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In Section 1.6, we construct the following mathematical model of measurement (see Fig. 1.4), which allows solving the task of the synthesis of the optimal measurement algorithms.

The essence of this task consists in the following. Suppose that on the segment AB, there is the “unknown” point X. The length of the segment AX is determined by using k “indicatory elements” (see Fig. 1.4).

As a result of the enclosing of jth IE (j = 1, 2, …, k) in the first step of the algorithm to the point C_j (j = 1, 2, …, j, j + 1, …, k), the “comparison” of the segment AX to the segment ACj(j = 1, 2, …, j, j + 1, …, k) is realized, i.e., the relations of the “less” (<) or the “greater than or equal to” between comparable segments AX and ACj (j = 1, 2, … + 1, …, k) are determined. Note that the indicated property of the IE is its main definition.

We assume that the “output signal” or “indication” of the jth IE at the point Cj (j = 1, 2, …, j, j + 1, …, k) takes the value 0 if the relation AX < ACj holds, and the value 1 if AX ACj. We will also assume that the jth IE is “to the right” relative to the point X, if as a result of its enclosing to the certain point Cj (j = 1, 2, …, j, j + 1, …, k), its “indication” signal takes the value 0, and is “to the left” (otherwise the “indication” signal takes the value 1).

Note that the task of measuring the segment AX in the “indicatory” model of the measurement (Fig. 1.4) reduces to narrowing the interval of uncertainty about the point X. The process of measurement consists in the fact that, at the first step of the measurement, the “indicatory elements” (IE) are enclosed to some points Cj (j = 1, 2, … + 1, …, k) of the initial uncertainty interval AB and, on the basis of the “indication” signals of the IE in these points, the uncertainty interval narrows down to some new segment A₁B₁ ⊂ AB; in all the subsequent steps, the “indicatory elements” enclose to the points of the uncertainty interval A₁B₁ chosen at the previous step.

Some conditions or restrictions S can be imposed on the measurement process. The system of formal rules, which determines for each segment AX at each of the n measurement steps the set of the points Cj (j = 1, 2, … + 1, …, k) of enclosing the k “indicatory elements”, based on their “indications” on the previous steps for the restrictions S, is called the (n, k, S)-algorithm.

Thus, according to the above constructive “indicatory” model of measurement (Fig. 1.4), the measurement process consists in the sequential enclosing of the IE to some points Cj (j = 1, 2, …,j, j + 1, …, k) of the segment AB and narrowing the uncertainty interval regarding the point X.

The constructed model of measurement reduces the task of measuring the measurable segment AX to the task of deterministic one-dimensional search of the coordinate of the point X on the segment AB by using k IE in n steps. Such a method makes it possible to use the ideas and principles of the optimum seeking methods [131] to the strict definition of the optimal (n, k, S)-algorithm.

As a result, the action of the (n, k, S)-algorithm on the segment AB is the certain interval of the uncertainty Δ, containing the point X at the last step of the algorithm. The length of this interval Δ determines the “exactness” of measuring the segment AX. By considering the operation of the (n, k, S)-algorithm for all possible points X of the segment AB and by highlighting all the intervals of uncertainty, that contain the corresponding points of X, we get the set of the uncertainty intervals:

which form the partition of the segment AB into N segments, wherein

It is clear that the partition (1.3), satisfying the relation (1.4), is an “integral” characteristic of the effectivenes of the (n, k, S)-algorithm, acting on the segment AB; therefore, it can be used to introduce the strict definition of the notion of the optimal (n, k, S)-algorithm.

Let’s consider the ratio which will be called the exactness of determining the point X, belonging to the interval Δ_i. Let’s select from the partition (1.3) the largest interval of uncertainty Δ_max.The segment Δ_max corresponds to the smallest (i.e., the “worst”) exactness of determining the point X

with the help of which we will evaluate the effectiveness of the operation of the (n, k, S)-algorithm on the segment AB. It is clear that each partition (1.3) is characterized by the only value of (1.5). We call this value, defined by the expression (1.5), the (n, k)-exactness of the (n, k, S)-algorithm.

By using the concept of the (n, k)-exactness (1.5) as the criterion of the “effectiveness” or “optimality” of the (n, k, S)-algorithms (for the given values n, k, and the condition S), we can compare different (n, k, S)-algorithms with respect to their (n, k)-exactness.

The above consideration gives us a possibility introducing the following rigorous definition of the optimal (n, k, S)-algorithm.

Definition 1.1. The (n, k,S)-algorithm is called optimal if, for all other equal conditions, that is, for the given parameters of n, k, S, this algorithm provides the highest value of the (n, k)-exactness (1.5).

Note that this definition essentially uses the so-called mini–max principle, which is widely used in the modern theory of the optimal systems [131]. According to this principle, we call a strategy, which ensures the maximum value of the criterion of the effectiveness (1.5) for the worst case, as “optimal”.

Let’s consider once again the partition (1.3), which is the result of the operation of the (n, k, S)-algorithm on the segment AB, and let’s estimate this partition from the point of view of the concept of the (n, k)-exactness (1.5) introduced above. It is easy to show that the highest value of the (n, k)-exactness (1.5), for other equal conditions, corresponds to the uniform partition of the segment, when all the intervals of the uncertainty (1.3) are equal to each other, that is,

By substituting in (1.5) instead Δ_max its value, given by (1.6), we obtain the following expression for the (n, k)-exactness of the measurement algorithm:

where N is the number of the quantization levels, ensured by the optimal (n, k, S)-algorithm. This means that the optimal (n, k, S)-algorithm always provides the uniform partition of the segment AB; at the same time, the optimal algorithm is an (n, k, S)-algorithm that provides (for the given parameters n, k, S of the algorithm) the partition of the initial segment AB into the most number N of equal intervals.

It is clear that for the given restriction S, the efficiency criterion of the (n, k, S)-algorithm, that is, the number of quantization levels N, depends on the number of steps n of the algorithm and the number of “indicatory elements” k, involved in the measurement, i.e.,

Further, the function F(n, k) will also be called the efficiency function of the measurement algorithm.