Parameterizing the Semantics of Fuzzy Attribute Implications by Systems of Isotone Galois Connections

We study the semantics of fuzzy if-then rules called fuzzy attribute implications parameterized by systems of isotone Galois connections. The rules express dependences between fuzzy attributes in object-attribute incidence data. The proposed parameterizations are general and include as special cases the parameterizations by linguistic hedges used in earlier approaches. We formalize the general parameterizations, propose bivalent and graded notions of semantic entailment of fuzzy attribute implications, show their characterization in terms of least models and complete axiomatization, and provide characterization of bases of fuzzy attribute implications derived from data.

[1]  Georg Struth,et al.  Residuated Lattices , 1938, Arch. Formal Proofs.

[2]  Manuel Ojeda-Aciego,et al.  Multi-adjoint concept lattices with heterogeneous conjunctors and hedges , 2014, Annals of Mathematics and Artificial Intelligence.

[3]  Radko Mesiar,et al.  Triangular norms on product lattices , 1999, Fuzzy Sets Syst..

[4]  Bernhard Ganter,et al.  Two Basic Algorithms in Concept Analysis , 2010, ICFCA.

[5]  Radim Belohlávek,et al.  Crisply Generated Fuzzy Concepts , 2005, ICFCA.

[6]  Lotfi A. Zadeh,et al.  Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic , 1997, Fuzzy Sets Syst..

[7]  Jan Pavelka,et al.  On Fuzzy Logic II. Enriched residuated lattices and semantics of propositional calculi , 1979, Math. Log. Q..

[8]  P. Hájek.  Fuzzy logic and arithmetical hierarchy , 1995 .

[9]  Petr Hájek,et al.  A dialogue on fuzzy logic , 1997, Soft Comput..

[10]  Vilém Vychodil,et al.  Attribute Dependencies for Data with Grades , 2014, ArXiv.

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

[12]  Vilém Vychodil,et al.  Fuzzy Attribute Implications and their Expressive Power , 2013, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[13]  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.

[14]  Manuel Ojeda-Aciego,et al.  Dual multi-adjoint concept lattices , 2013, Inf. Sci..

[15]  Maria Adler,et al.  Mathematical Principles Of Fuzzy Logic , 2016 .

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

[17]  Vilém Vychodil,et al.  Fuzzy attribute logic over complete residuated lattices , 2006, J. Exp. Theor. Artif. Intell..

[18]  Vilém Vychodil,et al.  Formal concept analysis and linguistic hedges , 2012, Int. J. Gen. Syst..

[19]  Vincent Duquenne,et al.  Familles minimales d'implications informatives résultant d'un tableau de données binaires , 1986 .

[20]  Lluis Godo,et al.  A logical approach to fuzzy truth hedges , 2013, Inf. Sci..

[21]  A. Tarski.  A LATTICE-THEORETICAL FIXPOINT THEOREM AND ITS APPLICATIONS , 1955 .

[22]  Vilém Vychodil,et al.  Codd's Relational Model from the Point of View of Fuzzy Logic , 2011, J. Log. Comput..

[23]  Robert LIN,et al.  NOTE ON FUZZY SETS , 2014 .

[24]  Lotfi A. Zadeh,et al.  The Concepts of a Linguistic Variable and its Application to Approximate Reasoning , 1975 .

[25]  Manuel Ojeda-Aciego,et al.  On multi-adjoint concept lattices based on heterogeneous conjunctors , 2012, Fuzzy Sets Syst..

[26]  Jan Pavelka,et al.  On Fuzzy Logic I Many-valued rules of inference , 1979, Math. Log. Q..

[27]  Richard Holzer.  Knowledge Acquisition under Incomplete Knowledge using Methods from Formal Concept Analysis: Part I , 2004, Fundam. Informaticae.

[28]  Jan Konecny,et al.  Isotone fuzzy Galois connections with hedges , 2011, Inf. Sci..

[29]  Brian A. Davey,et al.  An Introduction to Lattices and Order , 1989 .

[30]  R. P. Dilworth.  Abstract Residuation over Lattices , 1938 .

[31]  David Maier,et al.  The Theory of Relational Databases , 1983 .

[32]  Vilém Vychodil,et al.  Fuzzy logic programming reduced to reasoning with attribute implications , 2015, Fuzzy Sets Syst..

[33]  Vilém Vychodil,et al.  On Proofs and Rule of Multiplication in Fuzzy Attribute Logic , 2007, IFSA.

[34]  Siegfried Gottwald,et al.  Mathematical Fuzzy Logics , 2008, Bulletin of Symbolic Logic.

[35]  J. A. Goguen,et al.  The logic of inexact concepts , 1969, Synthese.

[36]  H. Ono,et al.  Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Volume 151 , 2007 .

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

[38]  Christian Eitzinger,et al.  Triangular Norms , 2001, Künstliche Intell..

[39]  Silke Pollandt,et al.  Fuzzy-Begriffe - formale Begriffsanalyse unscharfer Daten , 1997 .

[40]  Vilém Vychodil,et al.  Fuzzy Closure Operators with Truth Stressers , 2005, Log. J. IGPL.

[41]  Robert A. Kowalski,et al.  The Semantics of Predicate Logic as a Programming Language , 1976, JACM.

[42]  Giangiacomo Gerla,et al.  Fuzzy Logic: Mathematical Tools for Approximate Reasoning , 2001 .

[43]  Vilém Vychodil,et al.  Attribute Implications in a Fuzzy Setting , 2006, ICFCA.

[44]  Andrei Popescu,et al.  Non-dual fuzzy connections , 2004, Arch. Math. Log..

[45]  Bernhard Ganter,et al.  Formal Concept Analysis: Mathematical Foundations , 1998 .

[46]  Bernard De Baets,et al.  Computing the Lattice of All Fixpoints of a Fuzzy Closure Operator , 2010, IEEE Transactions on Fuzzy Systems.

[47]  Abraham Kandel,et al.  Data Mining and Computational Intelligence , 2001 .

[48]  Eyke Hüllermeier,et al.  A systematic approach to the assessment of fuzzy association rules , 2006, Data Mining and Knowledge Discovery.

[49]  Anna Maria Radzikowska,et al.  Double Residuated Lattices and Their Applications , 2001, RelMiCS.

[50]  Petr Hájek,et al.  Handbook of mathematical fuzzy logic , 2011 .

[51]  Petr Hájek,et al.  Residuated fuzzy logics with an involutive negation , 2000, Arch. Math. Log..

[52]  A. Jaoua,et al.  Discovering knowledge from fuzzy concept lattice , 2001 .

[53]  Manuel Ojeda-Aciego,et al.  On basic conditions to generate multi-adjoint concept lattices via Galois connections , 2014, Int. J. Gen. Syst..

[54]  W. W. Armstrong,et al.  Dependency Structures of Data Base Relationships , 1974, IFIP Congress.

[55]  Wolfgang Wechler,et al.  Universal Algebra for Computer Scientists , 1992, EATCS Monographs on Theoretical Computer Science.

[56]  Vilém Vychodil,et al.  Reducing the Size of Fuzzy Concept Lattices by Hedges , 2005, The 14th IEEE International Conference on Fuzzy Systems, 2005. FUZZ '05..

[57]  C.J.H. Mann,et al.  Fuzzy Relational Systems: Foundations and Principles , 2003 .

[58]  Radim Bělohlávek,et al.  Fuzzy Relational Systems: Foundations and Principles , 2002 .

[59]  Grant DA-AR.  A Fuzzy-Set-Theoretic Interpretation of Linguistic Hedges , 2015 .

[60]  J. Goguen.  L-fuzzy sets , 1967 .

[61]  M. Baaz.  Infinite-valued Gödel logics with $0$-$1$-projections and relativizations , 1996 .

[62]  Jan Pavelka,et al.  On Fuzzy Logic III. Semantical completeness of some many-valued propositional calculi , 1979, Math. Log. Q..

[63]  Jonathan Lawry,et al.  Symbolic and Quantitative Approaches to Reasoning with Uncertainty , 2009 .

[64]  Peter Jipsen,et al.  Residuated lattices: An algebraic glimpse at sub-structural logics , 2007 .

[65]  Lluis Godo,et al.  Monoidal t-norm based logic: towards a logic for left-continuous t-norms , 2001, Fuzzy Sets Syst..

[66]  Petr Hájek,et al.  On very true , 2001, Fuzzy Sets Syst..

[67]  J. Lloyd.  Foundations of Logic Programming , 1984, Symbolic Computation.

[68]  Satoko Titani,et al.  Globalization of intui tionistic set theory , 1987, Ann. Pure Appl. Log..

[69]  Luís Moniz Pereira,et al.  Monotonic and Residuated Logic Programs , 2001, ECSQARU.

[70]  Manuel Ojeda-Aciego,et al.  Formal concept analysis via multi-adjoint concept lattices , 2009, Fuzzy Sets Syst..

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

[72]  Silke Pollandt.  Fuzzy Begriffe: formale Begriffsanalyse von unscharfen Daten , 1997 .