Functional Decomposition - Motivation For Decomposition

Motivation For Decomposition

As to why the decomposition is valuable, the reason is twofold. Firstly, decomposition of a function into non-interacting components generally permits more economical representations of the function. For example, on a set of quaternary (i.e., 4-ary) variables, representing the full function requires storing values, the value of the function for each element in the Cartesian product, i.e., each of the 1024 possible combinations for . However, if the decomposition into given above is possible, then requires storing 4 values, requires storing values, and again requires storing just 4 values. So in virtue of the decomposition, we need store only values rather than 1024 values, a dramatic savings.

Intuitively, this reduction in representation size is achieved simply because each variable depends only on a subset of the other variables. Thus, variable only depends directly on variable, rather than depending on the entire set of variables. We would say that variable screens off variable from the rest of the world. Practical examples of this phenomenon surround us, as discussed in the "Philosophical Considerations" below, but let's just consider the particular case of "northbound traffic on the West Side Highway." Let us assume this variable takes on three possible values of {"moving slow", "moving deadly slow", "not moving at all"}. Now let's say variable depends on two other variables, "weather" with values of {"sun", "rain", "snow"}, and "GW Bridge traffic" with values {"10mph", "5mph", "1mph"}. The point here is that while there are certainly many secondary variables that affect the weather variable (e.g., low pressure system over Canada, butterfly flapping in Japan, etc.) and the Bridge traffic variable (e.g., an accident on I-95, presidential motorcade, etc.) all these other secondary variables are not directly relevant to the West Side Highway traffic. All we need (hypothetically) in order to predict the West Side Highway traffic is the weather and the GW Bridge traffic, because these two variables screen off West Side Highway traffic from all other potential influences. That is, all other influences act through them.

Outside of purely mathematical considerations, perhaps the greatest value of functional decomposition is the insight it provides into the structure of the world. When a functional decomposition can be achieved, this provides ontological information about what structures actually exist in the world, and how they can be predicted and manipulated. For example, in the illustration above, if it is learned that depends directly only on, this means that for purposes of prediction of, it suffices to know only . Moreover, interventions to influence can be taken directly on, and nothing additional can be gained by intervening on variables, since these only act through in any case.