About the Family of Closure Systems Preserving Non-unit Implications in the Guigues-Duquenne Base

  • Alain Gély
  • Lhouari Nourine
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3874)


Consider a Guigues-Duquenne base \(\Sigma^{\mathcal{F}} = \Sigma^{\mathcal{F}}_{\mathcal{J}} \cup \Sigma^{\mathcal{F}}_{\downarrow}\) of a closure system \(\mathcal{F}\), where \(\Sigma_{\mathcal{J}}\) the set of implications \(P \rightarrow P^{\Sigma^{\mathcal{F}}}\) with |P|=1, and \(\Sigma^{\mathcal{F}}_{\downarrow}\) the set of implications \(P \rightarrow P^{\Sigma^{\mathcal{F}}}\) with |P| >1. Implications in \(\Sigma^{\mathcal{F}}_{\mathcal{J}}\) can be computed efficiently from the set of meet-irreducible \(\mathcal{M}(\mathcal{F})\); but the problem is open for \(\Sigma^{\mathcal{F}}_{\downarrow}\). Many existing algorithms build \(\mathcal{F}\) as an intermediate step.

In this paper, we characterize the cover relation in the family \(\mathcal{C}_{\downarrow}(\mathcal{F})\) with the same Σ, when ordered under set-inclusion. We also show that \(\mathcal{M}(\mathcal{F}_{\perp})\) the set of meet-irreducible elements of a minimal closure system in \(\mathcal{C}_{\downarrow}(\mathcal{F})\) can be computed from \(\mathcal{M}(\mathcal{F})\) in polynomial time for any \(\mathcal{F}\) in \(\mathcal{C}_{\downarrow}(\mathcal{F})\). Moreover, the size of \(\mathcal{M}(\mathcal{F}_{\perp})\) is less or equal to the size of \(\mathcal{M}(\mathcal{F})\).


Closure system Guigues-Duquenne base Equivalence relation 


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  1. 1.
    Eiter, T., Gottlob, G.: Identifying the minimal transversals of a hypergraph and related problems. SIAM Journal on Computing 24(6), 1278–1304 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Eiter, T., Gottlob, G.: Hypergraph transversal computation and related problems in logic and ai. In: Flesca, S., Greco, S., Leone, N., Ianni, G. (eds.) JELIA 2002. LNCS (LNAI), vol. 2424, pp. 549–564. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  3. 3.
    Ibaraki, T., Kogan, A., Makino, K.: Inferring minimal functional dependencies in horn and q-horn theories. Technical report, Rutcor Research Report, RRR 35-2000 (2000)Google Scholar
  4. 4.
    Maier, D.: The theory of relational data bases. Computer Science Press, Rockville (1983)Google Scholar
  5. 5.
    Fredman, M.L., Khachiyan, L.: On the complexity of dualization of monotone disjunctive normal forms. Journal of Algorithms, 618–628 (1996)Google Scholar
  6. 6.
    Guigues, J., Duquenne, V.: Familles minimales d’implications informatives résultant d’un tableau de données binaires. Math. Sci. hum 95, 5–18 (1986)MathSciNetGoogle Scholar
  7. 7.
    Bordalo, G., Monjardet, B.: The lattice of strict completions of a finite poset. Algebra Universalis 47, 183–200 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Nation, J.B., Pogel, A.: The lattice of completions of an ordered set. Order 14, 1–7 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Ganter, B.: Two basic algorithms in concept analysis. Technical report, No 831, Technische Hochschule Darmstadt (1984)Google Scholar
  10. 10.
    Beeri, C., Berstein, P.: Computational problems related to the design of normal form relational schemas. ACM Trans. on database systems 1, 30–59 (1979)CrossRefGoogle Scholar
  11. 11.
    Wild, M.: Implicational bases for finite closure systems. Technical report, No 1210, Technische Hochschule Darmstadt (1989)Google Scholar
  12. 12.
    Shock, R.C.: Computing the minimum cover of functional dependencies. Information Processing Letters 3, 157–159 (1986)CrossRefMathSciNetGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Alain Gély
    • 1
  • Lhouari Nourine
    • 1
  1. 1.Université Clermont II Blaise Pascal, LIMOSAubièreFrance

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