A 2-dimension linguistic lattice implication algebra(2DL-LIA)can build a bridge between logical algebra and 2-dimension fuzzy linguistic information.In this paper,the notion of a Boolean element is proposed in a 2DL-L...A 2-dimension linguistic lattice implication algebra(2DL-LIA)can build a bridge between logical algebra and 2-dimension fuzzy linguistic information.In this paper,the notion of a Boolean element is proposed in a 2DL-LIA and some properties of Boolean elements are discussed.Then derivations on 2DL-LIAs are introduced and the related properties of derivations are investigated.Moreover,it proves that the derivations on 2DL-LIAs can be constructed by Boolean elements.展开更多
To protect against algebraic attacks, a high algebraic immunity is now an important criterion for Boolean functions used in stream ciphers. In this paper, a new method based on a univariate polynomial representation o...To protect against algebraic attacks, a high algebraic immunity is now an important criterion for Boolean functions used in stream ciphers. In this paper, a new method based on a univariate polynomial representation of Boolean functions is proposed. The proposed method is used to constmct Boolean functions with an odd number of variables and with maximum algebraic immunity. We also discuss the nonlinearity of the constructed functions. Moreover, a lower bound is deter- mined for the number of Boolean functions with rmximum algebraic immunity.展开更多
Single input single output system was studied. With proportion, differential, integral results of deviation between given input and output as controller input, the logic rules in control process was analyzed, these lo...Single input single output system was studied. With proportion, differential, integral results of deviation between given input and output as controller input, the logic rules in control process was analyzed, these logic rule with Pan-Boolean algebra was described, therefore a PID Pan-Boolean algebra control algorithm was obtained. The simulation results indicates that the new control algorithm is more effective compared to the traditional PID algorithm, having advantages such as more than 3 adjustable parameters of controllers, better result, and so on.展开更多
In this paper, we review some of their related properties of derivations on MValgebras and give some characterizations of additive derivations. Then we prove that the fixed point set of Boolean additive derivations an...In this paper, we review some of their related properties of derivations on MValgebras and give some characterizations of additive derivations. Then we prove that the fixed point set of Boolean additive derivations and that of their adjoint derivations are isomorphic.In particular, we prove that every MV-algebra is isomorphic to the direct product of the fixed point set of Boolean additive derivations and that of their adjoint derivations. Finally we show that every Boolean algebra is isomorphic to the algebra of all Boolean additive(implicative)derivations. These results also give the negative answers to two open problems, which were proposed in [Fuzzy Sets and Systems, 303(2016), 97-113] and [Information Sciences, 178(2008),307-316].展开更多
This letter proposes algebraic attacks on two kinds of nonlinear filter generators with symmetric Boolean functions as the filter fimctions. Different fxom the classical algebraic attacks, the proposed attacks take th...This letter proposes algebraic attacks on two kinds of nonlinear filter generators with symmetric Boolean functions as the filter fimctions. Different fxom the classical algebraic attacks, the proposed attacks take the advantage of the combinational property of a linear feedback shift register （LFSR） and the symmetric Boolean function to obtain a tow-degree algebraic relation, and hence the complexities of the proposed attacks are independent of the algebraic immunity （AI） of the filter functions. It is shown that improper combining of the LFSR with the filter function can make the filter generator suffer from algebraic attacks. As a result, the bits of the LFSR must be selected properly to input the filter function with large AI in order to withstand the proposed algebraic attacks.展开更多
There are so many existing methods to obtain system reliability like re-generating point function technique, supplementary variables technique etc., but all these techniques are full of complicated calculations. Keepi...There are so many existing methods to obtain system reliability like re-generating point function technique, supplementary variables technique etc., but all these techniques are full of complicated calculations. Keeping above these facts in mind, the authors in this paper have evaluated some reliability parameters for tele-communication system by using Boolean functions technique and algebraic method. Reliability of considered tele-communication system has been evaluated by considering the fact that failures follow arbitrary time distribution. In particular, the reliability expression has also been calculated for Exponential and Weibull distributions. Further, an important reliability parameter namely M.T.T.F. (mean time to failure) has also been calculated. A numerical example with graphical illustrations has been appended at the end to highlight the important results and practical utility of the model.展开更多
This survey article illustrates many important current trends and perspectives for the field and their applications, of interest to researchers in modern algebra, mathematical logic and discrete mathematics. It covers...This survey article illustrates many important current trends and perspectives for the field and their applications, of interest to researchers in modern algebra, mathematical logic and discrete mathematics. It covers a number of directions, including completeness theorem and compactness theorem for hyperidentities;the characterizations of the Boolean algebra of n-ary Boolean functions and the bounded distributive lattice of n-ary monotone Boolean functions;the functional representations of finitely-generated free algebras of various varieties of lattices via generalized Boolean functions, etc.展开更多
We prove that the adjoint semigroup of an implicative BCK algebra is an upper semilattice, and the adjoint semigroup of an implicative BCK algebra with condition(s) is a generalized Boolean algebra. Moreover we prov...We prove that the adjoint semigroup of an implicative BCK algebra is an upper semilattice, and the adjoint semigroup of an implicative BCK algebra with condition(s) is a generalized Boolean algebra. Moreover we prove the adjoint semigroup of a bounded implicative BCK algebra is a Boolean algebra.展开更多
The Monty Hall problem has received its fair share of attention in mathematics. Recently, an entire monograph has been devoted to its history. There has been a multiplicity of approaches to the problem. These approach...The Monty Hall problem has received its fair share of attention in mathematics. Recently, an entire monograph has been devoted to its history. There has been a multiplicity of approaches to the problem. These approaches are not necessarily mutually exclusive. The design of the present paper is to add one more approach by analyzing the mathematical structure of the Monty Hall problem in digital terms. The structure of the problem is described as much as possible in the tradition and the spirit—and as much as possible by means of the algebraic conventions—of George Boole’s Investigation of the Laws of Thought (1854), the Magna Charta of the digital age, and of John Venn’s Symbolic Logic (second edition, 1894), which is squarely based on Boole’s Investigation and elucidates it in many ways. The focus is not only on the digital-mathematical structure itself but also on its relation to the presumed digital nature of cognition as expressed in rational thought and language. The digital approach is outlined in part 1. In part 2, the Monty Hall problem is analyzed digitally. To ensure the generality of the digital approach and demonstrate its reliability and productivity, the Monty Hall problem is extended and generalized in parts 3 and 4 to related cases in light of the axioms of probability theory. In the full mapping of the mathematical structure of the Monty Hall problem and any extensions thereof, a digital or non-quantitative skeleton is fleshed out by a quantitative component. The pertinent mathematical equations are developed and presented and illustrated by means of examples.展开更多
基金Supported by the National Natural Science Foundation of China(11501523,61673320)。
文摘A 2-dimension linguistic lattice implication algebra(2DL-LIA)can build a bridge between logical algebra and 2-dimension fuzzy linguistic information.In this paper,the notion of a Boolean element is proposed in a 2DL-LIA and some properties of Boolean elements are discussed.Then derivations on 2DL-LIAs are introduced and the related properties of derivations are investigated.Moreover,it proves that the derivations on 2DL-LIAs can be constructed by Boolean elements.
基金This work was supported by the National Natural Science Foundation of China under Grants No. 61103191, No. 61070215 the Funds of Key Lab of Fujian Province University Network Security and Cryptology under Crant No. 2011003 and the Open Research Fund of State Key Laboratory of Inforrmtion Security.
文摘To protect against algebraic attacks, a high algebraic immunity is now an important criterion for Boolean functions used in stream ciphers. In this paper, a new method based on a univariate polynomial representation of Boolean functions is proposed. The proposed method is used to constmct Boolean functions with an odd number of variables and with maximum algebraic immunity. We also discuss the nonlinearity of the constructed functions. Moreover, a lower bound is deter- mined for the number of Boolean functions with rmximum algebraic immunity.
基金Project (J51801) supported by Shanghai Education Commission Key DisciplineProject(08ZY79)supported by Shanghai Education Commission Research FundProject(DZ207004)supported by Shanghai Second Polytechnic University Fund
文摘Single input single output system was studied. With proportion, differential, integral results of deviation between given input and output as controller input, the logic rules in control process was analyzed, these logic rule with Pan-Boolean algebra was described, therefore a PID Pan-Boolean algebra control algorithm was obtained. The simulation results indicates that the new control algorithm is more effective compared to the traditional PID algorithm, having advantages such as more than 3 adjustable parameters of controllers, better result, and so on.
基金Supported by a grant of National Natural Science Foundation of China(12001243,61976244,12171294,11961016)the Natural Science Basic Research Plan in Shaanxi Province of China(2020JQ-762,2021JQ-580)。
文摘In this paper, we review some of their related properties of derivations on MValgebras and give some characterizations of additive derivations. Then we prove that the fixed point set of Boolean additive derivations and that of their adjoint derivations are isomorphic.In particular, we prove that every MV-algebra is isomorphic to the direct product of the fixed point set of Boolean additive derivations and that of their adjoint derivations. Finally we show that every Boolean algebra is isomorphic to the algebra of all Boolean additive(implicative)derivations. These results also give the negative answers to two open problems, which were proposed in [Fuzzy Sets and Systems, 303(2016), 97-113] and [Information Sciences, 178(2008),307-316].
基金Supported by the National Basic Research Program of China （No. 2007CB311201）, the National Natural Science Foundation of China （No.60833008 No.60803149）, and the Foundation of Guangxi Key Laboratory of Information and Communication （No.20902）.
文摘This letter proposes algebraic attacks on two kinds of nonlinear filter generators with symmetric Boolean functions as the filter fimctions. Different fxom the classical algebraic attacks, the proposed attacks take the advantage of the combinational property of a linear feedback shift register （LFSR） and the symmetric Boolean function to obtain a tow-degree algebraic relation, and hence the complexities of the proposed attacks are independent of the algebraic immunity （AI） of the filter functions. It is shown that improper combining of the LFSR with the filter function can make the filter generator suffer from algebraic attacks. As a result, the bits of the LFSR must be selected properly to input the filter function with large AI in order to withstand the proposed algebraic attacks.
文摘There are so many existing methods to obtain system reliability like re-generating point function technique, supplementary variables technique etc., but all these techniques are full of complicated calculations. Keeping above these facts in mind, the authors in this paper have evaluated some reliability parameters for tele-communication system by using Boolean functions technique and algebraic method. Reliability of considered tele-communication system has been evaluated by considering the fact that failures follow arbitrary time distribution. In particular, the reliability expression has also been calculated for Exponential and Weibull distributions. Further, an important reliability parameter namely M.T.T.F. (mean time to failure) has also been calculated. A numerical example with graphical illustrations has been appended at the end to highlight the important results and practical utility of the model.
文摘This survey article illustrates many important current trends and perspectives for the field and their applications, of interest to researchers in modern algebra, mathematical logic and discrete mathematics. It covers a number of directions, including completeness theorem and compactness theorem for hyperidentities;the characterizations of the Boolean algebra of n-ary Boolean functions and the bounded distributive lattice of n-ary monotone Boolean functions;the functional representations of finitely-generated free algebras of various varieties of lattices via generalized Boolean functions, etc.
文摘We prove that the adjoint semigroup of an implicative BCK algebra is an upper semilattice, and the adjoint semigroup of an implicative BCK algebra with condition(s) is a generalized Boolean algebra. Moreover we prove the adjoint semigroup of a bounded implicative BCK algebra is a Boolean algebra.
文摘The Monty Hall problem has received its fair share of attention in mathematics. Recently, an entire monograph has been devoted to its history. There has been a multiplicity of approaches to the problem. These approaches are not necessarily mutually exclusive. The design of the present paper is to add one more approach by analyzing the mathematical structure of the Monty Hall problem in digital terms. The structure of the problem is described as much as possible in the tradition and the spirit—and as much as possible by means of the algebraic conventions—of George Boole’s Investigation of the Laws of Thought (1854), the Magna Charta of the digital age, and of John Venn’s Symbolic Logic (second edition, 1894), which is squarely based on Boole’s Investigation and elucidates it in many ways. The focus is not only on the digital-mathematical structure itself but also on its relation to the presumed digital nature of cognition as expressed in rational thought and language. The digital approach is outlined in part 1. In part 2, the Monty Hall problem is analyzed digitally. To ensure the generality of the digital approach and demonstrate its reliability and productivity, the Monty Hall problem is extended and generalized in parts 3 and 4 to related cases in light of the axioms of probability theory. In the full mapping of the mathematical structure of the Monty Hall problem and any extensions thereof, a digital or non-quantitative skeleton is fleshed out by a quantitative component. The pertinent mathematical equations are developed and presented and illustrated by means of examples.
