Interval additive generators of interval t-norms and interval t-conorms

Abstract The aim of this paper is to introduce the concepts of interval additive generators of interval t-norms and interval t-conorms, as interval representations of additive generators of t-norms and t-conorms, respectively, considering both the correctness and the optimality criteria. The formalization of interval fuzzy connectives in terms of their interval additive generators provides a more systematic methodology for the selection of interval t-norms and interval t-conorms in the various applications of fuzzy systems. We also prove that interval additive generators satisfy the main properties of additive generators discussed in the literature.

[1]  K. Hofmann,et al.  Continuous Lattices and Domains , 2003 .

[2]  Tsau Young Lin,et al.  Combination of interval-valued fuzzy set and soft set , 2009, Comput. Math. Appl..

[3]  Humberto Bustince,et al.  Mathematical analysis of interval-valued fuzzy relations: Application to approximate reasoning , 2000, Fuzzy Sets Syst..

[4]  Peter Viceník,et al.  Additive generators of associative functions , 2005, Fuzzy Sets Syst..

[5]  Glad Deschrijver,et al.  Additive and Multiplicative Generators in Interval-Valued Fuzzy Set Theory , 2007, IEEE Transactions on Fuzzy Systems.

[6]  Joan Torrens,et al.  S- and R-implications from uninorms continuous in ]0, 1[2 and their distributivity over uninorms , 2009, Fuzzy Sets Syst..

[7]  P. Mostert,et al.  On the Structure of Semigroups on a Compact Manifold With Boundary , 1957 .

[8]  Dorel Miheţ.  Fixed point theorems in probabilistic metric spaces , 2009 .

[9]  Chris Cornelis,et al.  The pseudo-linear semantics of interval-valued fuzzy logics , 2009, Inf. Sci..

[10]  Susanne Saminger-Platz,et al.  The dominance relation in some families of continuous Archimedean t-norms and copulas , 2009, Fuzzy Sets Syst..

[11]  Humberto Bustince,et al.  On averaging operators for Atanassov's intuitionistic fuzzy sets , 2011, Inf. Sci..

[12]  I. Turksen.  Interval valued fuzzy sets based on normal forms , 1986 .

[13]  Mohammad Hossein Fazel Zarandi,et al.  Systematic image processing for diagnosing brain tumors: A Type-II fuzzy expert system approach , 2011, Appl. Soft Comput..

[14]  Benjamín R. C. Bedregal,et al.  On interval fuzzy negations , 2010, Fuzzy Sets Syst..

[15]  R. Callejas Bedregal,et al.  Intervals as Domain Constructor , 2001 .

[16]  Radko Mesiar,et al.  Generated triangular norms , 2000, Kybernetika.

[17]  Humberto Bustince,et al.  Comment on: "Image thresholding using type II fuzzy sets". Importance of this method , 2010, Pattern Recognit..

[18]  Klaus Meer.  On the Complexity of Some Problems in Interval Arithmetic , 2003, MFCS.

[19]  M. H. van Emden,et al.  Interval arithmetic: From principles to implementation , 2001, JACM.

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

[21]  E. Walker,et al.  Some comments on interval valued fuzzy sets , 1996 .

[22]  R. Baker Kearfott,et al.  Introduction to Interval Analysis , 2009 .

[23]  Jian-Bo Yang,et al.  On the combination and normalization of interval-valued belief structures , 2007, Information Sciences.

[24]  Balasubramaniam Jayaram,et al.  (U, N)-Implications and Their Characterizations , 2009, EUSFLAT Conf..

[25]  Liu Lianzhen,et al.  States on finite monoidal t-norm based algebras , 2011 .

[26]  Ivor Grattan-Guinness,et al.  Fuzzy Membership Mapped onto Intervals and Many-Valued Quantities , 1976, Math. Log. Q..

[27]  J. Mendel.  Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions , 2001 .

[28]  Glad Deschrijver,et al.  Arithmetic operators in interval-valued fuzzy set theory , 2007, Inf. Sci..

[29]  Ronald R. Yager,et al.  Level sets and the extension principle for interval valued fuzzy sets and its application to uncertainty measures , 2008, Inf. Sci..

[30]  Radko Mesiar,et al.  Quasi- and pseudo-inverses of monotone functions, and the construction of t-norms , 1999, Fuzzy Sets Syst..

[31]  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.

[32]  M. Gorzałczany.  A method for inference in approximate reasoning based on interval-valued fuzzy sets , 1987 .

[33]  Radko Mesiar,et al.  A characterization of fuzzy implications generated by generalized quantifiers , 2008, Fuzzy Sets Syst..

[34]  V. Kreinovich.  Computational Complexity and Feasibility of Data Processing and Interval Computations , 1997 .

[35]  Glad Deschrijver,et al.  Generalized arithmetic operators and their relationship to t-norms in interval-valued fuzzy set theory , 2009, Fuzzy Sets Syst..

[36]  Benjamín R. C. Bedregal,et al.  Formal Aspects of Correctness and Optimality of Interval Computations , 2006, Formal Aspects of Computing.

[37]  Renata Reiser,et al.  Analyzing the Relationship between Interval-valued D-Implications and Interval-valued QL-Implications , 2010 .

[38]  W. M. Faucett.  Compact semigroups irreducibly connected between two idempotents , 1955 .

[39]  Bijan Davvaz,et al.  Applications of interval valued t , 2008, Inf. Sci..

[40]  Lotfi A. Zadeh,et al.  Fuzzy Sets , 1996, Inf. Control..

[41]  Ramon E. Moore,et al.  Interval analysis and fuzzy set theory , 2003, Fuzzy Sets Syst..

[42]  Glad Deschrijver,et al.  A representation of t-norms in interval-valued L-fuzzy set theory , 2008, Fuzzy Sets Syst..

[43]  Glad Deschrijver,et al.  Characterizations of (weakly) Archimedean t-norms in interval-valued fuzzy set theory , 2009, Fuzzy Sets Syst..

[44]  A. Mesiarová.  Generators of triangular norms , 2005 .

[45]  Chris Cornelis,et al.  Advances and challenges in interval-valued fuzzy logic , 2006, Fuzzy Sets Syst..

[46]  Syuan-Yi Chen,et al.  Interval type-2 fuzzy neural network control for X-Y-Theta motion control stage using linear ultrasonic motors , 2009, Neurocomputing.

[47]  Jerry M. Mendel,et al.  Advances in type-2 fuzzy sets and systems , 2007, Inf. Sci..

[48]  Vladik Kreinovich,et al.  Interval-Valued Degrees of Belief: Applications of Interval Computations to Expert Systems and Intelligent Control , 1997, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[49]  Rafik A. Aliev,et al.  Systemic approach to fuzzy logic formalization for approximate reasoning , 2011, Inf. Sci..

[50]  Renata Reiser,et al.  Analyzing Properties of Fuzzy Implications Obtained via the Interval Constructor , 2006 .

[51]  Janusz T. Starczewski.  Efficient triangular type-2 fuzzy logic systems , 2009, Int. J. Approx. Reason..

[52]  Andrea Mesiarová-Zemánková,et al.  Ranks of additive generators , 2009, Fuzzy Sets Syst..

[53]  Mirko Navara,et al.  A survey on different triangular norm-based fuzzy logics , 1999, Fuzzy Sets Syst..

[54]  Didier Dubois,et al.  Random sets and fuzzy interval analysis , 1991 .

[55]  Yun Shi,et al.  On the First Place Antitonicity in QL-implications , 2008, EUSFLAT Conf..

[56]  Kaj Madsen,et al.  Existence Test for Asynchronous Interval Iteration , 1997, Reliab. Comput..

[57]  Hamid R. Tizhoosh,et al.  Image thresholding using type II fuzzy sets , 2005, Pattern Recognit..

[58]  Jerry M. Mendel,et al.  On answering the question "Where do I start in order to solve a new problem involving interval type-2 fuzzy sets?" , 2009, Inf. Sci..

[59]  Radko Mesiar,et al.  Triangular norms. Position paper II: general constructions and parameterized families , 2004, Fuzzy Sets Syst..

[60]  Joan Torrens,et al.  A characterization of (U, N), RU, QL and D-implications derived from uninorms satisfying the law of importation , 2010, Fuzzy Sets Syst..

[61]  Radko Mesiar,et al.  On special fuzzy implications , 2009, Fuzzy Sets Syst..

[62]  A. Mesiarová,et al.  H-transformation of t-norms , 2006 .

[63]  Peter Viceník,et al.  On a class of generated triangular norms and their isomorphisms , 2010, Fuzzy Sets Syst..

[64]  Marc Roubens,et al.  Fuzzy Preference Modelling and Multicriteria Decision Support , 1994, Theory and Decision Library.

[65]  Peter Viceník,et al.  Additive generators of border-continuous triangular norms , 2008, Fuzzy Sets Syst..

[66]  Oscar Castillo,et al.  A hybrid learning algorithm for a class of interval type-2 fuzzy neural networks , 2009, Inf. Sci..

[67]  Benjamín R. C. Bedregal,et al.  Interval Valued QL-Implications , 2007, WoLLIC.

[68]  Feng Feng,et al.  Application of level soft sets in decision making based on interval-valued fuzzy soft sets , 2010, Comput. Math. Appl..

[69]  Chris Cornelis,et al.  Classification Of Intuitionistic Fuzzy Implicators: An Algebraic Approach , 2002, JCIS.

[70]  Hao Ying,et al.  Interval type-2 Boolean fuzzy systems are universal approximators , 2010, 2010 Annual Meeting of the North American Fuzzy Information Processing Society.

[71]  Robert Ivor John,et al.  An Interval Type-2 Fuzzy multiple echelon supply chain model , 2010, Knowl. Based Syst..

[72]  Ramon E. Moore.  Methods and applications of interval analysis , 1979, SIAM studies in applied mathematics.

[73]  Benjamín R. C. Bedregal,et al.  Xor-Implications and E-Implications: Classes of Fuzzy Implications Based on Fuzzy Xor , 2009, LSFA.

[74]  R. Mesiar,et al.  Logical, algebraic, analytic, and probabilistic aspects of triangular norms , 2005 .

[75]  Lotfi A. Zadeh,et al.  The concept of a linguistic variable and its application to approximate reasoning-III , 1975, Inf. Sci..

[76]  E. Walter,et al.  Applied Interval Analysis: With Examples in Parameter and State Estimation, Robust Control and Robotics , 2001 .

[77]  Benjamín R. C. Bedregal,et al.  Interval Valued Versions of T-Conorms, Fuzzy Negations and Fuzzy Implications , 2006, 2006 IEEE International Conference on Fuzzy Systems.

[78]  Jerry M. Mendel,et al.  Uncertainty measures for interval type-2 fuzzy sets , 2007, Inf. Sci..

[79]  Benjamín R. C. Bedregal,et al.  Interval t-norms as Interval Representations of t-norms , 2005, The 14th IEEE International Conference on Fuzzy Systems, 2005. FUZZ '05..

[80]  B. Schweizer,et al.  Statistical metric spaces. , 1960 .

[81]  Berthold Schweizer,et al.  Probabilistic Metric Spaces , 2011 .

[82]  Radko Mesiar,et al.  A characterization of the ordering of continuous t-norms , 1997, Fuzzy Sets Syst..

[83]  Yao Ouyang.  On the construction of boundary weak triangular norms through additive generators , 2007 .

[84]  Joan Torrens,et al.  Two types of implications derived from uninorms , 2007, Fuzzy Sets Syst..

[85]  Humberto Bustince,et al.  Interval-valued fuzzy sets constructed from matrices: Application to edge detection , 2009, Fuzzy Sets Syst..

[86]  Benjamín R. C. Bedregal,et al.  A Quasi-Metric Topology Compatible with Inclusion Monotonicity on Interval Space , 1997, Reliab. Comput..

[87]  Benjamín R. C. Bedregal,et al.  On interval fuzzy S-implications , 2010, Inf. Sci..

[88]  K. Jahn.  Intervall‐wertige Mengen , 1975 .

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

[90]  Zhenjiang Zhao,et al.  A generalization of additive generator of triangular norms , 2008, Int. J. Approx. Reason..

[91]  Benjamín R. C. Bedregal,et al.  Interval Additive Generators of Interval T-Norms , 2008, WoLLIC.

[92]  Hung T. Nguyen,et al.  A First Course in Fuzzy Logic , 1996 .

[93]  Janusz T. Starczewski.  Extended triangular norms , 2009, Inf. Sci..

[94]  Benjamín R. C. Bedregal,et al.  The best interval representations of t-norms and automorphisms , 2006, Fuzzy Sets Syst..

[95]  Michal Baczynski,et al.  QL-implications: Some properties and intersections , 2010, Fuzzy Sets Syst..

[96]  Gaspar Mayor,et al.  Additive Generators of Discrete Conjunctive Aggregation Operations , 2007, IEEE Transactions on Fuzzy Systems.

[97]  Jerry M. Mendel,et al.  A comparative study of ranking methods, similarity measures and uncertainty measures for interval type-2 fuzzy sets , 2009, Inf. Sci..

[98]  Ricardo Martínez-Soto,et al.  Optimization of Interval Type-2 Fuzzy Logic Controllers for a Perturbed Autonomous Wheeled Mobile Robot Using Genetic Algorithms , 2009, Soft Computing for Hybrid Intelligent Systems.

[99]  Joan Torrens,et al.  A characterization of residual implications derived from left-continuous uninorms , 2010, Inf. Sci..

[100]  Wenyi Zeng,et al.  Normalized distance, similarity measure, inclusion measure and entropy of interval-valued fuzzy sets and their relationship , 2008, Inf. Sci..

[101]  Michal Baczynski,et al.  Intersections between some families of (U, N)- and RU-implications , 2011, Fuzzy Sets Syst..

[102]  John Leventides,et al.  An approach to the selection of fuzzy connectives in terms of their additive generators , 2002, Fuzzy Sets Syst..

[103]  Sankar K. Pal,et al.  Fuzzy sets in pattern recognition and machine intelligence , 2005, Fuzzy Sets Syst..

[104]  Joan Torrens,et al.  On a new class of fuzzy implications: h-Implications and generalizations , 2011, Inf. Sci..

[105]  Michal Baczynski,et al.  (S, N)- and R-implications: A state-of-the-art survey , 2008, Fuzzy Sets Syst..

[106]  Didier Dubois,et al.  Interval-valued Fuzzy Sets, Possibility Theory and Imprecise Probability , 2005, EUSFLAT Conf..

[107]  Krassimir T. Atanassov,et al.  Intuitionistic fuzzy sets , 1986 .

[108]  Oscar Castillo,et al.  Interval type-2 fuzzy logic and modular neural networks for face recognition applications , 2009, Appl. Soft Comput..

[109]  R. Mesiar,et al.  Triangular norms: Basic notions and properties , 2005 .

[110]  Chris Cornelis,et al.  On the representation of intuitionistic fuzzy t-norms and t-conorms , 2004, IEEE Transactions on Fuzzy Systems.

[111]  Petr Hájek,et al.  Metamathematics of Fuzzy Logic , 1998, Trends in Logic.

[112]  R. B. Kearfott.  Rigorous Global Search: Continuous Problems , 1996 .