Verification of realizability of boolean functions by a neural element with a threshold activation function
DOI:
https://doi.org/10.15587/1729-4061.2017.90917Keywords:
tolerance matrix, convex linear shell, structure vector, activation functionAbstract
A widespread application of neural network circuits from neural elements with a threshold activation function would be possible if efficient methods for the verification of realizability of functions of the algebra of logic by one neural element are devised, as well as the synthesis of these elements with a large number of inputs. The article examines algebraic structure of kernels and reduced kernels of Boolean functions. A connection is established between the kernels of Boolean functions that are implemented by one neural element with a threshold activation function and tolerance matrices. Based on the convex linear combination of kernel elements of functions of the algebra of logic, we proved a criteria of their realizability by one neural element with a threshold activation function. By using algebraic properties of kernels in Boolean functions and the representations of their reduced kernels by tolerance matrices, we obtained a number of easily verified necessary conditions for the realizability of functions of the algebra of logic by one neural element. These necessary conditions in many cases make it possible not to perform complicated calculations by the methods of approximation of different orders and by the iterative methods, in which, by means of limit cycles, the realizability or non-realizability of Boolean functions by one neural element with a threshold activation function is determined. Based on the sufficient conditions, obtained in the work, for the realizability of functions of the algebra of logic by one neural element, we devised an effective method for the synthesis of integer neural elements with a large number of inputs.
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