Zygote and its Progeny

In his book, ''A New Kind of Science" (1), Stephen Wolfram describes a new modeling tool, called Cellular Automat (CA). Simple programs evolve in an unpredicted  fashion and become extremely complex. CA is particularly suitable for illustrating some characteristics of life, which cannot be modeled with other Artificial Life (AL) tools, e.g., neural networks (NN), or genetic algorithms (GA).

Life is complex, creative , optimal , and continually moves (changes). These characteristics will be illustrated here with CAs . specified by Wolfram. The following examples will apply two totalistic CAs with the respective rules , #357, #600. The first image illustrates the structure of a rule #600 CA. It originates in a seed which is always a 1. Each row represents a state of the CA. The last row is its present state. The picture depicts a CA trajectory which is also its history.

Rule #357

The picture depicts four CA histories. The first (marked by a 0), depicts a history of a single CA, which gradually grows, yet will soon die. The next history depicts two CAs whose seeds were 5 units apart from each other. After fusing, they became immortal. The next history depicts two CAs whose seeds were 22 units apart from each other. Both  grow along each other, and shortly before they die they interact (fertilize) each other and become immortal. In the last history the two CAs interact, gain mass (strength), and die.

The next graph illustrates how the distance between two seeds determines survival.

Rule #600

The picture depicts 5 CA histories. When the distance between two seeds is 30 units, the CAs are immortal. Yet when the seeds are placed 8 units apart, they die young (annihilation). The third history (marked by 13) shows that the two CAs fused, thrived for a while and then died. The following history starts with a single CA. At time = 30 a second seed is planted at a distance = 8. This interaction (fertilization) caused the CA to move to the left and establish a new niche where it remained until the end of time. . A less pronounced movement to the left  is observed also in the fourth history (13). The last CA was fertilized at t=30. It thrived for a while, gained in mass and then died.


CA rules are global and deterministic. After a seed is planted, its progeny oscillates in a strange attractor which it never leaves. This bleak fate changes dramatically when two CAs interact. A mortal #357 CA becomes immortal (5), or accumulates mass and lives somewhat longer (23). On the other hand  an immortal #600 CA becomes mortal. Here fertilization leads to extinction.

Fertilization may cause a CA to move and establish a new strange attractor (#600-8). Its fate was changed when a pollen coming from another dimension hit it.  The CA responded creatively and established a new niche.

Time series

Each  history (trajectory) is specified by the following functions, f[rule, x, t]. Actually it is a time series. The interaction occurring in the last history can be expressed as f[600,50,0] * f[600,58,30] (* stands for interaction). Although it specifies exactly the CA trajectory, it cannot predict it. The outcome of two interactions  f*f  is unpredictable. It may turn two mortals into immortals (#357 distance = 22), or vice versa (#600 distance = 13).

Cause and effect

In the interaction, f[600,50,0] * f[600,58,30], it appears as if the pollen f[600,58,30], caused a f[600,50,0] displacement. Did it really cause it? Obviously it is not the same cause which brings about a displacement of an elastic ball hit by another one.  One tends to describe this phenomenon in terms of triggering, or  initiation. Aristotle said that this type of interaction has four causes:

1. Material cause:  CA structure.
2. Formal cause:  Its rule
3. Efficient cause: The second CA (here the pollen)
4. Final cause: The "purpose of the collision". Better, the benefit of the collision to the CA, which  will  be implemented in the future.

Modern physics rejects three causes and keeps the effective cause, which causes a ball to move when hit by another one. Yet effective cause is inadequate to describe biological phenomena. Here all four causes are required.

The same applies to CAs. Hitherto the final cause was not implemented. Although  CAs tend to settle in strange attractors, we are unable to explain why they choose a special niche. The living organism is governed by an optimality principle (Wisdom of the Body (WOB)) which drives it to specific strange attractors called homeorhesis. The implementation of the final cause in the CA design may assist us in this endeavor..

Interaction between CA with different rules

The first two histories depict two non interacting CAs. CA#357 is short lived. CA#600 was planted at t=50 and is immortal. The last history depicts their interaction. CA#357 was planted first. At the age t-50 CA#600 was planted at a distance of 10 units from the first seed.. Ca#600 died after at t=70. Before dying it fused with CA#547 and initiated in it the following changes: CA#547 moved to a new niche, became immortal, and changed its structure. The last is known also as differentiation.

Please note that in its new niche, the differentiated CA#357 adhered to its rule, while none other structure adhered to rule #600

1. Wolfram S. A New Kind of Science  ISBN 1-57955-008-8

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