Abstract
Although there is an increasing amount of experimental research on learning concepts expressed in first-order logic, there are still relatively few formal results on the polynomial learnability of first-order representations from examples. Most previous analyses in the pac-model have focused on subsets of Prolog, and only a few highly restricted subsets have been shown to be learnable. In this paper, we will study instead the learnability of the restricted first-order logics known as “description logics”, also sometimes called “terminological logics” or “KL-ONE-type languages”. Description logics are also subsets of predicate calculus, but are expressed using a different syntax, allowing a different set of syntactic restrictions to be explored. We first define a simple description logic, summarize some results on its expressive power, and then analyze its learnability. It is shown that the full logic cannot be tractably learned. However, syntactic restrictions exist that enable tractable learning from positive examples alone, independent of the size of the vocabulary used to describe examples. The learnable sublanguage appears to be incomparable in expressive power to any subset of first-order logic previously known to be learnable.
Article PDF
Similar content being viewed by others
References
Angluin, D. (1987). Learning regular sets from queries and counterexamples.Information and Control, 75:87–106.
Banerji, R. (1988). Learning theories in a subset of polyadic logic. InProceedings of the 1988 Workshop on Computational Learning Theory (pp. 267–278), Boston, Massachusetts.
Beck, H. (1991). Language acquisition from cases. InProceedings of the 1991 DARPA Case-Based Reasoning Workshop (pp. 159–169), Washington, D.C.
Beck, H., Gala, H., & Navathe, S. (1989). Classification as a query processing technique in the CANDIDE semantic model. InProceedings of the Data Engineering Conference (pp. 572–581), Los Angeles, California.
Blum, A. (1990). Learning boolean functions in a infinite attribute space. In22nd Annual Symposium on the Theory of Computing (pp. 373–386). ACM Press.
Blumer, A., Ehrenfeucht, A., Haussler, D., & Warmuth, M. (1989). Classifying learnable concepts with the Vapnik-Chervonenkis dimension.Journal of the Association for Computing Machinery, 36(4):929–965.
Bobrow, D. & Winograd, T. (1977). An overview of KRL, a knowledge representation language.Cognitive Science, 1(1):3–46.
Borgida, A. (1992). Description logics are not just for the fightless-birds: a new look at the utility and foundations of description logics. Technical Report DCS-TR-295, Rutgers University Department of Computer Science.
Borgida, A. (1993). On the relationship between description logics and predicate logic queries. Technical Report DCS-TR-295a, Rutgers University Department of Computer Science.
Borgida, A. & Patel-Schneider, P. F. (1994). A semantics and complete algorithm for subsumption in the CLASSIC description logic.Journal of Artificial Intelligence Research, 1:277–308.
Brachman, R. J. (1979). On the epistomological status of semantic networks. In Findler, N. V., editor,Associative networks: representation and use of knowledge by computers. Academic Press.
Buntine, W. (1988). Generalized subsumption and its application to induction and redundancy.Artificial Intelligence, 36(2):149–176.
Cohen, W. & Hirsh, H. (1992). Learnability of the CLASSIC 1.0 knowledge representation language. AT&T Bell Labs Technical Memorandum. Available from the author on request.
Cohen, W. W. (1993a). Cryptographic limitations on learning one-clause logic programs. InProceedings of the Tenth National Conference on Artificial Intelligence (pp. 80–85), Washington, D.C.
Cohen, W. W. (1993b). A pac-learning algorithm for a restricted class of recursive logic programs. InProceedings of the Tenth National Conference on Artificial Intelligence (pp. 86–92), Washington, D.C.
Cohen, W. W., Borgida, A., & Hirsh, H. (1992). Computing least common subsumers in description logics. InProceedings of the Tenth National Conference on Artificial Intelligence (pp. 754–760), San Jose, California. MIT Press.
Conklin, D. & Gasglow, J. (1992). Spatial analogy and subsumption. InProceedings of the Ninth International Conference on Machine Learning (pp. 111–116), Aberdeen, Scotland. Morgan Kaufmann.
Devanbu, P., Brachman, R. J., Selfridge, P., & Ballard, B. (1991). LaSSIE: A knowledge-based software information system.Communications of the ACM, 34(5), 34–49.
Dietterich, T. & Michalski, R. (1983). A comparative review of selected methods for learning from examples. InMachine Learning: An Artificial Intelligence Approach. Morgan Kaufmann.
Dietterich, T. G., London, B., Clarkson, K., & Dromey, G. (1982). Learning and inductive inference. In Cohen, P. and Feigenbaum, E. A., editors,The Handbook of Artificial Intelligence, Volume III. William Kaufmann, Los Altos, CA.
Džeroski, S., Muggleton, S., & Russell, S. (1992). Pac-learnability of determinate logic programs. InProceedings of the 1992 Workshop on Computational Learning Theory (pp. 128–135), Pittsburgh, Pennsylvania.
Frisch, A. & Page, C. D. (1990). Generalization with taxonomic information. InProceedings of the Eighth National Conference on Artificial Intelligence (pp. 755–761), Boston, Massachusetts. MIT Press.
Frisch, A. & Page, C. D. (1991). Learning constrained atoms. InProceedings of the Eighth International Workshop on Machine Learning (pp. 427–431), Ithaca, New York. Morgan Kaufmann.
Gold, M. (1967). Language identification in the limit.Information and Control, 10, 447–474.
Haussler, D. (1989). Learning conjunctive concepts in structural domains.Machine Learning, 4(1), 7–40.
Helmbold, D., Sloan, R., & Warmuth, M. (1990). Learning nested differences of intersection-closed concept classes.Machine Learning, 5(2), 165–196.
Hirsh, H. (1990).Incremental Version Space Merging: A General Framework for Concept Learning. Kluwer Academic Publishers.
Hopcroft, J. E. & Ullman, J. D. (1979).Introduction to Automata Theory, Languages, and Computation. Addison-Wesley.
Idestam-Almquist, P. (1993). Generalization under implication by recursive anti-unification. InProceedings of the Ninth International Conference on Machine Learning (pp. 151–158), Amherst, Massachusetts. Morgan Kaufmann.
Kearns, M., Li, M., Pitt, L., & Valiant, L. (1987). Recent results in boolean concept learning. InProceedings of the Fourth International Workshop on Machine Learning (pp. 337–352), Ithaca, New York. Morgan Kaufmann.
Kearns, M. & Valiant, L. (1989). Cryptographic limitations on learning Boolean formulae and finite automata. In21th Annual Symposium on the Theory of Computing (pp. 433–444). ACM Press.
Kietz, J.-U. (1993). Some computational lower bounds for the computational complexity of inductive logic programming. InProceedings of the 1993 European Conference on Machine Learning (pp. 115–123), Vienna, Austria.
Littlestone, N. (1988). Learning quickly when irrelevant attributes abound: A new linear-threshold algorithm.Machine Learning, 2(4), 285–318.
MacGregor, R. M. (1991). The evolving technology of classification-based knowledge representation systems. In Sowa, J., editor,Principles of semantic networks: explorations in the representation of knowledge. Morgan Kaufmann.
Mays, E., Apte, C., Griesmer, J., & Kastner, J. (1987). Organizing knowledge in a complex financial domain.IEEE Expert, pages 61–70.
Minsky, M. (1975). A framework for representing knowledge. In Winston, P., editor,The psychology of computer vision. McGraw-Hill.
Morik, K. (1989). A bootstrapping approach to conceptual clustering. InProceedings of the Sixth International Workshop on Machine Learning (pp. 503–504), Ithaca, New York. Morgan Kaufmann.
Muggleton, S. & Buntine, W. (1988). Machine invention of first order predicates by inverting resolution. InProceedings of the Fifth International Conference on Machine Learning (pp. 339–352), Ann Arbor, Michigan. Morgan Kaufmann.
Muggleton, S. & Feng, C. (1992). Efficient induction of logic programs. InInductive Logic Programming. Academic Press.
Muggleton, S. H., editor, (1992).Inductive Logic Programming. Academic Press.
Natarajan, B. K. (1987). On learning boolean functions. In19th Annual Symposium on the Theory of Computing (pp. 296–352), ACM Press.
Pfenning, F. (1991). Unification and anti-unification in the oalculus of constructions. InSixth Annual IEEE Symposium on Logic in Computer Science (pp. 74–85). Amsterdam: The Netherlands. Also available as Ergo Report 91-096, School of Computer Science, CMU, Pittsburgh.
Pitt, L. & Valiant, L. (1988). Computational limitations on learning from examples.Journal of the ACM, 35(4):965–984.
Pitt, L. & Warmuth, M. (1990). Prediction-preserving reducibility.Journal of Computer and System Sciences, 41:430–467.
Plotkin, G. D. (1969). A note on inductive generalization.Machine Intelligence, 5 153–163.
Quillian, M. R. (1967). Word concepts: a theory and simulation of some basic semantic capabilities.Behavioral Science, 12:410–430.
Quinlan, J. R. (1990). Learning logical definitions from relations.Machine Learning, 5(3), 239–266.
Rivest, R. L. (1987). Learning decision lists.Machine Learning, 2(3), 229–246.
Schewe, K. D. (1989). Variant construction using constraint propagation techniques over semantic networks. InProceedings of the 5th Austrian AI Conference, Insbruck.
Shapiro, E. (1982).Algorithmic Program Debugging. MIT Press.
Valiant, L. G. (1984). A theory of the learnable.Communications of the ACM, 27(11), 1134–1142.
Vilain, M., Koton, P., & Chase, M. (1990). On analytical and similarity-based classification. InProceedings of the Eighth National Conference on Artificial Intelligence (pp. 867–874), Boston, Massachusetts: MIT Press.
Winston, P. (1975). Learning structural descriptions from examples. In Winston, P., editor,The psychology of computer vision, pages 157–209, McGraw-Hill.
Woods, W. A. & Schmolze, J. G. (1992). The KL-ONE family.Computers and Mathematics with Applications, 23(2–5).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Cohen, W.W., Hirsh, H. The learnability of description logics with equality constraints. Mach Learn 17, 169–199 (1994). https://doi.org/10.1007/BF00993470
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00993470