Archive for Mathematical Logic

, Volume 54, Issue 7–8, pp 785–802 | Cite as

A logic of graded attributes

  • Radim Belohlavek
  • Vilem Vychodil


We present a logic for reasoning about attribute dependencies in data involving degrees such as a degree to which an object is red or a degree to which two objects are similar. The dependencies are of the form AB and can be interpreted in two ways: first, in data tables with entries representing degrees to which objects (rows) have attributes (columns); second, in database tables where each domain is equipped with a similarity relation. We assume that the degrees form a scale equipped with operations representing many-valued logical connectives. If 0 and 1 are the only degrees, the algebra of degrees becomes the two-element Boolean algebra and the two interpretations become well-known dependencies in Boolean data and functional dependencies of relational databases. In a setting with general scales, we obtain a new kind of dependencies with naturally arising degrees of validity, degrees of entailment, and related logical concepts. The deduction rules of the proposed logic are inspired by Armstrong rules and make it possible to infer dependencies to degrees—the degrees of provability. We provide a soundness and completeness theorem for such a setting asserting that degrees of entailment coincide with degrees of provability, prove the independence of deduction rules, and present further observations.


Attribute implication Fuzzy logic Graded-style completeness Pavelka-style logic 

Mathematics Subject Classification

03B50 03B52 68T37 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abiteboul S. et al.: The Lowell database research self-assessment. Commun. ACM 48(5), 111–118 (2005)CrossRefGoogle Scholar
  2. 2.
    Armstrong, W.W.: Dependency structures in data base relationships. IFIP Congress, Geneva, Switzerland, pp. 580–583 (1974)Google Scholar
  3. 3.
    Belohlavek R., Vychodil V.: Attribute implications in a fuzzy setting. Lect. Notes Artif. Intell. 3874, 45–60 (2006)Google Scholar
  4. 4.
    Belohlavek R., Vychodil V.: Codd’s relational model from the point of view of fuzzy logic. J. Logic Comput. 21(5), 851–862 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Belohlavek, R., Vychodil, V.: Attribute dependencies for data with grades (submitted, arXiv:1402.2071 [cs.LO]).
  6. 6.
    Carpineto C., Romano G.: Concept Data Analysis: Theory and Applications. Wiley, New Jersey (2004)CrossRefGoogle Scholar
  7. 7.
    Cintula, P., Hájek, P., Noguera, C.: Handbook of Mathematical Fuzzy Logic, vol. I, II, College Publ., London 2011 (vols. 37 and 38 of series Studies in Logic)Google Scholar
  8. 8.
    Esteva, F., Godo, L., Marchioni, E.: Fuzzy logics with enriched language. In [7], vol. II, pp. 627–711Google Scholar
  9. 9.
    Fagin R.: Functional dependencies in a relational database and propositional logic. IBM J. Res. Dev. 21(6), 543–544 (1977)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fagin R.: Combining fuzzy information: an overview. SIGMOD Rec 31(2), 109–118 (2002)CrossRefGoogle Scholar
  11. 11.
    Galatos N., Jipsen P., Kowalski T., Ono H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Elsevier, New York (2007)Google Scholar
  12. 12.
    Ganter B., Wille R.: Formal Concept Analysis. Mathematical Foundations. Springer, Berlin (1999)CrossRefzbMATHGoogle Scholar
  13. 13.
    Gerla G.: Inferences in probability logic. Artif. Intell. 70(1–2), 33–52 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gerla G.: Fuzzy Logic. Mathematical Tools for Approximate Reasoning. Kluwer, Dordrecht (2001)zbMATHGoogle Scholar
  15. 15.
    Goguen, J.A.: The logic of inexact concepts. Synthese 18, 325–373 (1968–1969)Google Scholar
  16. 16.
    Gottwald, S.: A Treatise on Many-Valued Logic. Studies in Logic and Computation, vol. 9, Research Studies Press: Baldock, Hertfordshire, England (2001)Google Scholar
  17. 17.
    Gottwald S.: Mathematical fuzzy logics. Bull. Symb. Logic 14, 210–239 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hájek P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (1998)CrossRefzbMATHGoogle Scholar
  19. 19.
    Hájek P.: On very true. Fuzzy Sets Syst. 124, 329–333 (2001)CrossRefzbMATHGoogle Scholar
  20. 20.
    Maier D.: The Theory of Relational Databases. Computer Science Press, Rockville (1983)zbMATHGoogle Scholar
  21. 21.
    Mayor G., Torrens J.: Triangular norms on discrete settings. In: Klement, E.P., Mesiar, R. (eds) Logical, Algebraic, Analytic and Probabilistic Aspects of Triangular Norms, pp. 189–230. Elsevier, New York (2005)Google Scholar
  22. 22.
    Novák V., Perfilieva I., Močkoř J.: Mathematical Principles of Fuzzy Logic. Kluwer, Dodrecht (1999)CrossRefzbMATHGoogle Scholar
  23. 23.
    Pavelka, J.: On fuzzy logic I, II, III. Z. Math. Logik Grundlagen Math. 25, 45–52, 119–134, 447–464 (1979)Google Scholar
  24. 24.
    Raju K.V.S.V.N., Majumdar A.K.: Fuzzy functional dependencies and lossless join decomposition of fuzzy relational database systems. ACM Trans. Database Syst. 13(2), 129–166 (1988)CrossRefGoogle Scholar
  25. 25.
    Takeuti G., Titani S.: Globalization of intuitionistic set theory. Ann. Pure Appl. Logic 33, 195–211 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Tyagi B.K., Sharfuddin A., Dutta R.N., Tayal D.K.: A complete axiomatization of fuzzy functional dependencies using fuzzy function. Fuzzy Sets Syst. 151(2), 363–379 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Zadeh L.A.: Fuzzy Logic. IEEE Comput. 21(4), 83–93 (1988)CrossRefGoogle Scholar
  28. 28.
    Zadeh L.A.: Fuzzy logic, neural networks, and soft computing. Commun. ACM 37(3), 77–84 (1994)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Zhang C., Zhang S.: Association Rule Mining. Models and Algorithms. Springer, Berlin (2002)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Computer SciencePalacký UniversityOlomoucCzech Republic

Personalised recommendations