MODELING COST FUNCTIONS - MATHEMATICS AND CYBERNETICS

The similarities of mathematics and cybernetics are mostly expressed by their metatheoretical nature: they both are used for, not to describe objects and parts of the world, but to understand and model the world. That makes them both metadisciplines which means they can be applied also to themselves: they consist of models about how to build and use models: metamodels (Van Gigh, 1986), but metamodel stays a model and it can be modeled by other metamodels as well as by itself. On the other hand there are differences between the mathematics and cybernetics: simplicity, regularity and invariance versus complexity, variety and process; the fact that systems separate into independent elements versus the fact that elements always interact; and the objective, context and value independent nature of knowledge versus the subjective, context and value dependent nature of knowledge. The cybernetics as such is reaching beyond by covering the phenomena which cannot be placed in a static, unchangeable, formal framework. In order to present these similarities and differences we will take into consideration a problem formally defined as follows. In order to formally define the problem considered here we have to define some notions. Let G(V,E,u) be an undirected weighted graph where V(G)={v(1),v(2),....,v(n)} is the set of nodes, E = E(G) is the set of edges connecting nodes and u is a weight function assigning a positive cost of traversing edges. Moreover, let V*(G) = {v(1)*,v*(2),....,v*(n)} be a sequence of priority nodes. For a walk W = w(0)w(1)w(2)...w(N) starting at v(0)* we define dw(vj) = min {Sigma(k)(i=1)u(w(i-1)w(i))/w(k) = w(j)} and alpha,beta(i)epsilon R+. If W does not meet v(j) then the set dw(v(j)) = infinity. Now the defined problem can be modeled as the Priority Constrained Chinese Postman Problem (PCCPP). INPUT: Undirected weighted graph G(V,E,u). TASK: The objective is to find a walk, which is the shortest walk W that traverses each edge at least once and minimizes the cost. COST: The cost function defined with respect to the above mentioned demands is as follows: COST(W) = alpha . Sigma(wi-1wi epsilon N*) u(w(i-1)w(i)) + Sigma(k)(i-1) beta(i) . d(w)(v(i)*) In the paper we will try to show the differences and similarities mathematics versus cybernetics using this purely mathematically defined model of the real world and try to present it using the cybernetic model.