Function, script and event approximation

Introduction

Thanks to Igor Katricek for a great question! At the forum devoted to projection modeling, he asked an interesting question:

If I look at the motor shaft, and its turns merge into one, it will be a function. If I count every turn of the shaft by 90 degrees or another angle, this will be an operation. And if I follow the position of a point on the shaft, for example, in order to automatically control its coordinates, what is it? For example, on the shaft of a radar antenna. There are no operations, since there is no beginning and end of movement, the turns of the shaft are not discrete, the required position of the antenna is constantly changing by the operator, and the actual changes from the wind. There are no functions either, since the turns of the shaft do not merge into a single rotation. What is it?

The question is so interesting that I decided to devote a separate article to him. This will help on a specific example to deal with the definitions of projection modeling. At the same time, I will tell you what my request is for mathematicians.

Formulation of the problem

So, let me remind the postulate What is hidden behind the term modeling : we do not know how the world works, but we know how we interpret it. This means that if I see the plane flying, then in reality I do not register EACH position of the plane in space. I register only a FEW provisions that I can interpret as a continuous movement. Whether the movement was continuous or not, we do not know. We are given:

• Several positions of the aircraft in space-time, measured with some error.
• The hypothesis that the plane was moving continuously in time.

Required:

• For a specific point in time, assess the possible positions of the aircraft. And ideally, give an algorithm for obtaining such estimates for any point in time with an indication of their probability.

Do you feel how the statement of the problem actually sounds? Do you understand that this is not exactly the statement we are used to at school? It takes a whole year for a first-year student to forget about school tasks and learn how to set real tasks. And that is not always possible. I regularly observe the difficulty of such a transition.
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Let's return to the motor. Given:

• We have the coordinates of the shaft, measured at some points in time with some accuracy. Suppose that each measurement in the framework of the problem being solved lasted so little that it can be considered not an operation on measurement, but an event. Let me remind you that an event is an operation whose duration can be neglected. From this sequence of events you can build a script: event1, event 2, event3, etc. Let me remind you that the script is a set of operations, in particular, a set of events. So, what we have at the entrance is the script.

Required:

• For any moment of time, estimate the possible positions of the shaft and their probability.

The solution of the problem

To solve this problem, we apply approximation, for example, in the form of functional dependence. Now whoever asks for us, we will be able to create a model of an event relating to any point in time, and for this event indicate the possible shaft positions and the estimated probability of such a position, based on the hypothesis that rotation exists. Therefore, our model will be, on the one hand, finite, because there are a limited number of points, and, on the other hand, it is expandable so that we can receive new events if we need to. The final model is described with the help of a scenario, and we get extensions in the form of hypotheses - we assume that such events could happen at such a time with such a probability.

Now about the definition of the function. I recall the definition of a function: a function is a construct of an infinite number of operations. Since an event is a special case of an operation, it can be said that part of the functions are constructs from an infinite number of events. It is clear that real events are always finite. However, I deliberately used the term infinity. In this context, infinity reflects our perception of function. We perceive it as an endless stream of continuous events. In the last article, the two competencies of the analyst, I showed that in order to do this, it is often necessary to abandon anthropomorphic representations. So we do, but not always aware of it. From the thesis about the infinity of operations it follows the thesis that the description of a function is possible ONLY by the description of classes of operations (events), but not of operations (events) of classes.

It's time to remember about mathematicians. They use axiomatics, in which points are the last indivisible part of matter, from which pieces of matter are obtained by multiplying by the continuum. It is clear that this axiomatics is poorly suited for our case, in which the point is a material body, the dimensions of which we neglect in the framework of the problem being solved. But within the framework of another task, we can remember that this body is material and has quite a finite size.

Similarly, with events - we consider them operations, the duration of which in the framework of the problem being solved, we neglect. But within the framework of another task, it is no longer possible to neglect their duration. Therefore, the explanation of Zeno’s paradox becomes as follows: in detailing events, sooner or later we come to a situation in which it becomes impossible to ignore the duration of the event — we have no means of registering such rapid changes. And then we won't be able to divide time into parts.

In this regard, all objects with which we are dealing are finite. But axiomatics for this yet. I am afraid that this axiom will have to be invented by ourselves.

findings

So, if we have a sequence of events - this is a scenario, if a class of events, then this is a function, if an approximation of events, then this is neither a scenario nor a function - this is a method of constructing models of the events we assume.

Announcement: In the article The concept of connection in projection modeling, I explain in more detail the definition of connection.