# 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. $f(X_N)=\frac{1}{\sigma\sqrt{2\pi}}\exp(–\frac{{X_N}^2}{2\sigma^2})$

Values $$f(X_N)\cdot 4^N$$
$$N$$ $$X_N$$ $$3\sigma$$
–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: