Abstract
In this paper, we show two hardness results for approximating the best function-free Horn clause by an element of the same class. Our first result shows that for some constant k > 0, the error rate of the best k-Horn clause cannot be approximated in polynomial time to within any constant factor by an element of the same class. Our second result is much stronger. Under some frequently encountered complexity hypothesis, we show that if we replace the constant number of Horn clauses by a small, poly-logarithmic number, the constant factor blows up exponentially to a quasi-polynomial factor n log k n, where n is the number of predicates of the problem, a measure of its complexity. Our main result links the difficulty of error approximation with the number of clauses allowed. We finally give an outline of the incidence of our result on systems that learn using ILP (Inductive Logic Programming) formalism.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
S. Arora. Probabilistic checking of proofs and hardness of approximation problems. Technical Report CS-TR-476-94, Princeton University, 1994.
W. Buntine. Generalized subsumption and its applications to induction and redundancy. Artificial Intelligence, 36:149–176, 1988.
W.W. Cohen. Pac-learning nondeterminate clauses. In Proceedings of the Twelfth National Conference on Artificial Intelligence, AAAI'94, pages 676–681, 1994.
W.W. Cohen. Pac-learning recursive logic programs: Efficient algorithms. Journal of Artificial Intelligence Research, 2:501–539, 1995.
W.W. Cohen. Pac-learning recursive logic programs: Negative results. Journal of Artificial Intelligence Research, 2:541–571, 1995.
S. Dzerovski, S.H. Muggleton, and S. Russel. Pac-learnability of determinate logic programs. In Proceedings of COLT-92, pages 128–137, 1992.
E.M. Gold. Language indentification in the limit. Information and Control, 10:447–474, 1967.
K-U. Höffgen and H.U. Simon. Robust trainability of single neurons. In Proc. of the 5 th International Conference on Computational Theory, 1992.
P. Jappy, R. Nock, and O. Gascuel. Negative robust learning results for horn clause programs. In Proceedings of ICML'96, pages 258–265, 1996.
M. Kearns, M. Li, L. Pitt, and L.G. Valiant. On the learnability of boolean for-mulae. In Proceedings of STOCS'87, pages 285–294, 1987.
J.U. Kietz and S. Dzeroski. Inductive logic programming and learnability. Sigart Bulletin, 5:22–32, 1994.
S.H. Muggleton. Bayesian inductive logic programming. In Proceedings of the Seventh Workshop on COmputational Learning Theory, 1994.
S.H. Muggleton and C. Feng. Efficient induction of logic programs. Inductive Logic Programming. Academic Press, New York, 1992.
R. Nock and P. Jappy. On the hardness of approximating function-free horn clauses. Technical Report LIRMM-RR-98017, LIRMM, 1998.
L.G. Valiant. A theory of the learnable. Association for Computing Machinery Communications, 27:1134–1142, 1984.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1998 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Nock, R., Jappy, P. (1998). Function-free Horn clauses are hard to approximate. In: Page, D. (eds) Inductive Logic Programming. ILP 1998. Lecture Notes in Computer Science, vol 1446. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0027323
Download citation
DOI: https://doi.org/10.1007/BFb0027323
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-64738-6
Online ISBN: 978-3-540-69059-7
eBook Packages: Springer Book Archive