Abstract
We can use a metric to measure the differences between elements in a domain or subsets of that domain (i.e. concepts). Which particular metric should be chosen, depends on the kind of difference we want to measure. The well known Euclidean metric on ℜn and its generalizations are often used for this purpose, but such metrics are not always suitable for concepts where elements have some structure different from real numbers. For example, in (Inductive) Logic Programming a concept is often expressed as an Herbrand interpretation of some firstorder language. Every element in an Herbrand interpretation is a ground atom which has a tree structure. We start by defining a metric d on the set of expressions (ground atoms and ground terms), motivated by the structure and complexity of the expressions and the symbols used therein. This metric induces the Hausdorff metric h on the set of all sets of ground atoms, which allows us to measure the distance between Herbrand interpretations. We then give some necessary and some sufficient conditions for an upper bound of h between two given Herbrand interpretations, by considering the elements in their symmetric difference.
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References
D. W. Aha, D. Kibler, and M. K. Albert. Instance-based learning algorithms. Machine Learning, 6:37–66, 1991.
R. Beale and T. Jackson Neural Computing, an Introduction Adam Hilger.
J. W. de Bakker and J. 1. Zucker.Processes and the denotational semantics of concurrency. Information and Control, 1/2, 1984.
J. Dieudonné. Foundations of Modern Analysis. Academic Press, 1969.
W. Emde and D. Wettschereck. Relational instance-based learning. In: L. Saitta, editor, Proceedings of the 13th International Conference on Machine Learning (ICML-96), pages 122–130. Morgan Kaufmann, 1996.
H. Bunke and B.T. Messmer. Similarity measure for strutured representations In: S. Wess, K.D. Althoff and M. Richter, editors, Topics in Case-Based Reasoning, First European workshop, EWCBR-93, pages 106–118, 1993. Springer-Verlag.
E. M. Gold. Language identification in the limit. Information and Control, 10:447–474, 1967.
A. Hutchinson. Metrics on Terms and Clauses. In: M. Someren, G. Widmer, editors. Proceedings of the 9th European Conference on Machine Learning (ECML-97), pages 138–145, 1997. Springer-Verlag.
D. T. Kao, R. D. Bergeron, M. J. Cullinane, and T. M. Sparr. Semantics and mathematics of science datasampliing. Technical Report 95–14, Department of Computer Science, University of New Hampshire, 1995.
M. J. Kearns and U. V. Vazirani. An Introduction to Computational Learning Theory. MIT Press, Cambridge (MA), 1994.
M. Li and P. Vitányi. An Introduction to Kolmogorov Complexity and Its Applications. Springer-Verlag, Berlin, second edition, 1997.
J. W. Lloyd. Foundations of Logic Programming. Springer-Verlag, Berlin, second edition, 1987.
S. H. Nienhuys-Cheng and R. de Wolf. A complete method for program specialization based on unfolding. In: W. Wahlster, editor, Proceedings of the 12th European Conference on Artificial Intelligence (ECAI-96), pages 438–442. Wiley, 1996.
S. H. Nienhuys-Cheng and R. de Wolf. Foundations of Inductive Logic Programming, LNAI Tutorial 1228, Springer-Verlag, May 1997.
G.D. Plotkin. A Note on Inductive Generalization. Machine Intelligence, 5:153–163, 1970.
J.C. Reynolds. Transformational Systems and the Algebraic Structure of Atomic Formulas. Machine Intelligence, 5:135–153, 1970.
C. Rouveirol. Flattening and saturation: Two representation changes for generalization. Machine Learning, 14:219–232, 1994.
E. Y. Shapiro. Inductive inference of theories from facts. Research Report 192, Yale University, 1981.
D. Wettschereck. A Study of Distance-Based Machine Learning Algorithms. PhD thesis, Oregon State University, 1994.
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© 1997 Springer-Verlag Berlin Heidelberg
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Nienhuys-Cheng, SH. (1997). Distance between Herbrand interpretations: A measure for approximations to a target concept. In: Lavrač, N., Džeroski, S. (eds) Inductive Logic Programming. ILP 1997. Lecture Notes in Computer Science, vol 1297. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3540635149_50
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DOI: https://doi.org/10.1007/3540635149_50
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