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О замкнутых классах функций в P_3, порожденных периодическими симметрическими функциями
Вестник Нижегородского университета им. Н.И. Лобачевского. 2013. № 1. С. 208-212.
Closed classes are considered of three-valued logic functions generated by symmetric functions taking values in the set {0, 1}. Criteria for existence of bases and for existence of finite generating systems are obtained for some classes generated by elementary periodic symmetric functions.
Mikhailovich A., / Cornell University. Series math "arxiv.org". 2016.
Closed classes of three-valued logic generated by periodic symmetric funtions that equal $1$ in tuples from $\{1,2\}^n$ and equal $0$ on the rest tuples are considered. Criteria for bases existence and finite bases existence for these classes is obtained. ...
Added: April 15, 2016
Mikhailovich A., / Cornell University. Series math "arxiv.org". 2015.
Closed classes of three-valued logic generated by symmetric funtions that equal 1 in almost all tuples from {1,2}n and equal 0 on the rest tuples are considered. Criteria for bases existence for these classes is obtained. ...
Added: March 28, 2015
Mikhailovich A., Прикладная дискретная математика 2015 № 1 С. 17-26
Closed classes of three-valued logic functions whose generating systems consist of symmetric functions taking values in the set {0, 1} and taking value 1 on bounded number of layers from {1, 2}^n are consideder. Criteria of existence of a basis and existence of finite basis are obtained for these classes. There shown how existence of ...
Added: March 11, 2015
Mikhailovich A., Moscow University Mathematics Bulletin 2012 Vol. 67 No. 1 P. 41-45
Closed classes of functions of three-valued logic whose generating systems include nonmonotone symmetric functions taking values in the set {0,1} are studied. It is shown that in some cases the problems of existence of a basis and existence of a finite basis can be reduced to a similar problem for reduced generated systems. ...
Added: October 30, 2012
Mikhailovich A., В кн. : Математические вопросы кибернетики. Вып. 18.: М. : Физматлит, 2013. С. 123-212.
Closed classes of functions of many-valued logic are studied. Problem on the basis existence is considered for some families of closed sets. Functions from generating systems are symmetric functions taking the values from the set {0,1} and equal to zero on the unit collection and collections containing at least one zero. Furthermore, closure of any ...
Added: March 25, 2014
Mikhailovich A., В кн. : Материалы X молодежной научной школы по дискретной математике и ее приложениям. : М. : Издательство ИПМ РАН, 2015. С. 51-55.
Closed classes of three-valued logic functions generated by quazi-symmetric functions that take values from the set {0,1} are considered. Criteria of basis existence and finite basis existence have been obtained. ...
Added: April 8, 2016
Mikhailovich A., В кн. : Материалы IX молодежной научной школы по дискретной математике и ее приложениям (Москва, 16-21 сентября 2013 г.). : М. : Издательство ИПМ РАН, 2013. С. 80-85.
Closed classes of functions of three-valued logic whose generating systems include nonmonotone symmetric functions taking values in the set {0,1} and taking value 1 on restricted number of layers are studied. Cryteria of existence of basis and existence of finite basis has been obtained. ...
Added: October 24, 2013
Mikhailovich A., В кн. : Материалы XII Международного семинара "Дискретная математика и её приложения" имени академика О.Б. Лупанова (Москва, МГУ, 20-25 июня 2016г.). : М. : Изд-во механико-математического факультета МГУ, 2016. С. 209-212.
Closed classes of three-valued logic, generated by periodical functions taking values from the set {0,1} are considered. Criteria of basis exitstence and finite basis existence for classes generated by periodical functions with period of the form p^k (p is fixed prime number, k is arbitrary natural number) are obtained. ...
Added: September 1, 2016
Mikhailovich A., В кн. : Проблемы теоретической кибернетики. Материалы XVII международной конференции. : Каз. : Отечество, 2014. С. 204-206.
Closed classes of three-valued logic functions whose generating systems consist of symmetric functions taking all values in the set {0, 1, 2} and taking values 1 and 2 on tuples from {1, 2}^n are consideder. Criteria of existence of a basis and existence of finite basis are obtained for these classes. ...
Added: March 12, 2015
Mikhailovich A., В кн. : Труды IX Международной конференции "Дискретные модели в теории управляющих систем". : М. : МАКС Пресс, 2015. С. 163-166.
Closed classes of multi-valued logic are observed. Families of closed classes generated by function with special properties are considered. Criteria for basis existence have been obtained for these classes. ...
Added: March 28, 2015
Mikhailovich A., Вестник Московского университета. Серия 1: Математика. Механика 2012 № 1 С. 58-62
Изучаются замкнутые классы функций трехзначной логики, порождающие системы которых содержат симметрические функции, принимающие значения из множества {0, 1}. Показано, что в некоторых случаях задачи о базируемости и конечной порожденности для таких классов сводятся к аналогичным задачам для классов, порождающие системы которых являются подмножествами порождающих систем исходных множеств. ...
Added: October 30, 2012
Mikhailovich A. V., Kochergin V. V., / Cornell University. Series math "arxiv.org". 2015.
The minimum number of NOT gates in a logic circuit computing a Boolean function is called the inversion complexity of the function. In 1957, A. A. Markov determined the inversion complexity of every Boolean function and proved that $\lceil\log_{2}(d(f)+1)\rceil$ NOT gates are necessary and sufficient to compute any Boolean function f (where d(f) is the ...
Added: October 20, 2015
Kochergin Vadim V., Mikhailovich Anna V., Discrete Mathematics and Applications 2017 Vol. 27 No. 5 P. 295-302
The paper is concerned with the complexity of realization of 𝑘-valued logic functions by logic circuits over an infinite complete bases containing all monotone functions; the weight of monotone functions (the cost of use) is assumed to be 0. The complexity problem of realizations of Boolean functions over a basis having negation as the only ...
Added: March 14, 2018
Mikhailovich A., Kochergin V., Дискретная математика 2016 Т. 28 № 4 С. 80-90
In this paper we consider the complexity of realization of k-valued logic functions by logic circuits over an infinite complete basis of special type. This basis contain all monotone functions with zero weight and non-monotone functions with non-zero weight. The problem of the complexity of a Boolean functions realization over basis containing all monotone functions ...
Added: February 25, 2017
Kochergin V., Mikhailovich A., Дискретный анализ и исследование операций 2018 Т. 25 № 1 С. 42-74
The complexity of realization of k-valued logic functions by circuits in a special infinite basis is invesigated. This basis consists of Post negation (i.e. function x+1(mod k)) and all monotone functions. The complexity of the circuit is the total number of elements of this circuit. The upper and the lower bounds of the complexity were ...
Added: September 28, 2017
Mikhailovich A., В кн. : Материалы 5-й Российской школы-семинара "Синтаксис и семантика логических систем". : Улан-Удэ : Издательство Бурятского госуниверситета, 2017. С. 91-95.
Lattice of all closed classes from closure of all functions from Janov and Muchnik examples has been described. ...
Added: September 22, 2017
Makarov I., / Logica Universalis. Series " ". 2015.
The article deals with finding finite total equivalence systems (FTES) for formulas based on an arbitrary closed class of functions of several variables defined on the set {0, 1, 2} and taking values in the set {0,1} with the property that the restrictions of its functions to the set {0, 1} constitutes a closed class ...
Added: October 17, 2013
Mikhailovich A.V., Kochergin V.V., Siberian Electronic Mathematical Reports 2017 Vol. 14 P. 1100-1107
The problem of the complexity of multi-valued logic functions realization by circuits in a special basis is investigated. This kind of basis consists of elements of two types. The first type of elements are monotone functions with zero weight. The second type of elements are non-monotone elements with unit weight. The non-empty set of elements ...
Added: September 28, 2017
Ilya Makarov, Olga Gerasimova, Logica Universalis 2017
We describe a method of finding the canonical types of formulas based on three-valued projection logic functions. The method focuses on a separation of all tuples of values for variables into disjoint sets and write indicators of these sets using only functions from the closed class under consideration. We obtain the required canonical type combining ...
Added: September 25, 2015
Existence of Finite Total Equivalence Systems for Certain Closed Classes of 3-Valued Logic Functions
Makarov I., Logica Universalis 2015 Vol. 9 No. 1 P. 1-26
The article deals with finding finite total equivalence systems for formulas based on an arbitrary closed class of functions of several variables defined on the set \{0, 1, 2\} and taking values in the set \{0,1\} with the property that the restrictions of its functions to the set \{0, 1\} constitutes a closed class of ...
Added: February 28, 2015
Kochergin V.V., Mikhailovich A.V., Journal of Applied and Industrial Mathematics (перевод журналов "Сибирский журнал индустриальной математики" и "Дискретный анализ и исследование операций") 2018 Vol. 12 No. 1 P. 40-58
The complexity of realization of k-valued logic functions by circuits in a special infinite basis is under study. This basis consists of Post negation (i.e. function x+1(mod k)) and all monotone functions. The complexity of the circuit is the total number of elements of this circuit. For an arbitrary function f, we find the lower and upper bounds ...
Added: March 11, 2018
V.V. Kochergin, A.V. Mikhailovich, Computational Mathematics and Modeling 2019 Vol. 30 No. 1 P. 13-25
We investigate the realization complexity of k-valued logic functions k ≥ 2 by combinational circuits in an infinite basis that includes the negation of the Lukasiewicz function, i.e., the function k−1−x, and all monotone functions. Complexity is understood as the total number of circuit elements. For an arbitrary function f, we establish lower and upper ...
Added: April 22, 2019
Улан-Удэ : Издательство Бурятского госуниверситета, 2017
The collection represents proceedings of the 5th school-seminar "Syntax and Semantics of Logic Systems" (Ulan-Ude, 08.08.2017 - 12.08.2017). The conference subject area includes: theory of models and universal algebra; theory of boolean and finite-valued functions; formal languages and logic calculus; mathematical logic in education. ...
Added: September 22, 2017
Киселева Л. Г., Talanov V. A., Математика в высшем образовании 2008 № 6 С. 67-76
Обсуждаются материалы тестового характера, используемые в течение ряда лет на факультете вычислительной математики и кибернетики Нижегородского государственного университета им. Н.И.Лобачевского при изучении начальных разделов линей ной алгебры. Авторы собрали достаточно большое число контрольных вопросов, отражаюжих логическую взаимосвязь между математическими понятиями. Ответы на вопросы базируются на небольшом числе теорем, которые входят в типовые программы курса геометрии ...
Added: January 14, 2013