12.1 General Systems Theory

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Table 12.1 Truth table for an

AND gate

Input

Output

0

0

1

0

2

1

Fig. 12.1 State structure of

the automaton represented by

Eq. (12.11) and Table 12.1

where normal ♢♦denotes that the additions in the matrix multiplication are to be carried out

using Boolean AND logic, i.e., according to Table 12.1. Enumerating all possible

starting values leads to the state structure shown in Fig. 12.1. The problem at the

end of this subsection will help the reader to be convinced that state structure is not

closely related to physical structure (the pattern of interconnexions). In fact, to study a

system one needs to determine the state structure and know both the interconnexions

and the functions of the individual objects (cells).

Most of the work on the evolution of automata (state structures) considers the

actual structure (interconnexions) and the individual cell functions to be immutable.

For biological systems, this appears to be an oversimplification. Relative to the con-

siderable literature on the properties of various kinds of networks, very little has been

done on evolving networks, however. ⁶

Problem. Determine the state structure of an automaton if (i) the functions of the

individual cells are changed from those represented by (12.11) such thatupper G 1G1 becomes

1 whenever upper G 2G2 is 1, upper G 2G2 becomes 1 whenever upper G 3G3 is 1, and upper G 3G3 becomes 1 whenever

upper G 1G1 andupper G 2G2 have the same value; (ii) keep these functions, but connectupper G 1G1’s output to

itself andupper G 3G3,upper G 2G2’s output to itself,upper G 1G1 andupper G 3G3, andupper G 3G3’s output toupper G 2G2; and (iii) keep

6 An exception is Érdi and Barna (1984) on a model of neuron interconnexions, simulating Hebb’s

rule (traffic on a synapse strengthens it, i.e., increases its capacity).