On minimal sets of graded attribute implications

We explore the structure of non-redundant and minimal sets consisting of graded if-then rules. The rules serve as graded attribute implications in object-attribute incidence data and as similarity-based functional dependencies in a similarity-based generalization of the relational model of data. Based on our observations, we derive a polynomial-time algorithm which transforms a given finite set of rules into an equivalent one which has the least size in terms of the number of rules.

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

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

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

[4]  Vilém Vychodil,et al.  Attribute dependencies for data with grades II*,† , 2016, Int. J. Gen. Syst..

[5]  Rudolf Kruse,et al.  Fuzzy-systems in computer science , 1994 .

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

[7]  Felix Distel,et al.  On the complexity of enumerating pseudo-intents , 2011, Discret. Appl. Math..

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

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

[10]  Deyu Li,et al.  Decision implication canonical basis: a logical perspective , 2015, J. Comput. Syst. Sci..

[11]  Mohammed J. Zaki.  Mining Non-Redundant Association Rules , 2004, Data Min. Knowl. Discov..

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

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

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

[15]  Angel Mora,et al.  Non-deterministic ideal operators: An adequate tool for formalization in Data Bases , 2008, Discret. Appl. Math..

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

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

[18]  Vilém Vychodil,et al.  Data Tables with Similarity Relations: Functional Dependencies, Complete Rules and Non-redundant Bases , 2006, DASFAA.

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

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

[21]  L. Zadeh.  A Fuzzy-Set-Theoretic Interpretation of Linguistic Hedges , 1972 .

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

[23]  Tomasz Imielinski,et al.  Mining association rules between sets of items in large databases , 1993, SIGMOD Conference.

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

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

[26]  Vilém Vychodil,et al.  Derivation digraphs for dependencies in ordinal and similarity-based data , 2014, Inf. Sci..

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

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

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

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

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

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

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

[34]  Petr Hájek.  On very true , 2001, Fuzzy Sets Syst..

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

[36]  Vilém Vychodil,et al.  Basic Algorithm for Attribute Implications and Functional Dependencies in Graded Setting , 2008, Int. J. Found. Comput. Sci..