Fuzzy logic programming reduced to reasoning with attribute implications

Abstract We present a link between two types of logic systems for reasoning with graded if–then rules: the system of fuzzy logic programming (FLP) in sense of Vojtas and the system of fuzzy attribute logic (FAL) in sense of Belohlavek and Vychodil. We show that each finite theory consisting of formulas of FAL can be represented by a definite program so that the semantic entailment in FAL can be characterized by correct answers for the program. Conversely, we show that for each definite program there is a collection of formulas of FAL so that the correct answers can be represented by the entailment in FAL. Using the link, we can transport results from FAL to FLP and vice versa which gives us, e.g., a syntactic characterization of correct answers based on Pavelka-style Armstrong-like axiomatization of FAL. We further show that entailment in FLP is reducible to reasoning with Boolean attribute implications and elaborate on related issues including properties of least models.

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

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

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

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

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

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

[7]  Vilém Vychodil,et al.  Query systems in similarity-based databases: logical foundations, expressive power, and completeness , 2010, SAC '10.

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

[9]  Vilém Vychodil,et al.  Representing Fuzzy Logic Programs by Graded Attribute Implications , 2012, MDAI.

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

[11]  P. Cousot,et al.  Constructive versions of tarski's fixed point theorems , 1979 .

[12]  Manuel Ojeda-Aciego,et al.  A Procedural Semantics for Multi-adjoint Logic Programming , 2001, EPIA.

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

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

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

[16]  Ulf Nilsson,et al.  Logic, programming and Prolog , 1990 .

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

[18]  Vilém Vychodil,et al.  Fuzzy Horn logic II , 2006, Arch. Math. Log..

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

[20]  Elliott Mendelson,et al.  Introduction to mathematical logic (3. ed.) , 1987 .

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

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

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

[24]  Elliott Mendelson,et al.  Introduction to Mathematical Logic , 1979 .

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

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

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

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

[29]  R. Belohlávek.  Fuzzy Relational Systems: Foundations and Principles , 2002 .

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

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

[32]  Manuel Ojeda-Aciego,et al.  Sorted Multi-adjoint Logic Programs: Termination Results and Applications , 2004, JELIA.

[33]  Peter Vojtás,et al.  Fuzzy logic programming , 2001, Fuzzy Sets Syst..

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

[35]  Vilém Vychodil,et al.  Implications from data with fuzzy attributes vs. scaled binary attributes , 2005, The 14th IEEE International Conference on Fuzzy Systems, 2005. FUZZ '05..

[36]  E. F. Codd,et al.  A relational model of data for large shared data banks , 1970, CACM.

[37]  John Wylie Lloyd,et al.  Foundations of Logic Programming , 1987, Symbolic Computation.

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

[39]  E. F. Codd,et al.  A relational model of data for large shared data banks , 1970, CACM.

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

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