One-dimensional two-state cellular automata - Wolfram's resultsWolfram's results

Consider the one-dimensional case with r = 1, i.e., using the closest neighbourhood. Then the next state of a cell depends on its own state and the states of its two neighbours: three consecutive cells' states determine it. In binary case, each of these cells can have at most two values, therefore there are 2^2·2·2 = 2⁸ = 256 possibilities, we can define 256 different systems (regarding their rules). Usually these systems are denoted by their binary code, or its decimal equivalent as follows.

Definition: Let CA = (Z,{0,1},N,d) be a one-dimensional cellular automata, where Z is the set of integers, N = {+1,-1} contains the positions of neighbour cells, d is the transition function. The transition function can be defined in the following way: consider the elements of {0,1}³ (they are the three letter long words over the alphabet {0,1}*) and order them alphabetically (using the relation 0<1) in decreasing way. To determine CA we need to add the new state (0 or 1) for these eight cases. Since the order of these eight cases is fixed, a word over the alphabet {0,1}* with length eight gives the transition function. This is an eight-bit binary word and it is called the Wolfram-number of CA. Usually (beside or instead of the binary representation) the decimal value is used (it is between 0 and 255).

Example (Wolfram-number of a one-dimensional automaton)

Let the number 122 be given. Its binary form (in eight bits) is 01111010. As a Wolfram-number its meaning is as follows. The cell with its two neighbours can have the following value-triplets (written with three-letter

Thus, CA = (Z,{0,1},{+1,-1},d) is a one-dimensional cellular automaton such that the next state of a cell will be 0 if its and both its neighbours' values are 1 in the previous time instant; the next state of a cell will be 1 if its and its left neighbour had value 1 and its right neighbour had value 0; 1 is the next state if the neighbours had value 1 while the cell itself had value 0; the value will be 1 if the left neighbour cell had value 1 and both the cell itself and its right neighbour had values 0; etc.

The dynamics of the one-dimensional cellular automata is usually depicted by a space-time diagram: the first row of these figures show configuration c0. Every other row shows the configuration of the next time instant as the previous row. In these diagrams the states are marked by various colours. Naturally, only a finite interval (both in space and time) can be graphically shown, but these diagrams still give a good background to start to analyse such systems.

Let us analyse the one-dimensional two-state cellular automata using the smallest neighbourhood. These systems are classified by Wolfram (with an experimental approach) to the next four classes based on their dynamic properties starting from a random configuration.

Class 1. The first class includes those systems that lead to a constant (also called steady) configuration, i.e., configuration in which all the cells are in the same state, from any initial configurations. Figure 3.6 shows an example for a space-time diagram of one of these systems: this one has Wolfram-number 160. It can be seen that the constant 0 (all cells are going to have value 0) is obtained soon after the initial configuration. (In this figure the time is also shown by lightening the colours of the states step by step.) The automata with Wolfram-number 0 and 255 are also belonging to this class: in these cases all the pixels (cells) are changed to 0 (white) or 1 (black or grey) in the first derivation step.

3.6. ábra - An example for the evolution of cellular automaton with Wolfram-number 160 from a given initial configuration.

Class 2. This class includes those models that develops to configurations that are periodic (in time) with a short period (maybe 1) from any initial configurations. Figure 3.7 shows a part of a computation of the automaton with Wolfram-number 4. In this model the value of a cell will be 1 if and only if its value was 1 and its neighbours had values 0 (i.e., at 010). The system obtains a configuration with period 1, i.e., constant in time, these configurations are also called stable. Figure 3.8 presents an example for the work of automaton with Wolfram-number 108. As one can observe the work of this automata is periodic (in time).

It is a very important property of the automata in this class that a given pattern survives, maybe in the same place, maybe in a shifted way (travelling), as at automaton with Wolfram number 2. The maximal speed of the movement is one cell in a derivation step. In two third of these automata the pattern survives in a fixed size, while in the half of the remaining cases the pattern grows up to the infinity in a nice regular, periodic way (e.g., automaton with Wolfram-number 50). About one seventh of the automata have a more complex dynamical property, as we detail in the next classes.

3.7. ábra - Automaton with Wolfram-number 4 obtains a stable configuration.

3.8. ábra - Automaton with Wolfram-number 108 leads to configurations that are periodic (in time).

Class 3. This class contains the automata that are obtaining ''random-like'' patterns. There are ten transition functions that describe these automata with chaotic dynamical properties. The patterns generated by them are fractal like, opposite to the regularity that the previous classes have. There are three different patterns of these automata. Figure 3.9 shows automaton with Wolfram-number 22. Here a cell is going to have value 1, if and only if there was exactly one 1's among the values of its neighbours (including itself). This automaton starting from a configuration including only one cell with value 1 generates (in the space-time diagram) a version of the fractal called Sierpinski-triangle.

3.9. ábra - The evolution of the automaton with Wolfram-number 22 starting from a random-like configuration.

Class 4. This class includes models that behave in a ''complex way'', e.g., the cellular automata with Wolfram-number 110. The other three automata that are in this class are equivalent to this one (interchanging the roles of left-right and/or the black/white, i.e., the 0-1). Figure 3.10 shows a small part of the evolution of automaton with Wolfram-number 110. This model mixes the property of stability with some chaotic properties. Starting with a configuration having only one cell in state 1, we can observe periodic patterns and also a lane where the evolution seems to be chaotic. These two parts has a complicated interaction at their border. This model is proven to be Turing-universal (and thus, it is considered as the simplest Turing-universal computing model).

3.10. ábra - An example for the evolution of automaton with Wolfram-number 110.

It is very interesting that such a simple algorithm can behave in a surprisingly complex manner.

It is usual to do experiments starting from a configuration that has only one cell with value 1 and all other cells have value 0. In this case the automata with even Wolfram-number (i.e., those ones for which the value 0 is assigned for the triplet 000 at the next time instant) can be seen that generates the binary form of integers:

configuration ci refers for the i-th element of the sequence.

Example (generating a sequence of numbers by cellular automaton)

Let us consider the automaton with Wolfram-number 220. It is 11011100 in binary representation. At the beginning (at time instant 0) the configuration is the code of 1. The transition function assigns 1 to the triplets 111, 110, 100, 011 and 010. Based on these facts, it is easy to show that after the first derivation step the code 11 (it is the binary representation of 3) is obtained; after the second derivation step 111 (it is the code of 7), then 1111 (it is the code of 15) etc. can be obtained. The i-th element of the sequence generated by this automaton is exactly the code of 2ⁱ⁺¹-1.

So far we have analysed the simplest two-state (or binary) cellular automata (in one dimension with the smallest neighbourhood). We note here that in the case of 3-state automata (when a third colour is also used), even the smallest neighbourhood is used in one dimension, there is a very huge amount of possible automata: their number is the same as the number of strings over an alphabet with three letters of length 3³ = 27. Wolfram analysed only the totalistic systems (using the more general definition of totalistic systems) among them and it is turned out that basically they have the same types of classes as the previously detailed two-state systems.