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О строении одного замкнутого класса функций трехзначной логики
С. 166-168.
All closed classes from Muchnik's example of closed class with infinite bases are described.
In book
М. : МАКС Пресс, 2017
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
Mikhailovich A., Kochergin V., В кн. : Материалы XIII Международного семинара "Дискретная математика и её приложения" имени академика О.Б. Лупанова. : Изд-во механико-математического факультета МГУ, 2019. С. 129-131.
Added: December 7, 2021
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
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
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
Kochergin V., Mikhailovich A., Ученые записки Казанского университета. Серия: Физико-математические науки 2020 Т. 162 № 3 С. 311-321
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 of this type is ...
Added: December 6, 2021
Kochergin V., Mikhailovich A., В кн. : Проблемы теоретической кибернетики. Материалы заочного семинара XIX международной конференции. : Издательство Казанского (Приволжского) федерального университета, 2021. С. 75-78.
В работе исследуется сложность реализации функций многозначной логики над базисами, содержащими все монотонные функции и конечное число немонотонных функций. Получены верхняя и нижняя оценка, отличающиеся на константу, не зависящую от базиса. ...
Added: December 6, 2021
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., / 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., 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
Podolskaya O., 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. 97-100.
We study Boolean circuit complexity in an infinite basis consisting of all Boolean functions that equal 1 only on sets of pair-wise incomparable tuples. It is known that lower bounds on the complexity of linear function, majority function and almost all boolean functions of $n$ variables are of the order $\sqrt n.$ We show that ...
Added: May 31, 2015
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. 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 V., Mikhailovich A., Mathematical notes 2023 Vol. 113 No. 5 P. 794-803
The problem of determining the nonmonotone complexity of the implementation ofk-valued logic functions by logic circuits in bases consisting of all monotone (with respect to thestandard order) functions and finitely many nonmonotone functions is investigated. In calculatingthe complexity measure under examination only those elements of the circuit which are assignednonmonotone basis functions are taken into ...
Added: November 19, 2023
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., 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
Mikhailovich A., Kochergin V., В кн. : Материалы 5-й Российской школы-семинара "Синтаксис и семантика логических систем". : Улан-Удэ : Издательство Бурятского госуниверситета, 2017. С. 48-52.
Problem of multi-valued function realization by logic circuits in special bases is investigated. These bases consist of all monotone functions with zero weight and finite number of non-monotone functions with unit weight. ...
Added: September 22, 2017
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
Kochergin V., Чебышевский сборник 2022 Т. 23 № 2(83) С. 121-150
В работе предпринята попытка не только дать обзор результатов, полученных
О. М. Касим–Заде, крупнейшим специалистом по дискретной математике и математической кибернетике, но и осознать его научное наследие в таких направлениях как исследование мер схемной сложности булевых функций, связанных с функционированием схем,
проблематика неявной и параметрической выразимости в конечнозначных логиках, вопросы глубины и сложности булевых функций и функций ...
Added: October 29, 2022
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., 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., 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
Михайлович А. В., Кочергин В. В., В кн. : Материалы XVIII международной конференции "Проблемы теоретической кибернетики" (Пенза, 19-23 июня 2017 г.). : М. : МАКС Пресс, 2017. С. 142-144.
The problem of multi-valued functions realization by circuits over special basis is inverstigated. The basis consis of Post negation and all monotone functions. ...
Added: September 21, 2017