Decision tree approximations of Boolean functions

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Abstract

Decision trees are popular representations of Boolean functions. We show that, given an alternative representation of a Boolean function f, say as a read-once branching program, one can find a decision tree T which approximates f to any desired amount of accuracy. Moreover, the size of the decision tree is at most that of the smallest decision tree which can represent f and this construction can be obtained in quasi-polynomial time. We also extend this result to the case where one has access only to a source of random evaluations of the Boolean function f instead of a complete representation. In this case, we show that a similar approximation can be obtained with any specified amount of confidence (as opposed to the absolute certainty of the former case.) This latter result implies proper PAC-learnability of decision trees under the uniform distribution without using membership queries.

MSC

05C60
20B25
68Q15
68Q25

Keywords

Decision trees
Representations of Boolean functions
Learning theory
Algorithms

Cited by (0)

1

Supported by NSF grant CCR-9988338.

2

Supported by NSF grant CCR-9820840.