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Grouping function is a special kind of aggregation function which measures the amount of evidence in favor of either of the two choices. Recently, complex fuzzy sets have been successfully used in many fields. This paper extends the concept of grouping functions to the complex-valued setting. We introduce the concepts of complex-valued grouping, complex-valued 0-grouping, complex-valued 1-grouping, and general complex-valued grouping functions. We present some interesting results and construction methods of general complex-valued grouping functions.
Ying Chen; Lvqing Bi; Bo Hu; Songsong Dai. General Complex-Valued Grouping Functions. Journal of Mathematics 2021, 2021, 1 -6.
AMA StyleYing Chen, Lvqing Bi, Bo Hu, Songsong Dai. General Complex-Valued Grouping Functions. Journal of Mathematics. 2021; 2021 ():1-6.
Chicago/Turabian StyleYing Chen; Lvqing Bi; Bo Hu; Songsong Dai. 2021. "General Complex-Valued Grouping Functions." Journal of Mathematics 2021, no. : 1-6.
This paper studies rough approximation via join and meet on a complete orthomodular lattice. Different from Boolean algebra, the distributive law of join over meet does not hold in orthomodular lattices. Some properties of rough approximation rely on the distributive law. Furthermore, we study the relationship among the distributive law, rough approximation and orthomodular lattice-valued relation.
Songsong Dai. Rough Approximation Operators on a Complete Orthomodular Lattice. Axioms 2021, 10, 164 .
AMA StyleSongsong Dai. Rough Approximation Operators on a Complete Orthomodular Lattice. Axioms. 2021; 10 (3):164.
Chicago/Turabian StyleSongsong Dai. 2021. "Rough Approximation Operators on a Complete Orthomodular Lattice." Axioms 10, no. 3: 164.
As a generalization of complex fuzzy set (CFS), interval-valued complex fuzzy set (IVCFS) is a new research topic in the field of CFS theory, which can handle two different information features with the uncertainty. Distance is an important tool in the field of IVCFS theory. To enhance the applicability of IVCFS, this paper presents some new interval-valued complex fuzzy distances based on traditional Hamming and Euclidean distances of complex numbers. Furthermore, we elucidate the geometric properties of these distances. Finally, these distances are used to deal with decision-making problem in the IVCFS environment.
Haifeng Song; Lvqing Bi; Bo Hu; Yingying Xu; Songsong Dai. New Distance Measures between the Interval-Valued Complex Fuzzy Sets with Applications to Decision-Making. Mathematical Problems in Engineering 2021, 2021, 1 -9.
AMA StyleHaifeng Song, Lvqing Bi, Bo Hu, Yingying Xu, Songsong Dai. New Distance Measures between the Interval-Valued Complex Fuzzy Sets with Applications to Decision-Making. Mathematical Problems in Engineering. 2021; 2021 ():1-9.
Chicago/Turabian StyleHaifeng Song; Lvqing Bi; Bo Hu; Yingying Xu; Songsong Dai. 2021. "New Distance Measures between the Interval-Valued Complex Fuzzy Sets with Applications to Decision-Making." Mathematical Problems in Engineering 2021, no. : 1-9.
Shor's quantum algorithm and other efficient quantum algorithms can break many public-key cryptographic schemes in polynomial time on a quantum computer. In response, researchers proposed postquantum cryptography to resist quantum computers. The multivariate cryptosystem (MVC) is one of a few options of postquantum cryptography. It is based on the NP-hardness of the computational problem to solve nonlinear equations over a finite field. Recently, Wang et al. (2018) proposed a MVC based on extended clipped hopfield neural networks (eCHNN). Its main security assumption is backed by the discrete logarithm (DL) problem over Matrics. In this brief, we present quantum cryptanalysis of Wang et al.'s eCHNN-based MVC. We first show that Shor's quantum algorithm can be modified to solve the DL problem over Matrics. Then we show that Wang et al.'s construction of eCHNN-based MVC is not secure against quantum computers; this against the original intention of that multivariate cryptography is one of a few options of postquantum cryptography.
Songsong Dai. Quantum Cryptanalysis on a Multivariate Cryptosystem Based on Clipped Hopfield Neural Network. IEEE Transactions on Neural Networks and Learning Systems 2021, PP, 1 -5.
AMA StyleSongsong Dai. Quantum Cryptanalysis on a Multivariate Cryptosystem Based on Clipped Hopfield Neural Network. IEEE Transactions on Neural Networks and Learning Systems. 2021; PP (99):1-5.
Chicago/Turabian StyleSongsong Dai. 2021. "Quantum Cryptanalysis on a Multivariate Cryptosystem Based on Clipped Hopfield Neural Network." IEEE Transactions on Neural Networks and Learning Systems PP, no. 99: 1-5.
Overlap function is a special type of aggregation function which measures the degree of overlapping between different classes. Recently, complex fuzzy sets have been successfully applied in many applications. In this paper, we extend the concept of overlap functions to the complex-valued setting. We introduce the notions of complex-valued overlap, complex-valued 0-overlap, complex-valued 1-overlap, and general complex-valued overlap functions, which can be regarded as the generalizations of the concepts of overlap, 0-overlap, 1-overlap, and general overlap functions, respectively. We study some properties of these complex-valued overlap functions and their construction methods.
Ying Chen; Lvqing Bi; Bo Hu; Songsong Dai. General Complex-Valued Overlap Functions. Journal of Mathematics 2021, 2021, 1 -6.
AMA StyleYing Chen, Lvqing Bi, Bo Hu, Songsong Dai. General Complex-Valued Overlap Functions. Journal of Mathematics. 2021; 2021 ():1-6.
Chicago/Turabian StyleYing Chen; Lvqing Bi; Bo Hu; Songsong Dai. 2021. "General Complex-Valued Overlap Functions." Journal of Mathematics 2021, no. : 1-6.
Qiu (Notes on automata theory based on quantum logic, Sci China Ser F-Inf Sci, 2007, 50(2)154–169) discovered that some basic issues in orthomodular lattice-valued automata rely on bi-implication operator satisfying following condition: and discovered that bi-implication operator based on Sasaki arrow satisfies this condition if and only if the truth-value lattice L is indeed a Boolean algebra, then asked a question of whether the result is also applied to other four quantum implication operators. We show that the answer is yes, and discuss several other conditions.
Songsong Dai. A note on implication operators of quantum logic. Quantum Machine Intelligence 2020, 2, 1 -6.
AMA StyleSongsong Dai. A note on implication operators of quantum logic. Quantum Machine Intelligence. 2020; 2 (2):1-6.
Chicago/Turabian StyleSongsong Dai. 2020. "A note on implication operators of quantum logic." Quantum Machine Intelligence 2, no. 2: 1-6.
Complex fuzzy set (CFS), as a generalization of fuzzy set (FS), is characterized by complex-valued membership degrees. By considering the complex-valued membership degree as a vector in the complex unit disk, we introduce the cosine similarity measures between CFSs. Then, we investigate some invariance properties of the cosine similarity measure. Finally, the cosine similarity measure is applied to measure the robustness of complex fuzzy connectives and complex fuzzy inference.
Wenping Guo; Lvqing Bi; Bo Hu; Songsong Dai. Cosine Similarity Measure of Complex Fuzzy Sets and Robustness of Complex Fuzzy Connectives. Mathematical Problems in Engineering 2020, 2020, 1 -9.
AMA StyleWenping Guo, Lvqing Bi, Bo Hu, Songsong Dai. Cosine Similarity Measure of Complex Fuzzy Sets and Robustness of Complex Fuzzy Connectives. Mathematical Problems in Engineering. 2020; 2020 ():1-9.
Chicago/Turabian StyleWenping Guo; Lvqing Bi; Bo Hu; Songsong Dai. 2020. "Cosine Similarity Measure of Complex Fuzzy Sets and Robustness of Complex Fuzzy Connectives." Mathematical Problems in Engineering 2020, no. : 1-9.
This paper investigates the geometric aggregation operators for aggregating the interval-valued complex fuzzy sets (IVCFSs) whose membership grades are a special set of complex numbers. We develop some geometric aggregation operators under the interval-valued complex fuzzy environment, namely, interval-valued complex fuzzy geometric (IVCFG), interval-valued complex fuzzy weighted geometric (IVCFWG), and interval-valued complex fuzzy ordered weighted geometric (IVCFOWG) operators. Then, we investigate the rotational and reflectional invariances of these operators. Further, a decision-making approach based on these operators is presented under the interval-valued complex fuzzy environment and an example is illustrated to demonstrate the efficiency of the proposed approach.
Songsong Dai; Lvqing Bi; Bo Hu. Interval-Valued Complex Fuzzy Geometric Aggregation Operators and Their Application to Decision Making. Mathematical Problems in Engineering 2020, 2020, 1 -10.
AMA StyleSongsong Dai, Lvqing Bi, Bo Hu. Interval-Valued Complex Fuzzy Geometric Aggregation Operators and Their Application to Decision Making. Mathematical Problems in Engineering. 2020; 2020 ():1-10.
Chicago/Turabian StyleSongsong Dai; Lvqing Bi; Bo Hu. 2020. "Interval-Valued Complex Fuzzy Geometric Aggregation Operators and Their Application to Decision Making." Mathematical Problems in Engineering 2020, no. : 1-10.
Songsong Dai. Logical foundation of symmetric implicational methods for fuzzy reasoning. Journal of Intelligent & Fuzzy Systems 2020, 39, 1089 -1095.
AMA StyleSongsong Dai. Logical foundation of symmetric implicational methods for fuzzy reasoning. Journal of Intelligent & Fuzzy Systems. 2020; 39 (1):1089-1095.
Chicago/Turabian StyleSongsong Dai. 2020. "Logical foundation of symmetric implicational methods for fuzzy reasoning." Journal of Intelligent & Fuzzy Systems 39, no. 1: 1089-1095.
Complex fuzzy sets (CFSs) are a new extension of fuzzy sets whose membership grades are vectors in the unit circle of the complex plane. Complex fuzzy operations are of high importance in both theoretical and applied studies of CFSs. The property of rotational invariance is highly pertinent to complex fuzzy operations. To enhance and extend the applicability of complex fuzzy operations, this paper develops new types of rotational invariance for complex fuzzy operations. These rotational invariances are raised by different rotation operations of the arguments of complex fuzzy operations. After that, rotational invariance of several complex fuzzy operations are studied.
Songsong Dai. A Generalization of Rotational Invariance for Complex Fuzzy Operations. IEEE Transactions on Fuzzy Systems 2020, 29, 1152 -1159.
AMA StyleSongsong Dai. A Generalization of Rotational Invariance for Complex Fuzzy Operations. IEEE Transactions on Fuzzy Systems. 2020; 29 (5):1152-1159.
Chicago/Turabian StyleSongsong Dai. 2020. "A Generalization of Rotational Invariance for Complex Fuzzy Operations." IEEE Transactions on Fuzzy Systems 29, no. 5: 1152-1159.
In this paper, we give a definition for fuzzy Kolmogorov complexity. In the classical setting, the Kolmogorov complexity of a single finite string is the length of the shortest program that produces this string. We define the fuzzy Kolmogorov complexity as the minimum classical description length of a finite-valued fuzzy language through a universal finite-valued fuzzy Turing machine that produces the desired fuzzy language. The classical Kolmogorov complexity is extended to the fuzzy domain retaining classical descriptions. We show that our definition is robust, that is to say, the complexity of a finite-valued fuzzy language does not depend on the underlying finite-valued fuzzy Turing machine.
Songsong Dai. Fuzzy Kolmogorov Complexity Based on a Classical Description. Entropy 2020, 22, 66 .
AMA StyleSongsong Dai. Fuzzy Kolmogorov Complexity Based on a Classical Description. Entropy. 2020; 22 (1):66.
Chicago/Turabian StyleSongsong Dai. 2020. "Fuzzy Kolmogorov Complexity Based on a Classical Description." Entropy 22, no. 1: 66.
Haifeng Song; Weiwei Yang; Songsong Dai; Lei Du; Yongchen Sun. Using dual-channel CNN to classify hyperspectral image based on spatial-spectral information. Mathematical Biosciences and Engineering 2020, 17, 3450 -3477.
AMA StyleHaifeng Song, Weiwei Yang, Songsong Dai, Lei Du, Yongchen Sun. Using dual-channel CNN to classify hyperspectral image based on spatial-spectral information. Mathematical Biosciences and Engineering. 2020; 17 (4):3450-3477.
Chicago/Turabian StyleHaifeng Song; Weiwei Yang; Songsong Dai; Lei Du; Yongchen Sun. 2020. "Using dual-channel CNN to classify hyperspectral image based on spatial-spectral information." Mathematical Biosciences and Engineering 17, no. 4: 3450-3477.
A complex fuzzy set is an extension of the fuzzy set, of which membership grades take complex values in the complex unit disk. We present two complex fuzzy power aggregation operators including complex fuzzy weighted power (CFWP) and complex fuzzy ordered weighted power (CFOWP) operators. We then study two geometric properties which include rotational invariance and reflectional invariance for these complex fuzzy aggregation operators. We also apply the new proposed aggregation operators to decision making and illustrate an example to show the validity of the new approach.
Bo Hu; Lvqing Bi; Songsong Dai. Complex Fuzzy Power Aggregation Operators. Mathematical Problems in Engineering 2019, 2019, 1 -7.
AMA StyleBo Hu, Lvqing Bi, Songsong Dai. Complex Fuzzy Power Aggregation Operators. Mathematical Problems in Engineering. 2019; 2019 ():1-7.
Chicago/Turabian StyleBo Hu; Lvqing Bi; Songsong Dai. 2019. "Complex Fuzzy Power Aggregation Operators." Mathematical Problems in Engineering 2019, no. : 1-7.
Songsong Dai; Wentao Cheng. Noncommutative symmetric difference operators for fuzzy logic. Journal of Intelligent & Fuzzy Systems 2019, 37, 8005 -8013.
AMA StyleSongsong Dai, Wentao Cheng. Noncommutative symmetric difference operators for fuzzy logic. Journal of Intelligent & Fuzzy Systems. 2019; 37 (6):8005-8013.
Chicago/Turabian StyleSongsong Dai; Wentao Cheng. 2019. "Noncommutative symmetric difference operators for fuzzy logic." Journal of Intelligent & Fuzzy Systems 37, no. 6: 8005-8013.
In [1], Dick introduced a partial ordering $\preceq$ on the unit circle of the complex plane D induced by the algebraic product and proved that (D, $\preceq$ ) is a complete lattice. In this paper, we point out that the relation $\preceq$ is not a partial order on D and that (D, $\preceq$ ) is not a lattice. We show that the relation $\preceq$ is a partial order over $\text{D}^{\circ}\cup\{1\}$ and a strict partial order over $\text{D}^{\circ} - \{0\}$ , where $\text{D}^{\circ}$ is the interior of D.
Songsong Dai. On Partial Orders in Complex Fuzzy Logic. IEEE Transactions on Fuzzy Systems 2019, 29, 698 -701.
AMA StyleSongsong Dai. On Partial Orders in Complex Fuzzy Logic. IEEE Transactions on Fuzzy Systems. 2019; 29 (3):698-701.
Chicago/Turabian StyleSongsong Dai. 2019. "On Partial Orders in Complex Fuzzy Logic." IEEE Transactions on Fuzzy Systems 29, no. 3: 698-701.
Bo Hu; Lvqing Bi; Songsong Dai. Approximate orthogonality of complex fuzzy sets and approximately orthogonality preserving operators. Journal of Intelligent & Fuzzy Systems 2019, 37, 5025 -5030.
AMA StyleBo Hu, Lvqing Bi, Songsong Dai. Approximate orthogonality of complex fuzzy sets and approximately orthogonality preserving operators. Journal of Intelligent & Fuzzy Systems. 2019; 37 (4):5025-5030.
Chicago/Turabian StyleBo Hu; Lvqing Bi; Songsong Dai. 2019. "Approximate orthogonality of complex fuzzy sets and approximately orthogonality preserving operators." Journal of Intelligent & Fuzzy Systems 37, no. 4: 5025-5030.
Complex fuzzy set (CFS) is a recent development in the field of fuzzy set (FS) theory. The significance of CFS lies in the fact that CFS assigned membership grades from a unit circle in the complex plane, i.e., in the form of a complex number whose amplitude term belongs to a [ 0 , 1 ] interval. The interval-valued complex fuzzy set (IVCFS) is one of the extensions of the CFS in which the amplitude term is extended from the real numbers to the interval-valued numbers. The novelty of IVCFS lies in its larger range comparative to CFS. We often use fuzzy distance measures to solve some problems in our daily life. Hence, this paper develops some series of distance measures between IVCFSs by using Hamming and Euclidean metrics. The boundaries of these distance measures for IVCFSs are obtained. Finally, we study two geometric properties include rotational invariance and reflectional invariance of these distance measures.
Songsong Dai; Lvqing Bi; Bo Hu. Distance Measures between the Interval-Valued Complex Fuzzy Sets. Mathematics 2019, 7, 549 .
AMA StyleSongsong Dai, Lvqing Bi, Bo Hu. Distance Measures between the Interval-Valued Complex Fuzzy Sets. Mathematics. 2019; 7 (6):549.
Chicago/Turabian StyleSongsong Dai; Lvqing Bi; Bo Hu. 2019. "Distance Measures between the Interval-Valued Complex Fuzzy Sets." Mathematics 7, no. 6: 549.
Lvqing Bi; Songsong Dai; Bo Hu; Sizhao Li. Complex fuzzy arithmetic aggregation operators. Journal of Intelligent & Fuzzy Systems 2019, 36, 2765 -2771.
AMA StyleLvqing Bi, Songsong Dai, Bo Hu, Sizhao Li. Complex fuzzy arithmetic aggregation operators. Journal of Intelligent & Fuzzy Systems. 2019; 36 (3):2765-2771.
Chicago/Turabian StyleLvqing Bi; Songsong Dai; Bo Hu; Sizhao Li. 2019. "Complex fuzzy arithmetic aggregation operators." Journal of Intelligent & Fuzzy Systems 36, no. 3: 2765-2771.
Complex fuzzy sets are characterized by complex-valued membership functions, whose range is extended from the traditional fuzzy range of [0,1] to the unit circle in the complex plane. In this paper, we define two kinds of entropy measures for complex fuzzy sets, called type-A and type-B entropy measures, and analyze their rotational invariance properties. Among them, two formulas of type-A entropy measures possess the attribute of rotational invariance, whereas the other two formulas of type-B entropy measures lack this characteristic.
Lvqing Bi; Zhiqiang Zeng; Bo Hu; Songsong Dai. Two Classes of Entropy Measures for Complex Fuzzy Sets. Mathematics 2019, 7, 96 .
AMA StyleLvqing Bi, Zhiqiang Zeng, Bo Hu, Songsong Dai. Two Classes of Entropy Measures for Complex Fuzzy Sets. Mathematics. 2019; 7 (1):96.
Chicago/Turabian StyleLvqing Bi; Zhiqiang Zeng; Bo Hu; Songsong Dai. 2019. "Two Classes of Entropy Measures for Complex Fuzzy Sets." Mathematics 7, no. 1: 96.
Bo Hu; Lvqing Bi; Songsong Dai; Sizhao Li. The approximate parallelity of complex fuzzy sets. Journal of Intelligent & Fuzzy Systems 2018, 35, 6343 -6351.
AMA StyleBo Hu, Lvqing Bi, Songsong Dai, Sizhao Li. The approximate parallelity of complex fuzzy sets. Journal of Intelligent & Fuzzy Systems. 2018; 35 (6):6343-6351.
Chicago/Turabian StyleBo Hu; Lvqing Bi; Songsong Dai; Sizhao Li. 2018. "The approximate parallelity of complex fuzzy sets." Journal of Intelligent & Fuzzy Systems 35, no. 6: 6343-6351.