Curried least general generalization: A framework for higher order concept learning

  • Srinivas Padmanabhuni
  • Randy Goebel
  • Koichi Furukawa
Invited Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1359)


Continued progress with research in inductive logic programming relies on further extensions of their underlying logics. The standard tactics for extending expressivity include a generalization to higher order logics, which immediately forces attention to the computational complexity of higher order reasoning.

A major thread of inductive logic programming research has focussed on the identification of preferred hypothesis sets, initiated by Plotkin's work on least general generalizations (LGGs). Within higher order frameworks, a relevant extension of LGG is Furukawa's hyper least general generalization (HLGG) [FIG97].

We present a relevant higher order extension of Furukawa's HLGG based on currying, which we call Curried Least General Generalization (CLGG). The idea is that the formal difficulties with the reasoning complexity of a higher order language can be controlled by forming new hypothetical terms restricted to those obtainable by Currying. This technique subsumes the inductive generalization power of HLGG, provides a basis for a significant extension of first order ILP, and is theoretically justified within a well understood formal foundation.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [BD77]
    R.M. Burstall and J. Darlington. A transformation system for developing recursive programts. Journal of the ACM, 24(1):44–67, 1977.zbMATHMathSciNetCrossRefGoogle Scholar
  2. [Cur30]
    Haskell B. Curry. Grundlagender kombinatorischen logik. Am. J. Math., 52:509–536, 1930.zbMATHMathSciNetCrossRefGoogle Scholar
  3. [FIG97]
    K. Furukawa, M. Imai, and R. Goebel. Hyper least general generalization and its application to higher-order concept learning. Research Memo IEI-RM 97001, SFC Research Institute, Keio University, 5322 Endo, Fujisawa-shi, Kanagawa 252, Japan, 1997.Google Scholar
  4. [FM92]
    C. Feng and S. Muggleton. Towards inductive generalisation in higher order logic. In D. Sleeman and P. Edwards, editors, Proceedings of the Ninth International Workshop on Machine Learning, San Mateo, California, 1992. Morgan Kaufman.Google Scholar
  5. [GM93]
    M.J.C. Gordon and T.F. Melham. Introduction to HOL: A theorem proving environment for higher order logic. Cambridge University Press, 1993.Google Scholar
  6. [KW92]
    J. Kietz and S. Wrobel. Controlling the Complexity of Learning in Logic through Syntactic and Task-Oriented Models. Inductive Logic Programming. Academic Press, 1992.Google Scholar
  7. [MF90]
    S. Muggleton and C. Feng. Efficient induction of logic programs. In Proceedings of the First Conference on Algorithmic Learning Theory, Tokyo, Japan, 1990.Google Scholar
  8. [MJ94]
    Stephen Muggleton and C.David Page Jr. Beyond first-order learning Inductive learning with higher-order logic. Technical Report PRG-TR-13-94, Oxford University, UK, 1994.Google Scholar
  9. [Mug92]
    S. Muggleton, editor. Inductive logic programming. Academic Press, New York, 1992.zbMATHGoogle Scholar
  10. [NM90]
    G. Nadathur and D. Miller. Higher-order horn clauses. Journal of the ACM, 37(4):777–814, 1990.zbMATHMathSciNetCrossRefGoogle Scholar
  11. [Plo70]
    G.D. Plotkin. A note on inductive generalization. In B. Meltzer and D. Michie, editors, Machine Intelligence, volume 6, pages 153–163. Edinburgh University Press, Edinburgh, 1970.Google Scholar
  12. [Plo71a]
    G.D. Plotkin. Automatic methods of inductive inference. Ph.d. dissertation, University of Edinburgh, Edinburgh, Scotland, 1971.Google Scholar
  13. [Plo71b]
    G.D. Plotkin. A further note on inductive generalization. In B. Meltzer and D. Michie, editors, Machine Intelligence, volume 6, pages 101–124. Edinburgh University Press, Edinburgh, 1971.Google Scholar
  14. [Pop70]
    R.J. Popplestone. An experiment in automatic deduction. In B. Meltzer and D. Michie, editors, Machine Intelligence, volume 5, pages ???-??? Edinburgh University Press, Edinburgh, 1970.Google Scholar
  15. [Rae93]
    Luc De Raedt. A brief introduction to inductive logic programming. In Proceedings of the 1993 International Symposium on Logic Programming, pages 45–51, Vancouver, Canada, October 26–29 1993.Google Scholar
  16. [RB92]
    L. De Raedt and M. Bruynooghe. Interactive theory revision: an inductive logic programming approach. Machine Learning, 8(2), 1992.Google Scholar
  17. [Sch24]
    M. Schonfinkel. Uber die baustine der matematischen logik. Math. Annalen, 92:305–316, 1924.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Srinivas Padmanabhuni
    • 1
  • Randy Goebel
    • 1
  • Koichi Furukawa
    • 2
  1. 1.Department of Computing ScienceUniversity of AlbertaEdmontonCanada
  2. 2.Graduate School of Media and GovernanceKeio UniversityKanagawaJapan

Personalised recommendations