### Abstract

A central topic in query learning is to determine which classes of Boolean formulas are efficiently learnable with membership and equivalence queries. We consider the class ℛ^{k} consisting of conjunctions of k unate DNF formulas. This class generalizes the class of k-clause CNF formulas and the class of unate DNF formulas, both of which are known to be learnable in polynomial time with membership and equivalence queries. We prove that ℛ^{2} can be properly learned with a polynomial number of polynomial-size membership and equivalence queries, but can be properly learned in polynomial time with such queries if and only if P = NP. Thus the barrier to properly learning ℛ^{2} with membership and equivalence queries is computational rather than informational. Few results of this type are known. In our proofs, we use recent results of Hellerstein et al. (1997, J. Assoc. Comput. Mach. 43 (5), 840-862), characterizing the classes that are polynomial-query learnable, together with work of Bshouty on the monotone dimension of Boolean functions. We extend some of our results to ℛ^{2} and pose open questions on learning DNF formulas of small monotone dimension. We also prove structural results for ℛ^{k}. We construct, for any fixed k ≥ 2, a class of functions f that cannot be represented by any formula in ℛ^{k}, but which cannot be "easily" shown to have this property. More precisely, for any function f on n variables in the class, the value of f on any polynomial-size set of points in its domain is not a witness that f cannot be represented by a formula in ℛ^{k}. Our construction is based on BCH codes.

Original language | English (US) |
---|---|

Pages (from-to) | 203-228 |

Number of pages | 26 |

Journal | Information and Computation |

Volume | 140 |

Issue number | 2 |

State | Published - Feb 1 1998 |

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### ASJC Scopus subject areas

- Computational Theory and Mathematics

### Cite this

*Information and Computation*,

*140*(2), 203-228.

**Conjunctions of Unate DNF Formulas : Learning and Structure.** / Feigelson, Aaron; Hellerstein, Lisa.

Research output: Contribution to journal › Article

*Information and Computation*, vol. 140, no. 2, pp. 203-228.

}

TY - JOUR

T1 - Conjunctions of Unate DNF Formulas

T2 - Learning and Structure

AU - Feigelson, Aaron

AU - Hellerstein, Lisa

PY - 1998/2/1

Y1 - 1998/2/1

N2 - A central topic in query learning is to determine which classes of Boolean formulas are efficiently learnable with membership and equivalence queries. We consider the class ℛk consisting of conjunctions of k unate DNF formulas. This class generalizes the class of k-clause CNF formulas and the class of unate DNF formulas, both of which are known to be learnable in polynomial time with membership and equivalence queries. We prove that ℛ2 can be properly learned with a polynomial number of polynomial-size membership and equivalence queries, but can be properly learned in polynomial time with such queries if and only if P = NP. Thus the barrier to properly learning ℛ2 with membership and equivalence queries is computational rather than informational. Few results of this type are known. In our proofs, we use recent results of Hellerstein et al. (1997, J. Assoc. Comput. Mach. 43 (5), 840-862), characterizing the classes that are polynomial-query learnable, together with work of Bshouty on the monotone dimension of Boolean functions. We extend some of our results to ℛ2 and pose open questions on learning DNF formulas of small monotone dimension. We also prove structural results for ℛk. We construct, for any fixed k ≥ 2, a class of functions f that cannot be represented by any formula in ℛk, but which cannot be "easily" shown to have this property. More precisely, for any function f on n variables in the class, the value of f on any polynomial-size set of points in its domain is not a witness that f cannot be represented by a formula in ℛk. Our construction is based on BCH codes.

AB - A central topic in query learning is to determine which classes of Boolean formulas are efficiently learnable with membership and equivalence queries. We consider the class ℛk consisting of conjunctions of k unate DNF formulas. This class generalizes the class of k-clause CNF formulas and the class of unate DNF formulas, both of which are known to be learnable in polynomial time with membership and equivalence queries. We prove that ℛ2 can be properly learned with a polynomial number of polynomial-size membership and equivalence queries, but can be properly learned in polynomial time with such queries if and only if P = NP. Thus the barrier to properly learning ℛ2 with membership and equivalence queries is computational rather than informational. Few results of this type are known. In our proofs, we use recent results of Hellerstein et al. (1997, J. Assoc. Comput. Mach. 43 (5), 840-862), characterizing the classes that are polynomial-query learnable, together with work of Bshouty on the monotone dimension of Boolean functions. We extend some of our results to ℛ2 and pose open questions on learning DNF formulas of small monotone dimension. We also prove structural results for ℛk. We construct, for any fixed k ≥ 2, a class of functions f that cannot be represented by any formula in ℛk, but which cannot be "easily" shown to have this property. More precisely, for any function f on n variables in the class, the value of f on any polynomial-size set of points in its domain is not a witness that f cannot be represented by a formula in ℛk. Our construction is based on BCH codes.

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M3 - Article

VL - 140

SP - 203

EP - 228

JO - Information and Computation

JF - Information and Computation

SN - 0890-5401

IS - 2

ER -