Qualitative mapping, inner product transformation of qualitative criterion, artificial neuron and pattern recognition

The qualitative mapping (QM) model for judging a property p(o) whose true value varies according to the qualitative criterion [α,β], τ_p(x, [α,β]) is presented in this paper. The inner product transformation of qualitative criterion w_[α,β] and the relation between w_[α,β] and artificial neuron is discussed. By the the intercept form of artificial neuron, we prove that an AN is just a boundary of a qualitative mapping, and if a neighborhood is closed by a group of artificial neurons induced by a group inner product transformations, then the qualitative mapping whose criterion is the closed neighborhood is equivalent to the artificial networks which consists of the group of them.