Hierarchical fuzzy modeling and jointly expandable functions

Hierarchical modeling is a promising tool to deal with too complex systems in the real world. A large-scale problem can be easily solved if the problem is equivalently partitioned into several independent smaller problems, which are locally solved and then combined into the globally optimal solution. This technique is what we call the “divide-and-conquer” method. Examples of hierarchical modeling include fuzzy classifier systems,2 datamining using genetic algorithms,11 approximate reasoning,3 neural networks,10 and fuzzy integral.5 Partitioning reduces the complexity of the given problem, which needs nonlinear running time to solve; i.e., O(n2), O(2n). The reductions of cost imply faster computation, saving resources such asmemory occupancy, and simplifying the problem space to search. However, we should notice that not all problems can be divided independently, or even if a problem was partitioned, subproblems might not be disjoint. Let us consider the case when a problem cannot be divided at all. For a boolean function f (x, y, z) = xy∨ x̄ ȳ∨ xz, there are no functions g and h such that: f (x, y, z) = g(x, h(y, z))