On fuzzy relational equations and the covering problem

Previous studies have shown that fuzzy relational equations (FREs) based on either the max-continuous Archimedean t-norm or the max-arithmetic mean composition can be transformed into the covering problem, which is an NP-hard problem. Exploiting the properties common to the continuous Archimedean t-norm and the arithmetic mean, this study proposes a generalization of them as the ''u-norm'', enabling FREs that are based on the max-continuous u-norm composition also to be transformed into the covering problem. This study also proposes a procedure for transforming the covering problem into max-product FREs. Consequently, max-continuous u-norm FREs can be solved by extending any procedure for solving either the covering problem or max-product FREs.

[1]  Paul P. Wang,et al.  Fuzzy Relation Equations (II): The Branch-point-solutions and the Categorized Minimal Solutions , 2006, Soft Comput..

[2]  G. Klir,et al.  Resolution of finite fuzzy relation equations , 1984 .

[3]  Edward Beltrami,et al.  Uncertainty and Information , 2020, What Is Random?.

[4]  D. Grant Fisher,et al.  Solution algorithms for fuzzy relational equations with max-product composition , 1998, Fuzzy Sets Syst..

[5]  George J. Klir,et al.  Fuzzy sets and fuzzy logic - theory and applications , 1995 .

[6]  Bih-Sheue Shieh,et al.  Solutions of fuzzy relation equations based on continuous t-norms , 2007, Inf. Sci..

[7]  Amin Ghodousian,et al.  Linear objective function optimization with fuzzy relation equation constraints regarding max-av composition , 2006, Appl. Math. Comput..

[8]  H. Zimmermann,et al.  Fuzzy Set Theory and Its Applications , 1993 .

[9]  Hong-Xing Li,et al.  Notes on "pseudo-t-norms and implication operators on a complete Brouwerian lattice" and "pseudo-t-norms and implication operators: direct products and direct product decompositions" , 2005, Fuzzy Sets Syst..

[10]  D. Dubois,et al.  Fundamentals of fuzzy sets , 2000 .

[11]  Witold Pedrycz,et al.  Data compression with fuzzy relational equations , 2002, Fuzzy Sets Syst..

[12]  Lotfi A. Zadeh,et al.  Is there a need for fuzzy logic? , 2008, NAFIPS 2008 - 2008 Annual Meeting of the North American Fuzzy Information Processing Society.

[13]  Yan-Kuen Wu,et al.  Optimization of fuzzy relational equations with max-av composition , 2007, Inf. Sci..

[14]  Shu-Cherng Fang,et al.  A survey on fuzzy relational equations, part I: classification and solvability , 2009, Fuzzy Optim. Decis. Mak..

[15]  Witold Pedrycz,et al.  A motion compression/reconstruction method based on max t-norm composite fuzzy relational equations , 2006, Inf. Sci..

[16]  S. Guu,et al.  Finding the complete set of minimal solutions for fuzzy relational equations with max-product composition , 2004 .

[17]  Elie Sanchez,et al.  Resolution of Composite Fuzzy Relation Equations , 1976, Inf. Control..

[18]  Salvatore Sessa,et al.  Fuzzy relation equations for coding/decoding processes of images and videos , 2005, Inf. Sci..

[19]  Yordan Kyosev,et al.  Algorithm for Solving Max-product Fuzzy Relational Equations , 2007, Soft Comput..

[20]  B. Baets.  Analytical solution methods for fuzzy relational equations. , 2000 .

[21]  Yandong Yu,et al.  Pseudo-t-norms and implication operators on a complete Brouwerian lattice , 2002, Fuzzy Sets Syst..

[22]  Toshihide Ibaraki,et al.  Complexity of Identification and Dualization of Positive Boolean Functions , 1995, Inf. Comput..

[23]  Yan-Kuen Wu,et al.  Reducing the search space of a linear fractional programming problem under fuzzy relational equations with max-Archimedean t-norm composition , 2008, Fuzzy Sets Syst..

[24]  Masaaki Miyakoshi,et al.  Composite Fuzzy Relational Equations with Non-Commutative Conjunctions , 1998, Inf. Sci..

[25]  Yan-Kuen Wu,et al.  Optimization of fuzzy relational equations with a linear convex combination of max-min and max-average compositions , 2007, 2007 IEEE International Conference on Industrial Engineering and Engineering Management.

[26]  W. Pedrycz,et al.  Fuzzy relation equations on a finite set , 1982 .

[27]  Irina Perfilieva,et al.  System of fuzzy relation equations with sup-* composition in semi-linear spaces: minimal solutions , 2007, 2007 IEEE International Fuzzy Systems Conference.

[28]  Lotfi A. Zadeh,et al.  Toward a generalized theory of uncertainty (GTU)--an outline , 2005, Inf. Sci..

[29]  A. V. Markovskii,et al.  On the relation between equations with max-product composition and the covering problem , 2005, Fuzzy Sets Syst..

[30]  Yan-Kuen Wu,et al.  An Efficient Procedure for Solving a Fuzzy Relational Equation With Max–Archimedean t-Norm Composition , 2008, IEEE Transactions on Fuzzy Systems.

[31]  W. Pedrycz,et al.  Fuzzy Relation Equations and Their Applications to Knowledge Engineering , 1989, Theory and Decision Library.

[32]  Shu-Cherng Fang,et al.  On the resolution and optimization of a system of fuzzy relational equations with sup-T composition , 2008, Fuzzy Optim. Decis. Mak..

[33]  Paul P. Wang,et al.  Fuzzy relation equations (I): the general and specialized solving algorithms , 2002, Soft Comput..

[34]  Vladimir Gurvich,et al.  On Generating the Irredundant Conjunctive and Disjunctive Normal Forms of Monotone Boolean Functions , 1999, Discret. Appl. Math..

[35]  Hong-Xing Li,et al.  Resolution of finite fuzzy relation equations based on strong pseudo-t-norms , 2006, Appl. Math. Lett..

[36]  Jun-Lin Lin,et al.  On the relation between fuzzy max-Archimedean t-norm relational equations and the covering problem , 2009, Fuzzy Sets Syst..

[37]  Jiranut Loetamonphong,et al.  An efficient solution procedure for fuzzy relation equations with max-product composition , 1999, IEEE Trans. Fuzzy Syst..

[38]  Yandong Yu,et al.  Pseudo-t-norms and implication operators: direct products and direct product decompositions , 2003, Fuzzy Sets Syst..

[39]  Spyros G. Tzafestas,et al.  Resolution of composite fuzzy relation equations based on Archimedean triangular norms , 2001, Fuzzy Sets Syst..

[40]  Esmaile Khorram,et al.  An algorithm for solving fuzzy relation equations with max-T composition operator , 2008, Inf. Sci..

[41]  Shu-Cherng Fang,et al.  Minimizing a linear fractional function subject to a system of sup-T equations with a continuous Archimedean triangular norm , 2009, J. Syst. Sci. Complex..

[42]  Chi-Tsuen Yeh,et al.  On the minimal solutions of max-min fuzzy relational equations , 2008, Fuzzy Sets Syst..

[43]  Witold Pedrycz,et al.  Fuzzy relational compression , 1999, IEEE Trans. Syst. Man Cybern. Part B.

[44]  Wen-June Wang,et al.  Matrix-pattern-based computer algorithm for solving fuzzy relation equations , 2003, IEEE Trans. Fuzzy Syst..