On the learnability of disjunctive normal form formulas and decision trees

The learnability of disjunctive normal form formulas and decision trees is investigated. Polynomial time algorithms are given, and nonlearnability results are obtained, for restricted versions of these general learning problems. Polynomial time algorithms are presented for exactly learning (with membership and equivalence queries) read-twice DNF and read-k-disjoint DNF. A read-twice DNF formula is a boolean formula in disjunctive normal form where each variable appears at most twice. A read-k disjoint DNF formula f is a DNF formula where each variable appears at most k times (for an arbitrary positive integer k) and every assignment to the variables satisfies at most one term of f. The read-k disjoint DNF result also applies for a generalization of this class, which we call read-k sat-j DNF. For a similar learning protocol, it is shown that, assuming NP $\not=$ co-NP, there does not exist a polynomial time algorithm for learning read-thrice DNF formulas-boolean formulas in disjunctive normal form where each variable appears at most three times. This result contrasts with our polynomial time algorithm for learning read-twice DNF, and adds evidence to the conjecture that DNF is hard to learn in the membership and equivalence query model. Nonlearnability results are also obtained for the class of read-k decision trees. It is shown that this class is hard to learn in the membership and equivalence query model, provided that the equivalence queries are also required to be read-k decision trees. It is also shown that read-k decision trees are hard to learn in the PAC model (without membership queries). A different type of nonlearnability result is obtained for the class of arbitrary DNF formulas. A natural approach for learning DNF formulas (suggested by Valiant in a seminal paper of learning theory) is to greedily collect the prime implicants of the hidden function. We show that no algorithm using such an approach can learn DNF in polynomial time. Results which suggest that DNF formulas are hard to learn rely on the construction of rare hard-to-learn formulas. This raises the question of whether most DNF formulas are learnable. For certain natural definitions of most DNF formulas, this question is answered affirmatively.