Requirements Analysis & Definition Techniques

Myth, Magic, or Methodology?

Methodologies Based on Decomposition

Decomposition concerns parts and wholes. Most practitioners (and even developers) of decomposition-based methodologies used in system requirements definition often and randomly violate the rules that are inherent in the very concept of decomposition without realizing that they do so. This happens because they do not really understand the elegant power of decomposition in the art of constructing concise expressions of well-structured thought. As a result, their pseudo-decompositions often are flawed.

The root of the problem lies in the word "decomposition" itself. Being defined in the negative, it seems to connote "breaking a whole into its parts", as though those parts already existed and needed merely to be disassembled. This never is the case. Any interesting whole has any number of possible decompositions. By the definition of the word "decomposition", each of those many decompositions must share a unique property: both individually and collectively each collection of parts must compose the same (unique) whole, with no gaps, excesses, or overlaps. Any collection of parts that violates this rule is not a decomposition at all.

The only way to ensure that a collection of parts is a decomposition of a given whole is to ensure that each part individually is itself a whole with an impenetrable boundary (no overlaps) so shaped that it exactly fits with its neighboring parts so as to leave no gaps, while the entire collection exactly matches the boundary of the intended whole (no excess). Anything less is by definition, not a decomposition. But on the other hand, once a decomposition is achieved, then if any part can itself be further decomposed, then the collection of those finer parts with all the other parts also will form a decomposition of the original whole -- by definition. But also on the other hand, if any part is replaced by a collection of subparts that is not a decomposition of itself, then no matter how "small" or seemingly inconsequential the change, the entire collection is not a decomposition of the original whole by definition.

The challenge of decomposition is not to "break a whole into its parts", but to define parts so that they compose the whole. This challenge is made tractable by adopting a way of defining parts in terms of their intended whole in such a way that by definition they do compose that whole. By definition there will be no excesses, gaps, or overlaps in any collection of alleged parts that satisfy the definitions. Therefore they must be parts and must compose the whole and be a decomposition of it.

The mode of part definition can be rigorously defined in terms of a few intuitive primitive concepts:

1. closed boundary (impenetrable but infinitely thin and flexible)

2. inside (of a closed boundary)

3. expansion about a point (inside a boundary)

4. naming of points, regions, and boundaries

5. attaching descriptions to named things

Then the following must be adhered to:

Dl: The substance of a whole is inside its boundary.

D2: A part is a (sub)whole inside its whole.

D3: The substance of a whole is the substance of its parts.

Thus, "substance", "whole", and "part" are descriptive terms attached to certain constructions of the primitive concepts. Notice that every part must have substance (because it is itself a whole), so every part participates in the composition of the whole. A whole considered to have only a single part (an atomic whole) is allowed, and since all of the substance is in that atomic part (which with its own boundary is inside the whole boundary) the boundary of a whole has no substance. Hence Dl - D3 guarantee that the parts are a decomposition of (the substance of) the whole.

The primitive concept of expansion about a point is only auxiliary, to assist in imagining how names and descriptions can be attached to interesting aspects of decompositions. The act of decomposing a whole into parts can universally be modeled as follows:

1. Consider an atomic whole.
2. Conceive of n descriptions of n parts of that whole.
3. Pick at least one point (center of interest) in the substance of the whole for each description.
4. Imagine a bubble of substance surrounding each center of interest such that the substance inside the bubble matches the corresponding part description.
5. Simultaneously expand each bubble until their boundaries collide in such a way that each expanded bubble becomes a part of the whole.

Notice that Step 3 allows parts to be collections of chunks of substance all of which match the same description. (Thus the nails in a house could be a single part.) Notice also that as long as some substance surrounding a center of interest point matches the part description, Step 5 does not specify how the boundary of each part is to expand, although clearly the intent is that the "reach" of each definition from each center of interest should be maximized, if the part definitions are to be useful for some purpose. Any non-atomic part can be further decomposed, of course, so that the boundaries can be made more precise by being wedged in more and more by specific centers of interest assigned to specific competing parts.

It is important to observe that, except for simple inventory examples (nails in a house, like raisons in a cake), in which some "everything else" part fills in all the non-interesting "rest" of the whole, there is no substance "between" the parts of a decomposition. All of the substance surrounding the bubbles has been absorbed into the competing parts leaving only the non-substantive boundary trapped between the expanded bubble boundaries. If a putative boundary ought to be substantive, it becomes another part between the original parts, just like any other part, defined again (by definition) by its non-substantive boundaries with them. This is inherent in the nature of decomposition. Nothing can be done about it -- except to treat the boundaries themselves as another, quite distinct whole.

Consider Step 4 in the decomposition procedure, when the bubbles have just been established and before any expansion takes place. Consider (only for the moment) the bubbles plus the "everything else" pseudo-part to be actual parts; i.e. freeze Step 4 as a proper decomposition of the whole (it needs no Step 5 expansion, by definition). Now consider the "everything else" part to be an entirely new whole -- a whole with many holes, one for each bubble, but nonetheless a whole with a closed boundary and an inside with (its own quite distinct) substance. This is no-man’s-land before the battle of the boundaries. Notice that it stretches from bubble boundary to bubble boundary before the battle is joined.

When two bubbles compete to determine the exact boundary between them they are mutually determining the interface between them. This can be thought of as either one making incursions into the fringe of the other, leaving a jagged boundary, or as a mustering and rearranging of forces so that a smooth, taut boundary results. Whatever the outcome, the new and separate substance of the no-man’s-land can be decomposed independently, as an overlay map, to record the history of the battle -- a commentary on, ‘though not a part of, the substance of the battle. In this way, the nature of the resultant interface, extending into the respective substances of the mutually bounded parts, is itself decomposed as a whole. The interface parts are a decomposition of the interface whole, and do not themselves overlap or intersect. They do, however, through the overlay concept (which may be thought of as projecting the map onto the battlefield itself), overlap and join together various portions of the original whole’s decomposition into parts. But with respect to the original whole, the interface boundaries have no substance at all. Once released, Step 5 definitively reduces no-man’s-land to nothing substantive at all. (Recall that, by definition, if the edge of a part were to become a substantive part of some "in—between" part, that would be merely another part. Interfaces are non substantive explanations of the nature of the border of a boundary, nothing more.)

It is significant to observe that the no-man’s-land interface domain does not include the bubbles, which are by definition atomic for both the substance decomposition and the interface overlay decomposition. This makes futile any attempt to employ the projections of the interface overlay onto the substantive decomposition to obtain a finer level of decomposition than otherwise would be possible by ordinary substantive decomposition. It is indeed possible, and often desirable, to employ such projections to guide the selection of points of interest, but the same parts thus discovered could be defined independently.

It also is significant to observe that, in theory, at least, Step 5 is to be unleashed afresh for every introduction of new points of interest, since even the highest level boundary may be changed by a change at the (new) atomic level.