Abstract
This paper characterizes the polynomial time learnability of TP k, the class of collections of at most k first-order terms. A collection in TPA k defines the union of the languages defined by each first-order terms in the set. Unfortunately, the class TP k not polynomial time learnable in most of learning frameworks under standard assumptions in computational complexity theory. To overcome this computational hardness, we relax the learning problem by allowing a learning algorithm to make membership queries. We present a polynomial time algorithm that exactly learns every concept in TP k using O(kn) equivalence and O(k 2 n · max{k, n}) membership queries, where n is the size of longest counterexample given so far. In the proof, we use a technique of replacing each restricted subset query by several membership queries under some condition on a set of function symbols. As corollaries, we obtain the polynomial time PAC-learnability and the polynomial time predictability of TP k when membership queries are available. We also show a lower bound Ω(kn) of the number of queries necessary to learn TP k using both types of queries. Further, we show that neither types of queries can be eliminated to achieve efficient learning of TP k. Finally, we apply our results in learning of a class of restricted logic programs, called unit clause programs.
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References
D. Angluin. Finding patterns common to a set of strings. Journal of Computer and System Sciences, 21:46–62, 1980.
D. Angluin. Queries and concept learning. Machine Learning, 2:319–342, 1988.
S. Arikawa, S. Kuhara, S. Miyano, A. Shinohara, and T. Shinohara. A learning algorithm for elementary formal systems and its experiments on identification of transmembrane domains. In Proc. the 25th HICSS, 675–684, 1992.
H. Arimura, T. Shinohara, and S. Otsuki. A polynomial time algorithm for finding finite unions of tree pattern languages. In Proc. the 2nd NIL, LNAI 659, Springer-Verlag, 118–131, 1991.
H. Arimura, T. Shinohara, and S. Otsuki. Finding minimal generalizations for unions of pattern languages and its application to inductive inference from positive data. In Proc. the 11th STACS, LNCS 775, Springer-Verlag, 649–660, 1994.
H. Arimura, H. Ishizaka, T. Shinohara, and S. Otsuki. A generalization of the least general generalization. Machine Intelligence, 13, 59–85, Oxford Univ. Press, 1994.
M. Frazier and L. Pitt. Classic Learning. In Proc. COLT'94, 23–34, 1994.
H. Ishizaka, H. Arimura, and T. Shinohara. Finding tree patterns consistent with positive and negative examples using queries In Proc. ALT'94, LNAI872, 317–332, 1994.
T. Jiang, A. Salomaa, K. Salomaa, and S. Yu. Inclusion is undecidable for pattern languages. In Proc. 20th ICALP, pp. 301–312, LNCS 700, Springer, 1993.
G. Kuper, K. McAloon, K. Palem, K. Perry. Efficient Parallel Algorithm for Anti-Unification and Relative Complement. In Proc. the 3rd LICS, 111–120, 1988.
N. Littlestone. Learning quickly when irrelevant attributes abound: A new linear-threshold algorithm. Machine Learning, 2, 285–318, 1988.
W. Maass and G. Turán. Lower bound methods and separation results for on-line learning models. Machine Learning, 9, 107–145, 1992.
C. D. Page Jr. and A. M. Frisch. Generalization and Learnability: A Study of Constrained Atoms. S. Muggleton, editor, Inductive Logic Programming, 29–61, 1992.
L. Pitt and L. G. Valiant. Computational limitations on learning from examples. JACM, 35:965–984, 1988.
L. Pitt and M. K. Warmuth. Prediction preserving reduction. JCSS, 41:430–467, 1990.
G. D. Plotkin. A note on inductive generalization. In Machine Intelligence 5, 153–163. Edinburgh University Press, 1970.
J. C. Reynolds. Transformational systems and the algebraic structure of atomic formulas. In Machine Intelligence 5, 135–152. Edinburgh University Press, 1970.
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© 1995 Springer-Verlag Berlin Heidelberg
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Arimura, H., Ishizaka, H., Shinohara, T. (1995). Learning unions of tree patterns using queries. In: Jantke, K.P., Shinohara, T., Zeugmann, T. (eds) Algorithmic Learning Theory. ALT 1995. Lecture Notes in Computer Science, vol 997. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60454-5_29
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DOI: https://doi.org/10.1007/3-540-60454-5_29
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