Complexity of computing Vapnik-Chervonenkis dimension

Ayumi Shinohara

doi:10.1007/3-540-57370-4_54

Complexity of computing Vapnik-Chervonenkis dimension

Ayumi Shinohara

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

3 Citations (Scopus)

Abstract

The Vapnik-Chervonenkis (VC) dimension is known to be the crucial measure of the polynomial-sample learnability in the PAC-learning model. This paper investigates the complexity of computing VC-dimension of a concept class over a finite learning domain. We consider a decision problem called the discrete VC-dimension problem which is, for a given matrix representing a concept class F and an integer K, to determine whether the VC-dimension of F is greater than K or not. We prove that (1) the discrete VC-dimension problem is polynomial-time reducible to the satisfiability problem of length J with O(log²J) variables, and (2) for every constant C, the satisfiability problem in conjunctive normal form with m clauses and Clog²m variables is polynomial-time reducible to the discrete VC-dimension problem. These results can be interpreted, in some sense, that the problem is “complete” for the class of nO(log n time computable sets.

Original language	English
Title of host publication	Algorithmic Learning Theory - 4th International Workshop, ALT 1993, Proceedings
Editors	Klaus P. Jantke, Shigenobu Kobayashi, Etsuji Tomita, Takashi Yokomori
Publisher	Springer Verlag
Pages	279-287
Number of pages	9
ISBN (Print)	9783540573708
DOIs	https://doi.org/10.1007/3-540-57370-4_54
Publication status	Published - 1993
Externally published	Yes
Event	4th Workshop on Algorithmic Learning Theory, ALT 1993 - Tokyo, Japan Duration: 1993 Nov 8 → 1993 Nov 10

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	744 LNAI
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Other

Other	4th Workshop on Algorithmic Learning Theory, ALT 1993
Country/Territory	Japan
City	Tokyo
Period	93/11/8 → 93/11/10

ASJC Scopus subject areas

Theoretical Computer Science
Computer Science(all)

Access to Document

10.1007/3-540-57370-4_54

Cite this

Shinohara, A. (1993). Complexity of computing Vapnik-Chervonenkis dimension. In K. P. Jantke, S. Kobayashi, E. Tomita, & T. Yokomori (Eds.), Algorithmic Learning Theory - 4th International Workshop, ALT 1993, Proceedings (pp. 279-287). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 744 LNAI). Springer Verlag. https://doi.org/10.1007/3-540-57370-4_54

Complexity of computing Vapnik-Chervonenkis dimension. / Shinohara, Ayumi.

Algorithmic Learning Theory - 4th International Workshop, ALT 1993, Proceedings. ed. / Klaus P. Jantke; Shigenobu Kobayashi; Etsuji Tomita; Takashi Yokomori. Springer Verlag, 1993. p. 279-287 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 744 LNAI).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Shinohara, A 1993, Complexity of computing Vapnik-Chervonenkis dimension. in KP Jantke, S Kobayashi, E Tomita & T Yokomori (eds), Algorithmic Learning Theory - 4th International Workshop, ALT 1993, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 744 LNAI, Springer Verlag, pp. 279-287, 4th Workshop on Algorithmic Learning Theory, ALT 1993, Tokyo, Japan, 93/11/8. https://doi.org/10.1007/3-540-57370-4_54

Shinohara A. Complexity of computing Vapnik-Chervonenkis dimension. In Jantke KP, Kobayashi S, Tomita E, Yokomori T, editors, Algorithmic Learning Theory - 4th International Workshop, ALT 1993, Proceedings. Springer Verlag. 1993. p. 279-287. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/3-540-57370-4_54

Shinohara, Ayumi. / Complexity of computing Vapnik-Chervonenkis dimension. Algorithmic Learning Theory - 4th International Workshop, ALT 1993, Proceedings. editor / Klaus P. Jantke ; Shigenobu Kobayashi ; Etsuji Tomita ; Takashi Yokomori. Springer Verlag, 1993. pp. 279-287 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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AB - The Vapnik-Chervonenkis (VC) dimension is known to be the crucial measure of the polynomial-sample learnability in the PAC-learning model. This paper investigates the complexity of computing VC-dimension of a concept class over a finite learning domain. We consider a decision problem called the discrete VC-dimension problem which is, for a given matrix representing a concept class F and an integer K, to determine whether the VC-dimension of F is greater than K or not. We prove that (1) the discrete VC-dimension problem is polynomial-time reducible to the satisfiability problem of length J with O(log2J) variables, and (2) for every constant C, the satisfiability problem in conjunctive normal form with m clauses and Clog2m variables is polynomial-time reducible to the discrete VC-dimension problem. These results can be interpreted, in some sense, that the problem is “complete” for the class of nO(log n time computable sets.

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Complexity of computing Vapnik-Chervonenkis dimension

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