We prove the following results. Any Boolean function of
O(log n) relevant variables can be exactly learned with
a set of non-adaptive membership queries alone and a minimum
sized decision tree representation of the function constructed,
in polynomial time. In contrast, such a function cannot be exactly
learned with equivalence queries alone using general decision trees
and other representation classes as hypotheses.
Our results imply others which may be of independent interest.
We show that truth-table minimization of decision trees can be done
in polynomial time, complementing the well-known result of Masek
that truth-table minimization of DNF formulas is NP-hard. The proofs
of our negative results show that general decision trees and
related representations are not learnable in polynomial time
using equivalence queries alone, confirming a folklore theorem.
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