Learnability and Definability in Trees and Similar Structures

Theory of Computing Systems

Volumn 37, Issue 1, 2004, Pages 193-220

  Grohe, Martin ^a   Turan Gy ^b,c  

^a HUMBOLDT UNIVERSITY   (Germany)

^b UNIVERSITY OF ILLINOIS AT CHICAGO   (United States)

^c INSTITUTE OF EXPERIMENTAL MEDICINE   (Hungary)

Author keywords

[No Author keywords available]

Indexed keywords

INDUCTIVE LOGIC PROGRAMMING; LEARNABILITY AND DEFINABILITY; PLANAR GRAPHS;

COMPUTATIONAL COMPLEXITY; FINITE AUTOMATA; FORMAL LOGIC; LEARNING SYSTEMS; LOGIC PROGRAMMING; MATHEMATICAL MODELS; QUERY LANGUAGES;

TREES (MATHEMATICS);

EID: 1342269749     PISSN: 14324350     EISSN: 14330490     Source Type: Journal    
DOI: 10.1007/s00224-003-1112-8     Document Type: Conference Paper

References (40)

1. H. Aizenstein, T. Hegedus, L. Hellerstein, and L. Pitt. Complexity-theoretic hardness results for query learning. Computational Complexity, 7:19-53, 1998.
2. D. Angluin. Queries and concept learning. Machine Learning, 2:319-342, 1988.
3. L. Babai and Gy. Turin. The complexity of defining a relation on a finite graph. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 33:277-288, 1987.
4. J.L. Balcázar. The consistency dimension, compactness and query learning. In J. Flum and M. Rodriguez-Artalejo, editors, Computer Science Logic, 13th International Workshop CSL '99, pages 2-13. Volume 1683 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 1999.
5. J.L. Balcázar, J. Castro, and D. Guijarro. A new abstract combinatorial dimension for exact learning via queries. In S.A. Goldman and N. Cesa-Bianchi, editors, Proceedings of the 13th Annual Conference on Computational Learning Theory (COLT 2000), pages 248-254. Morgan-Kaufmann, San Mateo, CA, 2000.
6. J.L. Balcázar, J. Castro, D. Guijarro, and H.U. Simon. The consistency dimension and distribution-dependent learning from queries. In O. Watanabe and T. Yokomori, editors, Algorithmic Learning Theory, 10th International Conference, ALT '99, pages 77-92. Volume 1720 of Lecture Notes in Computer Science, Springer-Verlag, Berlin, 1999.
7. M. Benedikt, L. Libkin, T. Schwentick, and L. Segoufin. A model-theoretic approach to regular string relations. In Proceedings of the 16th IEEE Symposium on Logic in Computer Science, pages 431-440, 2001.
8. A. Blumer, A. Ehrenfeucht, D. Haussler, and M.K. Warmuth. Learnability and the Vapnik-Chervonenkis dimension. Journal of the ACM, 36:929-965, 1989.
9. B. Courcelle, J. Engelfriet, and G. Rozenberg. Handle-rewriting hypergraph grammars. Journal of Computer and System Sciences, 46:218-270, 1993.
10. B. Courcelle and S. Olariu. Upper bounds to the clique-width of graphs. Discrete Applied Mathematics, 101:77-114, 2000.
11. H.-D. Ebbinghaus and J. Flum. Finite Model Theory, 2nd edition. Springer-Verlag, New York, 1999.
12. H.-D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic, 2nd edition. Springer-Verlag, New York, 1994.
13. D. Eppstein. Diameter and treewidth in minor-closed graph families. Algorithmica, 27:275-291, 2000.
14. M. Frazier and L. Pitt. CLASSIC learning. Machine Learning, 25:151-193, 1996.
15. M. Prick and M. Grohe. The complexity of first-order and monadic second-order logic revisited. In Proceedings of the 17th IEEE Symposium on Logic in Computer Science, pages 215-224, 2002.
16. H. Gaifman. On local and non-local properties. In J. Stern, editor, Proceedings of the Herbrand Symposium, Logic Colloquium '81, pages 105-135. North Holland, Amsterdam, 1982.
17. M.C.Golumbic and U. Roties. On the clique-width of some perfect graph classes. International Journal of Foundations of Computer Science, 11:423-443, 2000.
18. M. Grohe. Generalized model-checking problems for first-order logic. In H. Reichel and A. Ferreira, editors, Proceedings of the 18th Annual Symposium on Theoretical Aspects of Computer Science, pages 12-26. Volume 2010 of Lecture Notes in Computer Science, Springer-Verlag, Berlin, 2001.
19. M. Grohe. Local tree-width, excluded minors, and approximation algorithms. Combinatorica. To appear.
20. T. Hegedus. Generalized teaching dimensions and the query complexity of learning. In Proceedings of the 8th Annual Conference on Computational Learning Theory (COLT 1995), pages 108-117, 1995.
21. L. Hellerstein, K. Pillaipakkamnatt, V. Raghavan, and D. Wilkins. How many queries are needed to learn? Journal of the ACM, 43:840-862, 1996.
22. W. Hodges. Model Theory. Cambridge University Press, Cambridge, 1993.
23. M.J. Kearns and U.V. Vazirani. An Introduction to Computational Learning Theory. MIT Press, Cambridge, MA, 1994.
24. M.C. Laskowski. Vapnik-Chervonenkis classes of definable sets. Journal of the London Mathematical Society (2), 45:377-384, 1992.
25. L. Libkin. Logics with counting and local properties. ACM Transactions on Computational Logic, 1:33-59, 2000.
26. W. Maass and Gy. Turán. Lower bound methods and separation results for on-line learning models. Machine Learning, 9:107-145, 1992.
27. W. Maass and Gy. Turán. On leamability and predicate logic. In Proceedings of the Bar-Ilan Symposium on the Foundations of Artificial Intelligence (BISFAI-95), pages 75-85, 1995.
28. S.-H. Nienhuys-Cheng and R. de Wolf. Foundations of Inductive Logic Programming. Volume 1228 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 1997.
29. D.N. Osherson, M. Stob, and S. Weinstein. New directions in automated scientific discovery. Information Sciences, 57-58:217-230, 1991.
30. K.P. Podewski and M. Ziegler. Stable graphs. Fundamenta Mathematicae, 100:101-107, 1978.
31. N. Robertson and P.D. Seymour. Graph minors, II. Algorithmic aspects of tree-width. Journal of Algorithms, 7:309-322, 1986.
32. N. Robertson and P.D. Seymour. Graph minors, V. Excluding a planar graph. Journal of Combinatorial Theory, Series B, 41:92-114, 1986.
33. N. Sauer. On the density of families of sets. Journal of Combinatorial Theory, Series A, 13:145-147, 1972.
34. S. Shelah. Stability, the f.c.p. and superstability. Annals of Mathematical Logic, 3:271-362, 1971.
35. S. Shelah. A combinatorial problem: stability and order for models and theories in inflnitary languages. Pacific Journal of Mathematics, 41:241-261, 1972.
36. L.J. Stockmeyer and A.R. Meyer. Word problems requiring exponential time. In Proceedings of the 5th ACM Symposium on Theory of Computing, pages 1-9, 1973.
37. J.W. Thatcher and J.B. Wright. Generalised finite automata theory with an application to a decision problem of second-order logic. Mathematical Systems Theory, 2:57-81, 1968.
38. I. Tsapara and Gy.Turán. Learning atomic formulas with prescribed properties. In Proceedings of the 11th Annual Conference on Computational Learning Theory (COLT '98), pages 166-174, 1998.
39. L. van den Dries. Tame Topology and O-minimal Structures. Cambridge University Press, Cambridge, 1998.
40. V. Vapnik and A. Chervonenkis. On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability and its Applications, 16:264-280, 1971.