Abstract
In this paper, we investigate the VC-dimensions of finite automata. We show that for a fixed positive integer k (1) the VC-dimension of DFAk,n, which is the class of dfas of an alphabet size k whose minimum states dfa has at most n states, is (k− 1+o(1))n log n, (2) the VC-dimension of NFAk,n, which is the class of nfas of an alphabet size k whose minimum states nfa has at most n states, is Q(n 2) and (3) the VC-dimension of CDFAk,n, which is the class of commutative dfas of an alphabet size k whose minimum states commutative dfas has at most n states, is (1+o(1))n. These results are applied to the problems in computational learning theory.
Partially supported by the Grant in Aid for Scientific Research of the Ministry of Education, Science and Culture of Japan
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References
Abe, N.: Learning Commutative Deterministic Finite State Automata in Polynomial Time, New Generation Computing 8, 1991, pp.319–335.
Andrews, G.: The Theory of Partitions, Addison-Wesley, Massachusetts (1976).
Angluin, D.: Learning regular sets from queries and counterexamples, Information and Computation, Vol. 75, 1987, pp.87–106.
Angluin, D.: Queries and concept learning, Machine Learning, Vol. 2, 1988, pp.319–342.
Angluin, D.: Negative results for equivalence queries, Machine Learning, Vol. 5, 1990, pp.121–150.
Angluin, D. and M. Kharitonov: When Won't Membership Queries Help? in Proc. of 23nd STOC, ACM, 1991, pp.444–454.
Balcazar, J. L., J. Diaz, R. Gavalda and O. Watanabe: A Note on the Query Complexity of Learning DFA, Proc. of the 3rd workshop on Algorithmic Learning Theory, Japanese Society for AI, 1992, pp.53–62.
Benedek, G. M. and A. Itai: Nonuniform learnability, International Colloquium on Automata, Languages and Programming 1988, Lecture Notes in Computer Science 317, Springer-Varlag, pp.82–92.
Blumer, A., A. Ehrenfeucht, D. Haussler, and M. K. Warmuth: Classifying learnable geometric concepts with Vapnik-Chervonenkis dimension, in Proc. 18th ACM Symp. on Theory of Computing, 1986, pp.273–282.
Champarnaud, J.-M. and J.-E. Pin: A Maximin problem on finite automata, Discrete Applied Mathematics 23, 1989, pp.91–96.
Hopcroft, J. E., and Ullman, J. D., Introduction to Automata Theory, Languages, and Computation, Addison-Wesley, Massachusetts (1979).
Ishigami, Y.: A Sperner set generating a k-dimensional Young tableaux, Preprint (1993).
Gaizer, T.: The Vapanik-Chervonenkis dimension of finite automata. Unpublished manuscript (1990).
Maass, T., and G. Turán: On the complexity of learning from counterexamples, in Proc. 30th IEEE Symp. on Foundations of Computer Science 1989, pp.262–167.
Maass, T., and G. Turán: On the Complexity of Learning from Counterexamples and Membership Queries, in Proc. 31st IEEE Symp. on Foundations of Computer Science, 1990, pp.203–210.
Salomaa, A.: Theory of Automata, Pergamon Press, Oxford, 1969.
Watanabe, O.: A formal study of learnability via queries, International Colloquium on Automata, Languages and Programming, 1990, Lecture Notes in Computer Science 443, Springer-Varlag.
Yokomori, T.: Learning Non-deterministic Finite Automata from Queries and Counterexamples. To appear in Machine Intelligence, Vol.13, Oxford Univ. Press.
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Ishigami, Y., Tani, S. (1993). The VC-dimensions of finite automata with n states. In: Jantke, K.P., Kobayashi, S., Tomita, E., Yokomori, T. (eds) Algorithmic Learning Theory. ALT 1993. Lecture Notes in Computer Science, vol 744. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57370-4_58
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DOI: https://doi.org/10.1007/3-540-57370-4_58
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