## 3.4.1  Waveforms

After interpreting the marked state - as 1, and unmarked state as 0, we can introduce a waveform arithmetic, working in the discrete set {0,1}, or in continuous set [0,1]. Treating f= as a recursive form defined on the set {0,1}, by taking successive replacements of f in its equivalent form =1-f, we obtain a periodic sequence with the period two, 0,1,0,1,0,1,0,1 ¼. On the other hand, working in the continuous set [0,1], from the equation f=1-f we can find the fixed point f=1/2 of the recursion f= . It is a solution of Liar paradox in polyvalent logic.

In order to generate sequences of period greater then 2 it is necessary to apply recursions on more then one variable. Working in the discrete set {0,1}, the recursion T(x,y)=([`y],[`([`x] y)]) produces a sequence with the period three (0,0),(1,1),(0,1),(0,0),(1,1),(0,1)... Working in the continuous set [0,1], and solving the system of equations x=1-y and y=1-(1-x)y that results from x= and , we obtain as a solution for the fixed point (x,y)=(1-j, j), where j is the golden ratio.

Duality holds for periodic sequences generated by recursions, and for their fixed points as well. Solving the dual recursion for a fixed point, we obtain the result (x,y)=(j,1-j). It can be obtained from the previous result by duality.

In the same way, working with discrete values from the set {0,1}, the recursion generates a periodic sequence with the period five, (0,1,0), (1,1,0), (1,0,1), (0,1,1), (1,1,1), (0,1,0), (1,1,0), (1,0,1), (0,1,1), (1,1,1), (0,1,0), (1,1,0), (1,0,1), (0,1,1), (1,1,1) .... Its fixed point (x,y,z)=([(3+Ö2)/7],[(3-Ö2)/2],2-Ö2) that belongs to the interval [0,1] is obtained by solving the system of equations x=1 - xz, y=1 - x y + x y z, z=1 - y + x y + y z - 2 x y z. Working in {0,1}, the dual recursion generates again a dual periodic sequence with the period five. Solving the system of equations x = 1 - x - z + x z, y = z - x z - y z + xy z, z=1 - x - y + x y - z + 2 x z + y z - 2 x y z we obtain the dual fixed point (x,y,z)=([(4-Ö2)/7],[(Ö2-1)/2],Ö2-1), that belongs to the interval [0,1]. In certain cases, a set of fixed points for a recursive operator can be the complete interval [0,1]. This holds, for example, for the operator with the set of fixed points (x,1-x), x Î [0,1]. It produces a stable state for every y=1-x, and a periodic sequence with the period 2 otherwise.

L.H. Kauffman and F.J. Varela described an algorithm for producing recursive operators that generate periodic sequences of a desired period (1980). Let a periodic sequence of period p with n variables be given, where 2n-2 < p £ 2n. The condition 2n-2 < p implies no variable is constant. Usually, we are trying to express all periodic sequences by with a minimal number of variables n. One period of such a sequence can be represented in the form of an n×p array. For example, the periodic sequence of period p=5 with n=3 variables considered before, can be represented as

 x
 y
 z
 0
 1
 0
 1
 1
 0
 1
 0
 1
 0
 1
 1
 1
 1
 1