Functional decomposition: Difference between revisions

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Revision as of 13:51, 16 September 2020

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In mathematics, '''functional decomposition''' is the process of resolving a [[Function (mathematics)|functional]] relationship into its constituent parts in such a way that the original function can be reconstructed (i.e., recomposed) from those parts by [[function composition]].

This process of decomposition may be undertaken to gain insight into the identity of the constituent components which may reflect individual physical processes of interest. Also functional decomposition may result in a compressed representation of the global function, a task which is feasible only when the constituent processes possess a certain level of ''modularity'' (i.e., independence or non-interaction).

{{clarify span|Interactions|reason=The notion of 'interaction' of mathematical functions is undefined, likewise for 'observable', 'perception' etc. I guess this paragraph confuses mathematical notions (like 'function') with physical intuitions (like 'process'); this should be fixed.|date=September 2020}} between the components are critical to the function of the collection. All interactions may not be {{clarify span|observable|date=September 2020}}, but possibly deduced through repetitive {{clarify span|perception|date=September 2020}}, synthesis, validation and verification of composite behavior.

== Basic mathematical definition ==

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:<math>f(x_1,x_2,\dots,x_n) = \phi(g_1(x_1,x_2,\dots,x_n), g_2(x_1,x_2,\dots,x_n), \dots g_m(x_1,x_2,\dots,x_n))</math>

where <math>\phi</math> is some other function.{{clarify|reason=This definition is horribly underspecified, similar to e.g. decomposing a number into the sum of serveral others. Unless the set of admitted g_i or/and phi is given, it does not make sense at all. One trivial solution is m=n, g_i(x_1,...,x_n)=x_i, and phi=f; another one is m=1, g_1=f, and phi(x)=x.|date=September 2020}} Thus, we would say that the function <math>f</math> is decomposed into functions <math>\{g_1, g_2, \dots g_m\}</math>. This process is intrinsically hierarchical in the sense that we can (and often do) seek to further decompose the functions <math>g_i</math> into a collection of constituent functions <math>\{h_1, h_2, \dots h_p\}</math> such that

:<math>g_i(x_1,x_2,\dots,x_n) = \gamma(h_1(x_1,x_2,\dots,x_n), h_2(x_1,x_2,\dots,x_n), \dots h_p(x_1,x_2,\dots,x_n))</math>