Abstract
We describe an alternative construction of an existing canonical representation for definite Horn theories, the Guigues-Duquenne basis (or GD basis), which minimizes a natural notion of implicational size. We extend the canonical representation to general Horn, by providing a reduction from definite to general Horn CNF. Using these tools, we provide a new, simpler validation of the classic Horn query learning algorithm of Angluin, Frazier, and Pitt, and we prove that this algorithm always outputs the GD basis regardless of the counterexamples it receives.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Angluin, D. (1987). Learning regular sets from queries and counterexamples. Information and Computation, 75(2), 87–106.
Angluin, D., & Kharitonov, M. (1995). When won’t membership queries help? Journal of Computer and System Sciences, 50(2), 336–355.
Angluin, D., Frazier, M., & Pitt, L. (1992). Learning conjunctions of Horn clauses. Machine Learning, 9, 147–164.
Arias, M., & Balcázar, J. L. (2008). Query learning and certificates in lattices. In LNAI: Vol. 5254. ALT 2008 (pp. 303–315).
Arias, M., & Balcázar, J. L. (2009). Canonical Horn representations and query learning. In LNAI: Vol. 5809. ALT 2009 (pp. 156–170).
Arias, M., & Khardon, R. (2002). Learning closed Horn expressions. Information and Computation, 178(1), 214–240.
Arias, M., Feigelson, A., Khardon, R., & Servedio, R. A. (2006). Polynomial certificates for propositional classes. Information and Computation, 204(5), 816–834.
Arias, M., Khardon, R., & Maloberti, J. (2007). Learning Horn expressions with LogAn-H. Journal of Machine Learning Research, 8, 587.
Balcázar, J. L. (2005). Query learning of Horn formulas revisited. In Computability in Europe conference.
Bertet, K., & Monjardet, B. (2010). The multiple facets of the canonical direct unit implicational basis. Theoretical Computer Science, 411(22–24), 2155–2166.
Chang, C. L., & Lee, R. (1973). Symbolic logic and mechanical theorem proving. Orlando: Academic Press.
Frazier, M., & Pitt, L. (1993). Learning from entailment: an application to propositional Horn sentences. In Proceedings of the international conference on machine learning (pp. 120–127), Amherst, MA. San Mateo: Morgan Kaufmann.
Frazier, M., & Pitt, L. (1996). CLASSIC learning. Machine Learning, 25, 151–193.
Gaintzarain, J., Hermo, M., & Navarro, M. (2005). On learning conjunctions of Horn⊃ clauses. In Computability in Europe conference.
Guigues, J. L., & Duquenne, V. (1986). Familles minimales d’implications informatives resultants d’un tableau de données binaires. Mathématiques Et Sciences Humaines, 95, 5–18.
Hellerstein, L., Pillaipakkamnatt, K., Raghavan, V., & Wilkins, D. (1996). How many queries are needed to learn? Journal of the ACM, 43(5), 840–862.
Hermo, M., & Lavín, V. (2002). Negative results on learning dependencies with queries. In International symposium on artificial intelligence and mathematics.
Horn, A. (1956). On sentences which are true of direct unions of algebras. The Journal of Symbolic Logic, 16, 14–21.
Khardon, R., & Roth, D. (1996). Reasoning with models. Artificial Intelligence, 87(1–2), 187–213.
Kivinen, J., & Mannila, H. (1995). Approximate inference of functional dependencies from relations. Theoretical Computer Science, 149(1), 129–149.
Kleine Büning, H., & Lettmann, T. (1999). Propositional logic: deduction and algorithms. Cambridge: Cambridge University Press.
Maier, D. (1980). Minimum covers in relational database model. Journal of the ACM, 27, 664–674.
McKinsey, J. C. C. (1943). The decision problem for some classes of sentences without quantifiers. The Journal of Symbolic Logic, 8, 61–76.
Rouveirol, C. (1994). Flattening and saturation: two representation changes for generalization. Machine Learning, 14(1), 219–232.
Selman, B., & Kautz, H. (1996). Knowledge compilation and theory approximation. Journal of the ACM, 43(2), 193–224.
Valiant, L. G. (1984). A theory of the learnable. Communications of the ACM, 27(11), 1134–1142.
Wang, H. (1960). Toward mechanical mathematics. IBM Journal for Research and Development, 4, 2–22.
Wild, M. (1994). A theory of finite closure spaces based on implications. Advances in Mathematics, 108, 118–139.
Author information
Authors and Affiliations
Corresponding author
Additional information
Editor: Avrim Blum.
Work partially supported by MICINN projects SESAAME-BAR (TIN2008-06582-C03-01) and FORMALISM (TIN2007-66523).
Rights and permissions
About this article
Cite this article
Arias, M., Balcázar, J.L. Construction and learnability of canonical Horn formulas. Mach Learn 85, 273–297 (2011). https://doi.org/10.1007/s10994-011-5248-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10994-011-5248-5