Mathematics of Harmony as a New Interdisciplinary Direction and “Golden” Paradigm of Modern Science. Alexey Stakhov

k, S)-algorithm, there is an important notion of the restriction S, which is superimposed on the measurement algorithm. It is clear that any restriction reduces the efficiency of the algorithm, but, on the other hand, each restriction, as will be shown in the following, leads to very unusual measurement algorithms that may be of theoretical and practical interest.

      Let’s start from the simplest restriction that can be found from a comparative analysis of the classical measurement algorithms, in particular, the “binary” algorithm (Fig. 1.6) and the counting algorithm (Fig. 1.5).

      Let’s compare the two 3-step (n, k, S)-algorithms presented in Figs. 1.5 and 1.6. Both algorithms are implemented in the same number of steps (n = 3) and use only one IE (k = 1).

      What is the difference between these (3, 1, S)-algorithms? As the comparative analysis shows, the difference between them consists in the character of the movement of IE on the segment AB. Indeed, in the example in Fig. 1.6, an arbitrary movement of IE on the segment [0, 8] is allowed, that is, after a particular step, the IE can be enclosing at the next step both to the right or to the left relative to the point X. It is appropriate for the above restriction to use the notation S = 0. Then, we can consider the so-called (n, k, 0)-algorithms, that is, the measurement algorithms, in which the movement of IE on the segment AB obeys the restriction S = 0.

      Let’s now consider the counting algorithm in Fig. 1.5. In this example, the movement of IE is subjected to the following strict restriction: in the process of operation of the algorithm, IE moves on the segment AB only in one direction: from point A to point B. In this case, IE participates in the measurement algorithm as long as it is “to the left’ from point X; as soon as IE “jumps” over point X, it stops its further participation in the measurement algorithm. Denote this restriction by S = 1. Then we also have the right to consider the so-called (n, k, 1)-algorithms, that is, the measurement algorithms, in which the movement of IE on the segment AB obeys the restriction S = 1.

      Note that the restrictions S = 0 and S = 1 are not the only possible restrictions. It is important to emphasize that each restriction S leads to the discovery of some new class of the optimal measurement algorithms having theoretical or practical interest; therefore, the search for reasonable restrictions, imposed on the measurement algorithm, is an important task of the algorithmic measurement theory [16].

      It is essential to emphasize once again that the classical measurement algorithms, discussed above, played an important role in the development of mathematics and computer science. In particular, the counting algorithm (Fig. 1.5) underlies the Euclidean definition of natural numbers (1.10) and the Eudoxus–Archimedes axiom;the “binary” algorithm (Fig. 1.6) underlies the “binary” system (1.12), without which it is impossible to imagine modern computer science, that is, these simplest measurement algorithms affect the foundations of mathematics and computer science.

1.9. Optimal (n, k, 0)-Algorithms

       1.9.1. Recursion method

      As mentioned above, the (n, k, 0)-algorithm is a n-step measurement algorithm, which uses the k IE, the movement of which on the segment AB that obeys the restriction S = 0. Recall that the restriction S = 0 essentially means the absence of some restrictions on the movement of IE, that is, each IE at a particular step of the measurement algorithm can be enclosed to the arbitrary point of the segment AB.

      We will solve the task of synthesis of the “optimal” (n, k, 0)-algorithm, that is, the measurement algorithm, which, for the given n, k and the restriction S = 0, divides the segment AB into the biggest number of equal intervals.

      To solve the problems of synthesizing the optimal (n, k, S)-algorithms, we will use the “method of recurrent relations” [110], which is widely used to solve combinatorial problems. The essence of this method is that the solution of some n-step combinatorial task reduces to solving the (n – 1)-step combinatorial task; at the same time, there is the recurrent relation, which connects both solutions. This continues until we come to some “simple” combinatorial task, the solution of which does not cause difficulties.

       1.9.2. Synthesis of the optimal (n, k, 0)-algorithm

      The essence of the method of the recurrent relations in connection with synthesizing of the optimal (n, k, 0)-algorithms is as follows. We do the so-called inductive hypothesis. Suppose that for some n and k, there exists the optimal (n, k, 0)-algorithm that implements on the segment AB the (n, k)-exactness equal to F(n, k). In other words, our inductive hypothesis consists in the fact that the optimal (n, k, 0)-algorithm divides the segment AB into F(n, k) equal intervals of the unit length Δ = 1, that is,

      Now, let the first step of the optimal (n, k, 0)-algorithm on the segment AB be in the enclosing of the k indicatory elements on the segment AB, as shown in Fig. 1.8.

Fig. 1.8. The first step of the optimal (n, k, 0)-algorithm.

      After the first step, by using the indications of k IE, we get the (k + 1)th situations:

      In this case, all the segments in Fig. 1.8 are connected by the following relationship:

      Let’s consider the situation For this situation, before taking the next step, we have (n – 1) steps because one step has already been used, and k IE, which, according to the restriction S = 0, can be enclosed to some points of the new uncertainty interval CjCj+₁, which contains point X. But then each of the situations (1.15), that arose after the first step, is not distinguished from the initial situation in Fig. 1.8. Only the number of steps for the realization of the measurement algorithm, which operates on the new segment CjCj+i, decreased to (n – 1), and then, in this situation, for searching the coordinate of point X, we can use the new uncertainty interval CjCj+i the optimal (n – 1, k, 0)-algorithm, which, according to the inductive hypothesis (1.14), divides the segment C_jC_j+1 on F(n – 1, k) equal intervals of the unit length Δ = 1, that is,

      The above reasoning is valid for