ICFCA 2019: Formal Concept Analysis pp 55-72

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

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11511)

Abstract

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 . 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.

Keywords

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

References

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).
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).
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)
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)
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)
7. 7.
Bertet, K., Monjardet, B.: The multiple facets of the canonical direct unit implicational basis. Theoret. Comput. Sci. 411, 2155–2166 (2010)
8. 8.
Boros, E., Elbassioni, K., Gurvich, V., Khachiyan, L.: Generating dual-bounded hypergraphs. Optim. Meth. Softw. 17, 749–781 (2002)
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)
10. 10.
Fredman, M., Khachiyan, L.: On the complexity of dualization of monotone disjunctive normal forms. J. Algorithms 21, 618–628 (1996)
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)
12. 12.
Mannila, H., Räihä, K.J.: The Design of Relational Databases. Addison-Wesley, Reading (1992)
13. 13.
McKinsey, J.: The decision problem for some classes of sentences without quantifiers. J. Symbolic Logic 8, 61–76 (1943)
14. 14.
Murakami, K., Uno, T.: Efficient algorithms for dualizing large scale hypergraphs. Disc. Appl. Math. 170, 83–94 (2014)
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)
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)
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).
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).
21. 21.
Wild, M.: The joy of implications. Theoret. Comput. Sci. 658, 264–292 (2017) 