TY - GEN

T1 - Complexity of computing Vapnik-Chervonenkis dimension

AU - Shinohara, Ayumi

N1 - Publisher Copyright:
© 1993, Springer Verlag. All rights reserved.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 1993

Y1 - 1993

N2 - 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.

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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U2 - 10.1007/3-540-57370-4_54

DO - 10.1007/3-540-57370-4_54

M3 - Conference contribution

AN - SCOPUS:0040666498

SN - 9783540573708

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 279

EP - 287

BT - Algorithmic Learning Theory - 4th International Workshop, ALT 1993, Proceedings

A2 - Jantke, Klaus P.

A2 - Kobayashi, Shigenobu

A2 - Tomita, Etsuji

A2 - Yokomori, Takashi

PB - Springer Verlag

T2 - 4th Workshop on Algorithmic Learning Theory, ALT 1993

Y2 - 8 November 1993 through 10 November 1993

ER -