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Structural Operations

Introduction


In the article on relationships , I gave the definition of a relationship:

Communication is a 4-D volume, common to the objects being linked (operations)

Since the 4-D volume can be projected onto space and for a while in any way, the connection can be viewed separately from the connected objects as we wish. In the article on communications, I gave an example of the relationship between the two functions “bearing production” and “bearing consumption” (read - total 4-D space), which I also presented as a function “bearing transfer and transmission”.
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Considering the relationship as a 4-D object allows us to introduce a useful formalism in the framework of projection modeling : operations on the construction elements (scenario). The elements of the structure can now be carried out the same operations as on the elements of the sets.

Sets can be put together, so you can combine structures together.

Sets can be subtracted, so another one can be subtracted from one construction.

You can search for intersections of sets, so you can search for intersections of structures.

Previously, it was impossible to do this, because there was no interpretation of the connections. How can one element be deleted if it is connected with another element: where should we connect? Since we have now defined a relationship as a 4-D object, common to 4-D objects to be linked, the connection remains in place even after removing one of the connected elements.

Types of links


In the example with the bearings and even after removing the two functions “bearing production” and “bearing consumption” from the model, the connection remains - the function “receiving transfer bearings”.

Connections of the type “above”, “to the right”, etc., which reflect the property of the space in which the objects are placed, do not disappear with the disappearance of the object from the model. After all, the spatial volume occupied by this object remains in the model. Therefore, the relationship remains too.

Relationships of the “preceding-follows” type, which are a temporal analogue of the “higher-lower” connection in space, also do not disappear with the disappearance of an object from the model, because it is a property of 4-D space and not an object placed in it.

Causal relationships of the type - “the result of an activity in one operation is used in another operation” also do not disappear with the disappearance of an object from the model, because these are properties of 4-D space, and not objects placed in it.

Structural Operations


What in practice means the possibility of carrying out operations of addition of subtraction and intersection over structural elements?

If we are talking about a 4-D volume projected on the space (structures), then:


If we are talking about 4-D volume projected on time (scenarios), then:


From the above follows the methodology for the design of structures, whether spatial, or scenarios. Consider it in detail.

Structural Design Methodology


If we build a spatial structure, then it does not rest in infinite space. God knows where. The design is surrounded by other elements. Full-fledged structural modeling includes modeling the links between structural elements and elements outside the structure. If we are talking about a spatial connection like “Above-below”, then for the elements of a given structure we can simulate these connections with those objects that are outside of our structure. Those who studied physics probably remember how in optics, when setting a task, they often draw the eye of the observer, or in mechanics they often draw the upper part of the upper support on which the blocks hang. This is a description of relationships with those objects that are not in the design model.

In system engineering, it is often mentioned that the description of a “system” must begin with a description of its interfaces. I do not specifically introduce the concept of a system, because it is not clear what system engineering means: an object, its design, function, or functional structure. But the message is clear - if you want to make a full description of the structure, describe the connection of objects with the external environment.

For a structure, this is the spatial position of objects that are not part of the structure model.

If we are talking about scenarios, then external temporal and causal relationships will be links with the external environment. These links "rest" with one end on the script operation, and on the other on the operations that are not in our model, but there is an assumption that they exist.

If we are talking about the functional structure, then the connections with the external environment will be the boundary functions. They can be seen on the diagram in IDEF0 notation in the form of arrows going to the outside world.

Structural Operations Methodology


Join operation


If you want to combine the two structures, then this task does not arise just like that. Behind it stands the need. This need is that we have reached the boundary of the description of the structure and want to move on. The border, as we remember, is communication. Since there are links in the description, all we have to do is to tell which links of the two structures are common to them. Thus, we connect one construction with another.

One may ask: why are connections the interface through which the structures are joined? Is it possible to use common objects for this? Yes, it does not matter what type of objects we use for docking, but these are those that are in both designs, and, looking at which, we can say - this is one and the same element, be it a connection, or an object.

If it is required to combine two functional structures, then the links between them will be common functions. When combined, we simply indicate these common functions, thereby forming connections. In the diagram in IDEF0 notation, this is a combination of arrows.
Is it possible to draw a border not on the “arrows” and on the “functions”? In the same way, as with designs - it is possible. You can simply say that these two functions, shown in these two diagrams, are one and the same function.

If you want to combine two scenarios, then the links between them will be common temporal, or cause-effect relationships. Similarly, you can do the docking through common operations.

Subtraction and intersection methodology


The need for subtraction arises when the author of the model wants to concentrate his attention on the part of the structure. When performing a subtraction or crossing operation, the rules remain the same, but things may appear strange from the point of view of common sense. For example, as a result of subtraction or intersection, the remainder may remain:


Will the objects listed above be construed? Answer: they will, if you look at them not as objects, but as sets of objects. Let me remind you that the design is a set of objects. Any variety has a composition. So, the composition of the set can be anything. The design is a set of objects. The composition of this set can be any, even counterintuitive - consisting of nothing, or only of links.

findings


Conclusions: the definition of a relationship allowed us to introduce on a set of elements that make up a construction, operations similar to operations on ordinary sets: addition, subtraction, intersection. This allowed a formal way to approach the transformation of models in problems related to the expansion or contraction of the simulated region.

Source: https://habr.com/ru/post/345134/

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