Informatics
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«The intentional splitting of the logical foundation into the notions of “no” and “yes” is the most serious obstacle on your path to comprehend existence»
(Unknown author)
Let a variable \(x_i\) takes one of two values: \[x_i=\pm\frac{1}{2}\] The sum of a pair of such variables in four combinations can take one of three values: –1, 0, +1. This corresponds to a logical foundation “yes / no / don’t know”, and “don’t know” is most probable due to two modes.
The sum of N pairs of the independent random binary values \(x_i\) can take a value from –N до +N: \[X_N=\sum_{i=1}^N{x_{2i-1}+x_{2i}}\] The sum of the uniformly distributed values has the following distribution, which is defined by the particular combination numbers from the total combinations number \(4^N\):
| \(N\) | \(X_N\) | \(4^N\) | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| –4 | –3 | –2 | –1 | 0 | +1 | +2 | +3 | +4 | ||
| 1 | 1 | 2 | 1 | 4 | ||||||
| 2 | 1 | 4 | 6 | 4 | 1 | 16 | ||||
| 3 | 1 | 6 | 15 | 20 | 15 | 6 | 1 | 64 | ||
| 4 | 1 | 8 | 28 | 56 | 70 | 56 | 28 | 8 | 1 | 256 |
According to the central limit theorem, this distribution is close to the normal (Gaussian). This is evident, when a particular combination probability is converted into a probability density. \[f(X_N)=\frac{1}{\sigma\sqrt{2\pi}}\exp(–\frac{{X_N}^2}{2\sigma^2})\]
| \(N\) | \(X_N\) | \(3\sigma\) | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| –4 | –3 | –2 | –1 | 0 | +1 | +2 | +3 | +4 | ||
| 1 | 0,8 | 2,3 | 0,8 | 2,12 | ||||||
| 2 | 0,1 | 0,9 | 3,9 | 6,4 | 3,9 | 0,9 | 0,1 | 3 | ||
| 3 | 0,1 | 1,0 | 5,5 | 14,9 | 20,8 | 14,9 | 5,5 | 1,0 | 0,1 | 3,67 |
| 4 | 1,3 | 7,6 | 26,5 | 56,3 | 72,2 | 56,3 | 26,5 | 7,6 | 1,3 | 4,24 |
The most probable states of the \(x_i\) variable system are the equilibrium states \((X_N=0)\). Denoting the number of equilibrium states by \(\Omega_N\), the value \(\ln\Omega_N\) is called here the information capacity or the system entropy.
First three values of \(\Omega_N\) are close to the Fibonacci sequence sums:
| \(N\) | \(\sum_{i=0}^{2N–1}{F_i}\) | \(\Omega_N\) | \(\ln\Omega_N\) |
|---|---|---|---|
| 1 | (1+1) = 2 | 2 | 0,69 |
| 2 | (3+2) + (1+1) = 7 | 6 | 1,79 |
| 3 | (8+5) + (3+2) + (1+1) = 20 | 20 | 3 |
| 4 | (21+13) + (8+5) + (3+2) + (1+1) = 54 | 70 | 4,25 |
The especially remarkable case is \(N=3\), when the capacity or entropy \(\ln\Omega_3=2,996\approx 3\). The state space for a system of three pairs of binary values could be represented as a simple 3D-matrix (a cube) and also a stellated octahedron:
| Number of the zero pairs | \(X_N\) | Number of combinations | Geometry |
|---|---|---|---|
| 3 | 0 | 8 | Faces of internal octahedron |
| 2 | ±1 | 24 | Сonvex edges |
| 1 | 0 | 12 | Faces |
| ±2 | 12 | ||
| 0 | ±1 | 6 | Vertices |
| ±3 | 2 |
The system of three pairs of binary values takes place in the nature and the culture. For example:
- DNA/RNA codon consists of 3 nucleotides and has \(4^3=64\) combinations for the redundant coding of 20 amino acids.
- The human language has about 20 consonants (see «Speech»). In many cases, especially in the Semitic languages, the consonants are more informative than less various vowels.
- \(2^6=64\) hexagrams I-Ching. But the Chinese divide these hexagrams into two triples, and not into three pairs.
- Some of first computers used the 6-bit binary words with \(2^6=64\) allowed values, which are sufficient enough for the Latin alphabet, the Arabic digits and a minimal set of the special characters.
- The vigesimal (base-20) numerical system is ancient and the most prevalent after the modern decimal system. It’s used in the culture of Maya, Ainu (see also «China and Japan»), Celts, Caucasians and Africans. The Indo-European languages have numerals from 11 to 19.
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