Generalization Algorithms for Second-Order Terms
In this paper, we study the generalization algorithms for second-order terms, which are treated as first-order terms with function variables, under an instantiation order denoted by≽. First, we extend the least generalization algorithm lg for a pair of first-order terms under≽, introduced by Plotkin and Reynolds, to the one for a pair of second-order terms. The extended algorithm lg, however, is insufficient to characterize the generalization for a pair of second-order terms, because it computes neither the least generalization under≽nor the structure-preserving generalization. Since the transformation rule for second-order matching algorithm consists of an imitation and a projection, in this paper, we introduce the imitation-free generalization algorithm ifg and the projection-free generalization algorithm pfg. Then, we show that ifg computes the least generalization under≽of any pair of second-order terms, whereas pfg computes the generalization equivalent to lg under≽. Nevertheless, neither ifg nor pfg preserves the structural information. Hence, we also introduce the algorithm spg and show that it computes a structure-preserving generalization. Finally, we show that the algorithms lg, pfg and spg are associative, while the algorithm ifg is not.
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- 1.Baxter, L.D.: The complexity of unification, Doctoral Thesis, Department of Computer Science, University of Waterloo (1977)Google Scholar
- 2.Dietzen, S., Pfenning, F.: Higher-order and modal logic as a framework for explanation-based generalization. Mach. Learn. 9, 23–55 (1992)Google Scholar
- 4.Feng, C., Muggleton, S.: Towards inductive generalisation in higher order logic. In: Proc. 9th Internat. Conf. Machine Learning, pp. 154–162 (1992)Google Scholar
- 7.Hasker, R.: The reply of program derivations, Ph.D. Thesis, Department of Computer Science, University of Illinois at Urbana-Champaign (1995)Google Scholar
- 12.Lassez, J.-L., Maher, M.J., Marriot, L.: Unification revisited. In: Minker, J. (ed.) Foundations of deductive databases and logic programming, pp. 587–625. Morgan-Kaufmann, San Francisco (1988)Google Scholar
- 15.Muggleton, S.: Inverse entailment and Progol, New Generat. Comput. 13, 245–286 (1995)Google Scholar
- 16.Nienhuys-Cheng, S.-H., de Wolf, R.: Foundations of Inductive Logic Programming. LNCS(LNAI), vol. 1228. Springer, Heidelberg (1997)Google Scholar
- 17.Pfenning, F.: Unification and anti-unification in the calculus of constructions. In: Proc. 6th Annual Symp. Logic in Computer Science, pp. 74–85 (1991)Google Scholar
- 20.Suzuki, Y., Inomae, K., Shoudai, T., Miyahara, T., Uchida, T.: A polynomial time matching algorithm of structured ordered tree patterns for data mining from semistractural data. In: Matwin, S., Sammut, C. (eds.) ILP 2002. LNCS (LNAI), vol. 2583, pp. 270–284. Springer, Heidelberg (2003)CrossRefGoogle Scholar