TY - JOUR

T1 - On the Depth of Decision Trees with Hypotheses

AU - Moshkov, Mikhail

N1 - KAUST Repository Item: Exported on 2022-01-18
Acknowledgements: Research reported in this publication was supported by King Abdullah University of Science and Technology (KAUST). The author is greatly indebted to the anonymous reviewers for their useful comments and suggestions.

PY - 2022/1/12

Y1 - 2022/1/12

N2 - In this paper, based on the results of rough set theory, test theory, and exact learning, we investigate decision trees over infinite sets of binary attributes represented as infinite binary information systems. We define the notion of a problem over an information system and study three functions of the Shannon type, which characterize the dependence in the worst case of the minimum depth of a decision tree solving a problem on the number of attributes in the problem description. The considered three functions correspond to (i) decision trees using attributes, (ii) decision trees using hypotheses (an analog of equivalence queries from exact learning), and (iii) decision trees using both attributes and hypotheses. The first function has two possible types of behavior: logarithmic and linear (this result follows from more general results published by the author earlier). The second and the third functions have three possible types of behavior: constant, logarithmic, and linear (these results were published by the author earlier without proofs that are given in the present paper). Based on the obtained results, we divided the set of all infinite binary information systems into four complexity classes. In each class, the type of behavior for each of the considered three functions does not change.

AB - In this paper, based on the results of rough set theory, test theory, and exact learning, we investigate decision trees over infinite sets of binary attributes represented as infinite binary information systems. We define the notion of a problem over an information system and study three functions of the Shannon type, which characterize the dependence in the worst case of the minimum depth of a decision tree solving a problem on the number of attributes in the problem description. The considered three functions correspond to (i) decision trees using attributes, (ii) decision trees using hypotheses (an analog of equivalence queries from exact learning), and (iii) decision trees using both attributes and hypotheses. The first function has two possible types of behavior: logarithmic and linear (this result follows from more general results published by the author earlier). The second and the third functions have three possible types of behavior: constant, logarithmic, and linear (these results were published by the author earlier without proofs that are given in the present paper). Based on the obtained results, we divided the set of all infinite binary information systems into four complexity classes. In each class, the type of behavior for each of the considered three functions does not change.

UR - http://hdl.handle.net/10754/674972

UR - https://www.mdpi.com/1099-4300/24/1/116

U2 - 10.3390/e24010116

DO - 10.3390/e24010116

M3 - Article

C2 - 35052142

VL - 24

SP - 116

JO - Entropy

JF - Entropy

SN - 1099-4300

IS - 1

ER -