### Abstract

In the PAC-learning, or the query learning model, it has been an important open problem to decide whether the class of DNF and CNF formulas is leamable. Recently, it was pointed out that the problem of PAC-learning for these classes with membership queries can be reduced to that of learning for the class of k-quasi Horn formulas with membership and equivalence queries. A k-quasi Horn formula is a CNF formula with each clause containing at most k unnegated literals. In this paper, notions of F-Horn formulas and l-F-Horn formulas, which are extensions of k-quasi formulas, are introduced, and it is shown that the problem of query learning for DNF and CNF formulas with membership and equivalence queries can be reduced to that for l-F-Horn formulas for an appropriate choice of F. It is shown that under a condition on F, the class of orthogonal F-Horn formulas is learnable with membership, equivalence and subset queries. Moreover, it is shown that under the same condition the class of orthogonal l-F-Horn formulas is learnable with membership and equivalence queries. For the latter result, the condition of orthogonality of F-Horn formulas is crucial because, if the statement held without the condition, then the result would imply that DNF and CNF are exactly learnable with membership and equivalence queries.

Original language | English |
---|---|

Pages (from-to) | 177-190 |

Number of pages | 14 |

Journal | Theoretical Computer Science |

Volume | 185 |

Issue number | 1 |

DOIs | |

Publication status | Published - Oct 10 1997 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Theoretical Computer Science*,

*185*(1), 177-190. https://doi.org/10.1016/S0304-3975(97)00020-0

**Learning orthogonal F-Horn formulas.** / Takimoto, Eiji; Miyashiro, Akira; Maruoka, Akira; Sakai, Yoshifumi.

Research output: Contribution to journal › Article

*Theoretical Computer Science*, vol. 185, no. 1, pp. 177-190. https://doi.org/10.1016/S0304-3975(97)00020-0

}

TY - JOUR

T1 - Learning orthogonal F-Horn formulas

AU - Takimoto, Eiji

AU - Miyashiro, Akira

AU - Maruoka, Akira

AU - Sakai, Yoshifumi

PY - 1997/10/10

Y1 - 1997/10/10

N2 - In the PAC-learning, or the query learning model, it has been an important open problem to decide whether the class of DNF and CNF formulas is leamable. Recently, it was pointed out that the problem of PAC-learning for these classes with membership queries can be reduced to that of learning for the class of k-quasi Horn formulas with membership and equivalence queries. A k-quasi Horn formula is a CNF formula with each clause containing at most k unnegated literals. In this paper, notions of F-Horn formulas and l-F-Horn formulas, which are extensions of k-quasi formulas, are introduced, and it is shown that the problem of query learning for DNF and CNF formulas with membership and equivalence queries can be reduced to that for l-F-Horn formulas for an appropriate choice of F. It is shown that under a condition on F, the class of orthogonal F-Horn formulas is learnable with membership, equivalence and subset queries. Moreover, it is shown that under the same condition the class of orthogonal l-F-Horn formulas is learnable with membership and equivalence queries. For the latter result, the condition of orthogonality of F-Horn formulas is crucial because, if the statement held without the condition, then the result would imply that DNF and CNF are exactly learnable with membership and equivalence queries.

AB - In the PAC-learning, or the query learning model, it has been an important open problem to decide whether the class of DNF and CNF formulas is leamable. Recently, it was pointed out that the problem of PAC-learning for these classes with membership queries can be reduced to that of learning for the class of k-quasi Horn formulas with membership and equivalence queries. A k-quasi Horn formula is a CNF formula with each clause containing at most k unnegated literals. In this paper, notions of F-Horn formulas and l-F-Horn formulas, which are extensions of k-quasi formulas, are introduced, and it is shown that the problem of query learning for DNF and CNF formulas with membership and equivalence queries can be reduced to that for l-F-Horn formulas for an appropriate choice of F. It is shown that under a condition on F, the class of orthogonal F-Horn formulas is learnable with membership, equivalence and subset queries. Moreover, it is shown that under the same condition the class of orthogonal l-F-Horn formulas is learnable with membership and equivalence queries. For the latter result, the condition of orthogonality of F-Horn formulas is crucial because, if the statement held without the condition, then the result would imply that DNF and CNF are exactly learnable with membership and equivalence queries.

UR - http://www.scopus.com/inward/record.url?scp=0347936112&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0347936112&partnerID=8YFLogxK

U2 - 10.1016/S0304-3975(97)00020-0

DO - 10.1016/S0304-3975(97)00020-0

M3 - Article

AN - SCOPUS:0347936112

VL - 185

SP - 177

EP - 190

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

IS - 1

ER -