Статья 'Способы реализации функций троичной логики' - журнал 'Кибернетика и программирование' - NotaBene.ru
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Cybernetics and programming
Reference:

Ways of implementing ternary logic functions

Giniyatullin Vakhit Mansurovich

PhD in Technical Science

Associate Professor of the Department of Computer Science and Engineering Cybernetics at the Ufa State Oil Technical University

450062, Rossiya, Ufa, ul. Kosmonavtov, d. 1

fentazer@mail.ru
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Arslanov Ismagil Ganievich

Doctor of Technical Science

Professor of the Department of Computer Science and Engineering Cybernetics at the Ufa State Oil Technical University

450062, Rossiya, Ufa, ul. Kosmonavtov, d. 1

tnsfloat@mail.ru
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Bogdanova Polina Dinarovna

undergraduate student, Ufa State Oil Technical University

450062, Rossiya, Ufa, ul. Kosmonavtov, d. 1

tnsfloat@mail.ru
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Gabitov Rustam Nailovich

Assistant of the Department of Computer Science and Engineering Cybernetics at the Ufa State Oil Technical University

450062, Russia, Ufa, ul. Kosmonavtov, d. 1

tnsfloat@mail.ru
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Salikhova Marina Akhatovna

Senior Lecturer of the Department of Computer Science and Engineering Cybernetics at the Ufa State Oil Technical University

450062, Russia, Ufa, ul. Kosmonavtov, d. 1

tnsfloat@mail.ru
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DOI:

10.7256/2306-4196.2014.2.11918

Review date:

17-03-2014


Publish date:

1-4-2014


Abstract: The source data is represented by three-dimensional truth table for 3-demensional functions of binary, ternary and mixed logics. The calculation of values of logic function is performed by geometric interpretations, disjunctive / conjunctive normal forms, non-fully connected artificial neural networks and perceptrons with hidden layer. The article reviews in detail the intermediate results of calculations in all given ways. The authors study properties of mixed logic functions: binary-ternary and 3-2 logics in cases of one, two and three dimensions. The article shows mutually equivalent implementations of logic functions in the form of DNF and non-fully connected artificial neural network. The authors perform replacement of continuous activation function on ternary threshold function. The study uses the methods of DNF constructing; direct synthesis of ANN weight matrices, perceptron is trained using Back Propagation algorithm, some conclusions are made by the laws of mathematical inductions. The paper shows that:

Minimizations of neuron quantity in the perceptron hidden layer implicitly leads to the use of multiple-valued logics;
Some functions of binary-ternary logics may be used for generating disjunctive forms;
There is a decisive way of converting DNF to ANN and vice versa;
In the one-dimension  3-2 logics there only 8 functions and all of them are listed;

The suggested ANN structure can implement any other function of ternary logics of any dimensions.


Keywords: three-valued logic, 3-2 logic, binary-ternary logic, perfect disjunctive form, activation function, separating hyperplane, perceptron, XOR problem, neural network training, Back Propagation algorithm
This article written in Russian. You can find full text of article in Russian here .

References
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Kodd E. F. Rasshirenie relyatsionnoi modeli dlya luchshego otrazheniya semantiki. // SUBD, 1996, №2. S. 141-160.
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Deit K.Dzh. Vvedenie v sistemy baz dannykh. Moskva. Vil'yams. 2000. 848 s.
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URL: http://dit.ipg.pt./MBP
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Giniyatullin V.M. Modelirovanie logicheskikh funktsii v neirosetevom bazise. // Neftegazovoe delo, 2008 № 1 C. 35-43.
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URL: http://ru.wikipedia.org/wiki/Troichnye_funktsii
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