### Abstract

We show the following: (a) For any ε>0, log(3+ε)n-term DNF cannot be polynomial-query learned with membership and strongly proper equivalence queries. (b) For sufficiently large t, t-term DNF formulas cannot be polynomial-query learned with membership and equivalence queries that use t1+ε-term DNF formulas as hypotheses, for some ε<1 (c) Read-thrice DNF formulas are not polynomial-query learnable with membership and proper equivalence queries. (d) logn-term DNF formulas can be polynomial-query learned with membership and proper equivalence queries. (This complements a result of Bshouty, Goldman, Hancock, and Matar that logn-term DNF can be so learned in polynomial time.) Versions of (a)-(c) were known previously, but the previous versions applied to polynomial-time learning and used complexity theoretic assumptions. In contrast, (a)-(c) apply to polynomial-query learning, imply the results for polynomial-time learning, and do not use any complexity-theoretic assumptions.

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

Pages (from-to) | 435-470 |

Number of pages | 36 |

Journal | Journal of Computer and System Sciences |

Volume | 70 |

Issue number | 4 |

DOIs | |

State | Published - Jun 2005 |

### Fingerprint

### Keywords

- Algorithms
- Boolean functions
- Certificates
- Complexity theory
- Computational learning theory
- Disjunctive normal form
- DNF

### ASJC Scopus subject areas

- Computational Theory and Mathematics

### Cite this

*Journal of Computer and System Sciences*,

*70*(4), 435-470. https://doi.org/10.1016/j.jcss.2004.10.001

**Exact learning of DNF formulas using DNF hypotheses.** / Hellerstein, Lisa; Raghavan, Vijay.

Research output: Contribution to journal › Article

*Journal of Computer and System Sciences*, vol. 70, no. 4, pp. 435-470. https://doi.org/10.1016/j.jcss.2004.10.001

}

TY - JOUR

T1 - Exact learning of DNF formulas using DNF hypotheses

AU - Hellerstein, Lisa

AU - Raghavan, Vijay

PY - 2005/6

Y1 - 2005/6

N2 - We show the following: (a) For any ε>0, log(3+ε)n-term DNF cannot be polynomial-query learned with membership and strongly proper equivalence queries. (b) For sufficiently large t, t-term DNF formulas cannot be polynomial-query learned with membership and equivalence queries that use t1+ε-term DNF formulas as hypotheses, for some ε<1 (c) Read-thrice DNF formulas are not polynomial-query learnable with membership and proper equivalence queries. (d) logn-term DNF formulas can be polynomial-query learned with membership and proper equivalence queries. (This complements a result of Bshouty, Goldman, Hancock, and Matar that logn-term DNF can be so learned in polynomial time.) Versions of (a)-(c) were known previously, but the previous versions applied to polynomial-time learning and used complexity theoretic assumptions. In contrast, (a)-(c) apply to polynomial-query learning, imply the results for polynomial-time learning, and do not use any complexity-theoretic assumptions.

AB - We show the following: (a) For any ε>0, log(3+ε)n-term DNF cannot be polynomial-query learned with membership and strongly proper equivalence queries. (b) For sufficiently large t, t-term DNF formulas cannot be polynomial-query learned with membership and equivalence queries that use t1+ε-term DNF formulas as hypotheses, for some ε<1 (c) Read-thrice DNF formulas are not polynomial-query learnable with membership and proper equivalence queries. (d) logn-term DNF formulas can be polynomial-query learned with membership and proper equivalence queries. (This complements a result of Bshouty, Goldman, Hancock, and Matar that logn-term DNF can be so learned in polynomial time.) Versions of (a)-(c) were known previously, but the previous versions applied to polynomial-time learning and used complexity theoretic assumptions. In contrast, (a)-(c) apply to polynomial-query learning, imply the results for polynomial-time learning, and do not use any complexity-theoretic assumptions.

KW - Algorithms

KW - Boolean functions

KW - Certificates

KW - Complexity theory

KW - Computational learning theory

KW - Disjunctive normal form

KW - DNF

UR - http://www.scopus.com/inward/record.url?scp=17444399680&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=17444399680&partnerID=8YFLogxK

U2 - 10.1016/j.jcss.2004.10.001

DO - 10.1016/j.jcss.2004.10.001

M3 - Article

VL - 70

SP - 435

EP - 470

JO - Journal of Computer and System Sciences

JF - Journal of Computer and System Sciences

SN - 0022-0000

IS - 4

ER -