
related topics 
{math, number, function} 
{theory, work, human} 
{system, computer, user} 
{@card@, make, design} 
{area, part, region} 

Functional decomposition refers broadly to the process of resolving a 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. In general, this process of decomposition is undertaken either for the purpose of gaining insight into the identity of the constituent components (which may reflect individual physical processes of interest, for example), or for the purpose of obtaining 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 noninteraction).
Contents
Basic mathematical definition
For a multivariate function , functional decomposition generally refers to a process of identifying a set of functions such that
where φ is some other function. Thus, we would say that the function f is decomposed into functions . This process is intrinsically hierarchical in the sense that we can (and often do) seek to further decompose the functions g_{i} into a collection of constituent functions such that
where γ is some other function. Decompositions of this kind are interesting and important for a wide variety of reasons. In general, functional decompositions are worthwhile when there is a certain "sparseness" in the dependency structure; that is, when constituent functions are found to depend on approximately disjoint sets of variables. Thus, for example, if we can obtain a decomposition of into a hierarchical composition of functions {g_{1},g_{2},g_{3}} such that x_{1} = g_{1}(x_{2}), x_{2} = g_{2}(x_{3},x_{4},x_{5}), x_{5} = g_{3}(x_{6}), as shown in the figure at right, this would probably be considered a highly valuable decomposition.
Full article ▸


related documents 
Complexity 
Abductive reasoning 
Planner (programming language) 
Theorem 
Relation (mathematics) 
John von Neumann 
Universal algebra 
Probability 
Infinite monkey theorem 
Number theory 
Arrow's impossibility theorem 
Reinforcement learning 
Hilbert's second problem 
Graph theory 
Universal quantification 
Prototypebased programming 
Optimality theory 
Optimization (mathematics) 
Henri Lebesgue 
Polish notation 
Supervised learning 
Tensor 
Sheffer stroke 
Integer (computer science) 
Ordered pair 
Partially ordered set 
NP (complexity) 
Normal space 
Greatest common divisor 
Net (mathematics) 
