Structural Operations

    Introduction


    In the article devoted to connections , I gave the definition of connection:

    Connection is a 4-D volume common to connected objects (operations)

    Since a 4-D volume can be projected onto space and temporarily in any way, the connection can be considered separately from the connected objects as as we wish. In the article on relations, I gave an example of the relationship between the two functions “production of bearings” and “consumption of bearings” (read - total 4-D space), which I also presented in the form of the function “reception-transmission of bearings”.

    Consideration of communication as a 4-D object allows within the framework of projection modelingintroduce a useful formalism: operations on structural elements (scripts). You can now perform the same operations on structural elements as on set elements.

    Many can be folded, so you can combine designs together.

    Many can be subtracted, so another can be subtracted from one design.

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

    Previously, this could not be done, because there was no interpretation of relations. How can one element be deleted if it is connected to another element: where to put the link? Since we have now defined the connection as a 4-D object common to 4-D objects to be connected, the connection remains in place even after removing one of the connected elements.

    Types of Links


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

    Relations 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 connection remains, too.

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

    Causal relationships such as “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 4-D volume projected onto the space (structure), then:

    • The operation of combining two structures allows a formal way to construct a projection of the combined 4-D volume in the form of a combined structure.
    • The intersection operation of two structures allows us to formally construct a projection of the combined 4-D volume in the form of a general structure for two structures.
    • The subtraction operation allows you to find the construction remaining from the subtraction.

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

    • The operation of combining two scenarios allows you to formally construct a projection of the combined 4-D volume in the form of a combined scenario.
    • The operation of intersecting two scenarios allows you to formally construct a projection of the combined 4-D volume in the form of a common scenario for two scenarios.
    • The subtraction operation allows you to find the script remaining from the subtraction.

    From the above, a methodology for designing structures, whether spatial or scenario, follows. Let's 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 involves modeling the relationships between structural elements and elements outside the structure. If we are talking about a spatial connection of the “Higher-lower” type, then for the elements of this design we can simulate these relationships with those objects that are outside the scope of our design. Those who studied physics probably remember how in optics when setting a task they often draw the eye of an 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 systems engineering, the fact that a description of a “system” should begin with a description of its interfaces is often mentioned. I specifically do not introduce the concept of a system, because it is not clear what in system engineering means: an object, its construction, function, or functional structure. But the message is clear - if you want to make a complete description of the structure, describe the relationship of objects with the external environment.

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

    If we are talking about a scenario, then external temporal and causal relationships will be links with the external environment. These connections “rest” with one end in the script operation, and with the other in the operation, which are not in our model, but there is an assumption that they exist.

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

    Structural Operations Methodology


    Merge operation


    If it is required to combine two constructions, then this task does not arise just like that. Behind it is a need. This need is that we have reached the boundary of the design description and want to move on. The border, as we recall, is communication. Since there are connections in the description, all we have to do is say which connections of the two structures are common to them. Thus, we fit one design with another.

    One may ask: why are bonds the interface through which structures dock? Is it possible to use common objects for this? Yes, it doesn’t matter what type of objects we use for docking, but these are those that are in both structures, and looking at them, we can say that this is the same element, whether it be a connection, or an object.

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

    If you want to combine the two scenarios, then the relationships between them will be common temporal, or causal relationships. In the same way, docking can be done through common operations.

    Subtraction and intersection methodology


    The need for subtraction arises when the author of the model wants to concentrate on parts of the structure. During the operation of subtraction or intersection, the rules remain the same, but strange things can arise from the point of view of common sense. For example, as a result of subtraction or intersection, the following may remain as the remainder:

    • Nothing
    • Relationships without objects
    • Objects without links

    Will the objects listed above be considered constructions? Answer: they will, if you look at them not as objects, but as sets of objects. Let me remind you that a design is a lot of objects. Any set has a composition. So, the composition of the set can be any. A design is a lot of objects. The composition of this set can be any, even counter-intuitive - consisting of nothing, or only of connections.

    conclusions


    Conclusions: the definition of the relationship allowed us to introduce on the set of elements that make up the structure, operations similar to operations on ordinary sets: addition, subtraction, intersection. This allowed us to formally approach the transformation of models in tasks related to the expansion or narrowing of the simulated area.

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