I Ching Algorithm: Outstanding advantages of Hexagram Model compared to Diagram Theory


The Genetic Algorithm used to be a pretty good algorithm because it maintained a population of solutions. Therefore, it significantly overcomes the local minima in the optimization problem compared to other methods, typically the hill climbing method. But that advantage of Genetic Algorithm can only be used for simple problems.
Today's life is as complicated as the COVID-19 pandemic shows. Now, looking at it from a broader perspective, Genetic Algorithms have one drawback, which is a big one.
The crucial disadvantage of Genetic Algorithm is that it derives from a purely technical solution. That is, its operating principle is Diagram Theory. Technical problems can never touch real life. Although a diagram covers a fairly large space of solutions and is much more powerful than the point-by-point search of the hill-climbing method, in turn, it becomes the local diagram for complex problems and Genetic algorithms cannot converge to global optimality.
I Ching Algorithm does not limit the problem to any technical areas, its principle is to find the Tao or path to the optimal solution. This is the basic thing. As we see in The hexagram space theorem, all data is compressed into hexagram data. In the opposite direction, the hexagram data is decoded back to the problem data whose union covers the entire space.

A diagram of length m is a sequence representing m-bit binary sequences whose bits are the same as the fixed bits of the diagram. The other bits whose positions correspond to the * character in the diagram can be any bit 0 or bit 1.
Example diagram with length m = 10 below
will match 2 sequences
1001011000 and 1001011010
If each binary sequence is considered a valid solution in the problem space, then a diagram is a set of solutions. The idea is that the less cardinality the diagram has, the more compact the space is.
1) Degree of the diagram
The degree of the diagram S, denoted o(S), is the number of fixed bits in the diagram. For example
S1 = *****1*1*0
S2 = 1**0*1*01*
S3 = 001001*1*0
We have
o(S1) = 3
o(S2) = 5
o(S3) = 8
Thus, diagram S3 has the highest degree, its space is considered the most compact.
Unfortunately, the higher the degree of the diagram, the more vulnerable it is to mutations. When a fixed bit is mutated, the diagram is no longer what it is, it is broken.
2) Specified length of the diagram
The specified length of a diagram S, denoted δ(S), is the distance between two fixed positions at the beginning and at the end. We have
(S1) = 10 – 6 = 4
(S2) = 9 – 1 = 8
(S3) = 10 – 1 = 9
The larger a diagram of a specified length, the easier it is to be cut out, i.e. broken by hybridization.
3) Diagram theorem
Short, low-degree, above-average diagrams receive a exponentially increasing number of sequences in successive generations of the genetic algorithm.
This is only possible at an early stage in evolution, when the diagrams are beginning to form and the average fitness of the population is low. When approaching the optimal point, long, high-degree diagrams are very vulnerable. Furthermore, the diagram is unlikely to have above-average fitness because it has become "conservative" while novel elements are often more adaptive.
Above all, the foregoing is not yet the biggest weakness of Diagram Theory.

To illustrate, we use the simplest possible problem: Find the maximum of the function y with x in the interval [0, 4194303] as shown in the graph below

Looking at the graph, it is easy to see that y reaches the maximum value of 8 when x is on the right bound i.e. at the maximum value of x is 4194303. The population should reach the top M of the mountain. But the Genetic Algorithm was misled by the diagram and ended up on hill H with x = 1929378 and y having a value of 5 below the optimal point.
In this problem, the right bound value of x, 4194303 is the decimal value of a 22-bit binary sequence consisting of all 1s.
11 1111 1111 1111 1111 1111
And the value x = 1929378 is the decimal value of the sequence
01 1101 0111 0000 1010 0010

We know that the prehistoric period is the longest period in the history of human development, spanning millions of years. People with herd lifestyle and arbitrary actions do not follow a social contract. Therefore, the evolutionary process is gradual, happening very slowly, unlike the cognitive process of the rules of the universe with the industrial revolutions today.
Genetic algorithm with arbitrary reproduction mechanism similar to human prehistory. Due to its gradual evolution it approaches hill H and does not have I Ching transformation mechanism to cross the valley on the right to reach the mountain M. As a result, it forms the diagram
0* **** **** **** **** ****
This is a degree-1 diagram with the fixed bit being the highest bit with a value of 0. Individuals with the highest bit being 1, for example
10 0000 0000 0000 0000 0000
corresponds to x = 2097152, does not belong to the diagram and is disqualified.
Thus, the population does not continue to go to the right of the point x = 2097152 and never reaches the top of M.
I Ching Algorithm, with a philosophical solution that combines modern data mining, it will find the CẤN hexagram at the end of the road. The hexagram number of CẤN is 7, which is the decimal value of the 3-bit binary sequence 111.
The 3-bit binary sequence 111, when mapped into the data space of the problem, corresponds to a set of 22-bit binary sequences that obey the encoding rules. They are seen through the convolution window as groups of yang bit, for example
11111011 11001111 11011011
11001111 11111001 11111101
00111111 11101111 11111100
One of them is bit groups
11111111 11111111 11111111
It belongs to a 22-bit binary sequence containing all 1s
11 1111 1111 1111 1111 1111
is the optimal solution required by the objective function of the problem.

Do you feel anything?
When we use the image of a mountain top to illustrate the optimal point, the expected hexagram is CẤN. CẤN means mountain. CẤN also means to stop, reaching the top of the mountain is also the end of the problem. Feeling interesting is the source of the enlightenment of the Hexagram Model!

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