Abstract
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.
Highlights
One may define recent years as a period of rapid development of technical means and information technologies with high performance efficiency that led to the creation and implementation of more effective methods of processing and analysis of data and new methods of solving complex applied problems
Significant resources that are invested in creating software and hardware implementation of artificial neural networks, as well as widespread use of neuro-like structures, indicate that the problem of synthesis of neural elements with different activation functions and the construction of logical circuits from them is relevant and practically significant
When recognizing discrete images, at the compression and transmission of discrete signals, it is necessary to be able to synthesize neural elements, that have a large number of inputs (≥100); in these cases, the classical methods of approximation of different orders and various iterative methods cannot be applied to the synthesis of neural elements for the realization of discrete functions
Summary
One may define recent years as a period of rapid development of technical means and information technologies with high performance efficiency that led to the creation and implementation of more effective methods of processing and analysis of data and new methods of solving complex applied problems. In this regard, there is a surge of theoretical and practical techniques in the field of neurocomputers and there is increased interest in neuro-like structures, which are widely applied in various areas of human activity – pattern recognition, forecasting, business, medicine, engineering. The results obtained in present work make it possible to synthesize neural elements with a large number of inputs for the implementation of Boolean functions under certain constraints on their kernels
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