For every 0 in h-column we are reading its preceding row and associating it to h, where h is a variable standing as a head of a column (h=x,y,z). Because we are working with cyclic order, the last column is the preceding for the first. In the x-column 0 appears in the first and fourth row, so the fifth and third row will be associated to x, i.e., x=(1 1 1)(1 0 1). In the y-column 0 appears in the third row, so we associate the second row to y and obtain y=(1 1 0). In the z-column 0 appears in the first and second row, so we associate the last and first row to z and obtain z=(1 1 1)(0 1 0). By replacing every 1 by h, and every 0 by , and overlining each n-tuple, we obtain , and the recursive operator . After that, by using the square-free polynomial method, we can reduce terms obtained, and search for fixed points. Hence, =T(1 - x z, 1 - x y + x y z, 1 - y + x y + y z -2 x y z). Its fixed point (x,y,z)=([(3+2)/7],[(3-2)/2],2-2) that belongs to the interval [0,1] we obtain as in the same way as before, by solving the system of equations x=1 - x z, y=1 - x y + x y z, z=1 - y + xy + y z - 2 x y z.

This algorithm enables us to derive all sequences with a given period p. However, it will be useful to identify essentially different ones among them: non-isomorphic, not mutually reverse, and not mutually dual sequences. It is clear that two sequences given by a cyclic permutation of rows in the array will be equal. Also, sequences that are the same up to a permutation of variables can be identified. Two dual sequences, where one can be obtained from the other by replacing 0 by 1 and vice versa can be treated as equivalent as well. Cyclically equivalent sequences will have the same fixed points; sequences with permuted variables will have fixed points permuted in the same way; dual sequences will have dual fixed points from the interval [0,1]. Sequences not derivable from previous sequences by the mentioned ways are basic sequences. For p=2 and n=1 we have one basic sequence 0,1,0,1 defined by the operator , without fixed points from the discrete set {0,1}, that has the fixed point x=1/2 from the interval [0,1]. This sequence embodies the Liar paradox. For p=2 and n=2 we obtain two basic sequences: (0,0),(1,1)..., defined by the operator =(1-xy,1-xy), with the fixed point (x,y)=(j, j) from the interval [0,1], and (0,1),(1,0)..., defined by =(1-x+xy,1-y+xy), with the fixed point (x,y)=(1,1).

For p=3, the minimal n is n=2. From the complete set (0,0), (0,1), (1,0), (1,1) we can produce 24 different sequences of period 3. Each of them satisfies the necessary condition that no variable is constant. Taking one representative from each equivalence class, we obtain three basic sequences of period 3: (0,0),(0,1),(1,0), (0,0),(0,1),(1,1), and (0,0),(1,1),(0,1) The first is defined by the operator = T(y,1- x - y + 2 x y), the second by =T(x + y - 2 x y, 1 - xy), and the third by The fixed points of the first are (x,y)=(1,1) and (x,y)=(1/2,1/2), of the second (x,y)=(1/2 ,2/3), and of the third (x,y)=(1-j,j).

For p=4, the minimal n is n=2, we obtain three basic sequences: (0,0),(0,1),(1,0),(1,1)... with =(x+y-2xy,1-y); (0,0),(0,1),(1,1),(1,0)... with ==(y,1-x); (0,0),(1,1),(0,1),(1,0)... with ==(1-x,1-x-y+2xy). All of them cannot be stabilized for any pair of values from the discrete set {0,1}, and in [0,1] they have the same fixed point (1/2, 1/2). Continuing in the same way, it is possible to enumerate all basic periodic sequences of a higher period. For example, for n=3 there are 45 sequences of the period p=4, 160 for p=5, 382 for p=6, 840 for p=7, and 840 for p=8.

An infinite self-referential form with two fixed parameters m and n, produces by iterating . Continuing in this way, T(x)=T2(X)=T3(x)=... Thus, . This form contains a copy of itself, so . As the result, every point 1-n+x0mn, where x0 is an initial state (x0 [0,1]), will be its fixed point.

The LinKnot function fBoolean (webMathematica fBoolean) calculates the corresponding square-free polynomial from a given logic formula. The function fKauffAlg (webMathematica fKauffAlg) according to Kauffman algorithm produces all different periodic sequences of the period p with n variables. The function fDiffSeq (webMathematica fDiffSeq) produces all different periodic sequences of the period p with n variables, giving as the output their complete list, where every periodic sequence is given by one period, Kauffman code, and the corresponding square-free polynomials. The function fBalanced (webMathematica fBalanced) calculates stable (balanced) states for a given periodic sequence.


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