Synthesis of generalized neural elements by means of the tolerance matrices - Geche, Fedir; Mulesa, Oksana; Buchok, Viktor (Eastern-European Journal of Enterprise Technologies, 2017)
Application of neuromorphic structures in various spheres of human activity on the basis of generalized neural elements will become possible if effective methods for verifying realizability of the logic algebra functions by one neuron element with a generalized threshold activation function and synthesis of such elements with a large number of edntries are developed. A notion of nucleus of Boolean functions in relation to a given system of characters was introduced and algebraic structure of nuclei and reduced nuclei of Boolean functions was investigated. Relation between the nuclei of the logic algebra functions which are realized by one generalized neural element and matrices of tolerance was established. It was shown that the Boolean function is realized by one generalized neuron element if and only if the nucleus of this function admits representation by the matrices of tolerance. If there is no nucleus relative to a specified system of characters for a Boolean function, then such a function is not realized by one generalized neural element in relation to a specified system of characters. On the basis of the properties of the matrices of tolerance, a number of necessary and sufficient conditions for realization of the logic algebra functions by one generalized neural element were obtained. Based on the sufficient conditions, an algorithm for synthesis of integer-valued generalized neural elements with a large number of entries was constructed. In the synthesis of integer-valued generic neural elements for realization of the logic algebra functions, a block representation of the Boolean function nucleus was used and based on the properties of the matrices of tolerance, coordinates of the integer vector of the structure of the generalized neural element were sequentially found