Classical measurement algorithms

But if we want to develop the new measurement theory, the algorithmic measurement theory, we should understand that means the concept of the "measurement algorithm" and to give the strong definition of this concept.

Let's start from the classical measurement algorithms used widely in measurement practice.

Note that if we want to measure something we have to know the object of measurement and the range of measurable values. It is clear that there are different things to measure the cosmic distances, for instance, the distance from the Earth to the Sun, and to measure the atomic distances. However for all cases of measurement we will present the range of measurement as some line segment AB. But our measurable value is one of the possible values belonging to the given range. This situation will be represented by means of the unknown point X situated at the line segment AB.

And now we can formulate the aim of measurement. The aim of measurement is to determine the length of the line segment AX. In practice this aim is realized by means of the special means, for instance, the balance. The balance compares the measurable value with some standard weight and depending on the comparison result gives us the information about the measurable value. Thus, the measurement is reduced to the successive comparisons of the measurable value with some standard weights, which we have. To simulate the measurement process on the line segment AB we introduce the important concept of the "Indicator Element" (IE). Each IE can be applied to any point C of the line segment AB. The "Indicator Element" gives information about the mutual disposition of the points X and C. If the IE is to the right of the point X it "indicates" the binary signal of 0; in the opposite case dose the binary signal of 1.

Now we can "simulate" the "binary" algorithm acting at the line segment [0,8] as the following (Fig.1). The "binary" algorithm considered in Fig.1 consists of 3 steps.

The first step is applying the IE to the middle of the initial line segment [0,8], i.e. to the point 4. After the first step there arise two situations in dependence on the IE "indication".

The second step.

  1. If the IE "indication" in the point 4 is equal to 0 (the IE has shown to the left) this means that the measured point X is on the line segment [0,4]. In this situation the second step of the measurement algorithm is applying the IE to the middle of the line segment [0,4], i.e. to the point 2.
  2. If the IE "indication" in the point 4 is equal to 1 (the IE has shown to the right) this means that the measured point X is on the line segment [4,8]. In this situation the second step of the measurement algorithm is applying the IE to the middle of the line segment [4,8], i.e. to the point 6.

One may arise 4 situations after the second measurement step.

The third step.

  1. If the IE "indication" in point 2 is equal to 0 (the IE has shown to the left) this means that the measured point X is on the line segment [0,2]. In this situation the third step of the measurement algorithm is applying the IE to the middle of the line segment [0,2], i.e. to the point 1.
  2. If the IE "indication" in the point 2 is equal to 1 (the IE has shown to the right) this means that the measured point X is on the line segment [2,4]. In this situation the third step of the measurement algorithm is the applying the IE to the middle of the line segment [2,4], i.e. to the point 3.
  3. If the IE "indication" in the point 6 is equal 0 (IE has showed to the left) this means that the measured point X is on the line segment [4,6]. In this situation the third step of the measurement algorithm is applying the IE to the middle of the line segment [4,6], i.e. to the point 5.
  4. If the IE "indication" in point 6 is equal to 1 (the IE has shown to the right) this means that the measured point X is on the line segment [6,8]. In this situation the third step of the measurement algorithm is applying the IE to the middle of the line segment [6,8], i.e. to the point 7.

It follows from Fig. 1 that the "binary" algorithm subdivides the line segment [0,8] into 8 equal parts [0,1], [1,2], [2,3], [3,4], [4,5], [5,6], [6,7], [7,8].

The 'binary' measurement algorithm
Figure 1. The 'binary' measurement algorithm.

It is clear that the above-considered "binary" algorithm divides the line segment [0,8] into T = 8 equal parts. We can use this numerical characteristic of the algorithm as the criterion of its effectiveness. It is clear that in general case the n-step binary algorithm divides the line segment AB into T = 2n equal parts.

It is important to stress that the given measurement algorithm is "optimal" in the above-considered sense. This means that it is impossible to find the better n-step measurement algorithm, which could divide the line segment AB for bigger number of parts.

Note that if we write the sequence of the IE "indications", for instance 1001101001 ..., then the indicated sequence would present itself the "binary" code of the number corresponding to the length of the measurable segment AX. This means that the "binary" measurement algorithm "generates" the "binary" method of number representation.

But the "binary" algorithm is not the only algorithm of measurement. In our daily life we use often so-called "counting" algorithm underlying the Eudoxus-Archimede axiom considered above.

For example we use the "counting" algorithm if we try to count up the number of our steps lying down in some distance, for instance, in the distance between our house and our university.

'Counting' algorithm of measurement
Figure 2. "Counting" algorithm of measurement.

The "counting" algorithm demonstrated in Fig. 2 consists of 3 steps and makes use only one IE . It subdivides the initial line segment [0,4] into 4 equal parts for 3 steps.

The first step is applying the IE to the point 1. After the first step there arise two situations in dependence on the IE "indication", the line segment [0,1] and the line segment [1,3].

The second step.

  1. If the IE "indication" in the point 1 is equal to 0 (the IE has shown to the left) this means that the measured point X is on the line segment [0,1]. In this situation the measurement process is over because the X-coordinate (X Î {0,1]) has been defined with "exactness" to the measurement unit equal to 1.
  2. If the IE "indication" in the point 1 is equal to 1 (the IE has shown to the right) this means that the measured point X is on the line segment [1,4]. In this situation the measurement process is going on and the IE is applied to the point 2.

The third step.

  1. If the IE "indication" in the point 2 is equal to 0 (the IE has shown to the left) this means that the measured point X is on the line segment [1,2]. In this situation the measurement process is over because the X-coordinate (X Î {1,2]) has been defined with "exactness" to the measurement unit equal to 1.
  2. If the IE "indication" in the point 2 is equal to 1 (the IE has shown to the right) this means that the measured point X is on the line segment [2,4]. In this situation the measurement process is going on and the IE is applied to the point 3. As a result we can get two line segments [2,3] and [3,4] in dependence from the IE "indication" on the last step.

It is clear that the above considered "counting" algorithm divides the line segment [0,4] into T = 4 equal parts. It is easy to prove that in the general case the n-step "counting" algorithm divides the initial line segment AB into T = n + 1 equal parts.

There arises a question: what measurement algorithm is the best? If we compare the "binary" and "counting" n-step algorithms according to their effectiveness (T = 2n - for the "binary" algorithm and T = n + 1 - for the counting algorithm) then we can conclude that the "binary" algorithm is more effective. But sometimes we cannot use the "binary" algorithm! Everything depends on character of measurable value and technical means used for measurement. For example, if we measure a time intervals for this case the "counting" algorithm is a "natural" algorithm. This means that for comparison of effectiveness of the measurement algorithms we should compare the algorithms, which satisfy to the same condition or "restrictions" S laying on to the measurement algorithms.

What difference between the "binary" and "counting" algorithms? We can answer this question if we compare a character of the IE movement in these algorithms. It is clear that in the "binary" algorithm the IE movement is "chaotic" because IE can move to the left or to the right regarding to the preceding step. Let's mark the "restrictions" on the "binary" algorithm as S=0.

Let's consider now the "counting" algorithm. It is clear that movement of the IE for this case is submitted to the strong "restriction": the IE moves in the only direction from the left to the right. Let's mark this "restriction" as S=1.

However until now we did discover nothing new! The mankind know these classical measurement algorithms many millenniums. However we ask our reader would be patient! New results are ahead! Follow us!