fact, when weconsidered only two parameters at a time, in all three cases, two-dimensionalplots show that complex rules are identified by means of critical parametersvalues, forming a straight line (see Fig. 5 and Fig. 6). Moreover, we verifiedthat when an internal slice is obtained, for example varying (a4, a5) and keep-ing a3 to an intermediate value, complex rules are found as expected in theintersection between the ”critical plane” and the plane a3 = constant.

This result also suggests that if we consider the direction perpendicular to thisideal plane, there will be a single parameter (a linear combination of the threeconsidered parameters), that identifies the critical transition with a uniquecritical value.

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(a)

(b)

Fig. 4. Mean input-entropy, varying a4 and a5 with a3 = a3max. (a) 3D view; (b)2D view. For each parameters setting, the height of the surface corresponds to themeasure averaged over a sample of 100 rules. To be noted is the abrupt transitionbetween the ordered region (mean input-entropy near 0) and the chaotic region(mean input-entropy near to the maximum value). The boundary between thesetwo regions describes a straight line, and corresponds to the edge of chaos.

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(a)

(b)

Fig. 5. Complexity index I1, varying a4 and a5 with a3 = a3max. (a) 3D view; (b)2D view. For each parameters setting, the height of the surface corresponds to theindex I1 averaged over a sample of 100 rules. To be noted is the sharp peak ofcomplexity in the same critical transition identified in Fig.4.

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(a)

(b)

(c)

Fig. 6. Complexity index I1: (a) varying a3 and a4 with a5 = a5max; (b) varying a3

and a5 with a4 = a4max. As in Fig.5, the position of complex rules suggests thatcritical parameter values correspond to a straight line. In (c), considering the threedimensional (a3, a4, a5) parameter-space, the critical parameter values correspondto a plane that intersects the parallelepiped. The three points which identify the”critical plane” are : P1 = (10, 4, 21), P2 = (6, 15, 21) and P3 = (10, 15, 0).

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7 Conclusions

As pointed out in a) and b), these empirical results confirm the existence of anedge of chaos in the considered cellular automata space. From the other hand,we have demonstrated in (1) that many rules, in the complexity peaks, allowthe formation of self-replicating structures from an initial chaotic condition,showing a strong relationship between self-replication and phase transitions.

References

[1] E. Bilotta, A. Lafusa, P. Pantano (2002) Is self-replication an embeddedcharacteristic of the artificial/living matter? ALife VIII.

[2] C. G. Langton (1986) Studying artificial life with cellular automata. Phys-ica D, (22):120-149

[3] C. G. Langton (1990) Computation at the edge of chaos: Phase transitionsand emergent computation. Physica D, (42):12-37

[4] H. A. Gutovitz, C. G. Langton (1995) Mean field theory of the edge ofchaos. ECAL 1995, 52-64

[5] H. A. Gutovitz, J. D. Victor, B. W. Knight (1987) Local structure theoryfor cellular automata. Phisica D, (28):18-48

[6] A. Wuensche (1999) Classifying cellular automata automatically; findinggliders, filtering, and relating space-time patterns, attractor basins, and theZ parameter. Complexity, (4): 47-66.

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