Direct and Binary Direct Bases for One-Set Updates of a Closure System

  • Kira AdarichevaEmail author
  • Taylor Ninesling
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11511)


We introduce a concept of a binary-direct implicational basis and show that the shortest binary-direct basis exists and it is known as the D-basis introduced in Adaricheva, Nation, Rand [4]. Using this concept we approach the algorithmic solution to the Singleton Horn Extension problem, as well as the one set removal problem, when the closure system is given by the canonical direct or binary-direct basis. In this problem, a new closed set is added to or removed from the closure system forcing the re-write of a given basis. Our goal is to obtain the same type of implicational basis for the new closure system as was given for original closure system and to make the basis update an optimal process.


Closure system Horn-to-Horn belief revision Singleton Horn Extension problem Direct basis Canonical direct basis The D-basis Ordered direct basis 



The first results of the paper were presented on the poster session of ICFCA-2017 in Rennes, France, and both authors’ participation in the conference was supported by the research fund of Hofstra University. We thank Sergey Obiedkov for pointing to the important publication of Marcel Wild [19], and we thank Justin Cabot-Miller from Hofstra University for his support in producing valuable test cases in the implementation phase.


  1. 1.
    Adaricheva, K., Nation, J.B.: Bases of closure systems. In: Grätzer, G., Wehrung, F. (eds.) Lattice Theory: Special Topics and Applications, vol. 2. Springer, Basel (2016). Scholar
  2. 2.
    Adaricheva, K., Nation, J.B.: Lattices of algebraic subsets and implicational classes. In: Grätzer, G., Wehrung, F. (eds.) Lattice Theory: Special Topics and Applications, vol. 2. Springer, Basel (2016). Scholar
  3. 3.
    Adaricheva, K., Nation, J.B.: Discovery of the \(D\)-basis in binary tables based on hypergraph dualization. Theoret. Comput. Sci. 658, 307–315 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Adaricheva, K., Nation, J.B., Rand, R.: Ordered direct implicational basis of a finite closure system. Disc. Appl. Math. 161, 707–723 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Adaricheva, K., Sloan, R., Szörenyi, B., Turan, G.: Horn belief contraction: remainders, envelopes and complexity. In: Proceedings of the KR 2012, pp. 107–115 (2012)Google Scholar
  6. 6.
    Caspard, N., Monjardet, B.: The lattices of closure systems, closure operators, and implicational systems on a finite set: a survey. Disc. Appl. Math. 127, 241–269 (2003)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bertet, K., Monjardet, B.: The multiple facets of the canonical direct unit implicational basis. Theoret. Comput. Sci. 411, 2155–2166 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Boros, E., Elbassioni, K., Gurvich, V., Khachiyan, L.: Generating dual-bounded hypergraphs. Optim. Meth. Softw. 17, 749–781 (2002)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dowling, W., Gallier, J.H.: Linear-time algorithms for testing the satisfiability of propositional Horn formulae. J. Logic Program. 3, 267–284 (1984)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fredman, M., Khachiyan, L.: On the complexity of dualization of monotone disjunctive normal forms. J. Algorithms 21, 618–628 (1996)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Khachiyan, L., Boros, E., Elbassioni, K., Gurvich, V.: An efficient implementation of a quasi-polynomial algorithmfor generating hypergraph transversals and its application in joint generation. Disc. Appl. Math. 154, 2350–2372 (2006)CrossRefGoogle Scholar
  12. 12.
    Mannila, H., Räihä, K.J.: The Design of Relational Databases. Addison-Wesley, Reading (1992)zbMATHGoogle Scholar
  13. 13.
    McKinsey, J.: The decision problem for some classes of sentences without quantifiers. J. Symbolic Logic 8, 61–76 (1943)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Murakami, K., Uno, T.: Efficient algorithms for dualizing large scale hypergraphs. Disc. Appl. Math. 170, 83–94 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Langlois, M., Sloan, R., Szörenyi, B., Turan, G.: Horn complements: towards Horn-to-Horn belief revision. In: Proceedings of the AAAI 2008, pp. 466–471 (2008)Google Scholar
  16. 16.
    Rudolph, S.: Succintness and tractability of closure operator representations. Theoret. Comput. Sci. 658, 327–345 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Rodríguez-Lorenzo, E., Adaricheva, K., Cordero, P., Enciso, M., Mora, A.: Formation of the D-basis from implicational systems using simplification logic. Int. J. Gen. Syst. 46(5), 547–568 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Valtchev, P., Missaoui, R.: Building concept (Galois) lattices from parts: generalizing the incremental methods. In: Delugach, H.S., Stumme, G. (eds.) ICCS-ConceptStruct 2001. LNCS (LNAI), vol. 2120, pp. 290–303. Springer, Heidelberg (2001). Scholar
  19. 19.
    Wild, M.: Computations with finite closure systems and implications, Preprint N 1708. Technische Hochschule Darmstadt, pp. 1–22 (1994)Google Scholar
  20. 20.
    Wild, M.: Computations with finite closure systems and implications. In: Du, D.-Z., Li, M. (eds.) COCOON 1995. LNCS, vol. 959, pp. 111–120. Springer, Heidelberg (1995). Scholar
  21. 21.
    Wild, M.: The joy of implications. Theoret. Comput. Sci. 658, 264–292 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsHofstra UniversityHempsteadUSA
  2. 2.Hofstra UniversityHempsteadUSA

Personalised recommendations