### Abstract

Consider a logical system with N entities which assume binary values of either TRUE (1) or FALSE (0). There are 2^{N} vectors, each with N components, of this type. Even when a modest value of N, e.g. N = 50, the number of such vectors exceeds one quadrillion. We assume that an 'expert' exists which can ascertain whether a particular vector (observation) such as (1, 1, 0, 0, 1, 0, ..., 1) is allowable or not. This expert can be a human expert or an unknown system whose rules have to be inferred. Further, we assume that a sampling of m observations has resulted in M_{1} instances which the expert has classified as allowable and M_{2} = m - M_{1} instances which are not allowable. We call these instances positive and negative examples, respectively. The objective of this research is to infer a set of logical rules for the entire system based upon the m, and possibly, additional sample observations. The proposed algorithm in this paper is based on an highly efficient branch-and-bound formulation. This algorithm configures a sequence of logical clauses in conjunctive normal form (CNF), that when are taken together, accept all the positive examples and reject all the negative examples. Some computational results indicate that the proposed approach can process problems that involve hundreds of positive and negative examples in a few CPU seconds and with small memory requirements.

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

Pages (from-to) | 185-197 |

Number of pages | 13 |

Journal | Computers and Operations Research |

Volume | 21 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 1994 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Information Systems and Management
- Management Science and Operations Research
- Applied Mathematics
- Modeling and Simulation
- Transportation

### Cite this

*Computers and Operations Research*,

*21*(2), 185-197. https://doi.org/10.1016/0305-0548(94)90051-5

}

*Computers and Operations Research*, vol. 21, no. 2, pp. 185-197. https://doi.org/10.1016/0305-0548(94)90051-5

**Generating logical expressions from positive and negative examples via a branch-and-bound approach.** / Triantaphyllou, Evangelos; Soyster, Allen L.; Tirupatikumara, Soundar Rajan.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Generating logical expressions from positive and negative examples via a branch-and-bound approach

AU - Triantaphyllou, Evangelos

AU - Soyster, Allen L.

AU - Tirupatikumara, Soundar Rajan

PY - 1994/1/1

Y1 - 1994/1/1

N2 - Consider a logical system with N entities which assume binary values of either TRUE (1) or FALSE (0). There are 2N vectors, each with N components, of this type. Even when a modest value of N, e.g. N = 50, the number of such vectors exceeds one quadrillion. We assume that an 'expert' exists which can ascertain whether a particular vector (observation) such as (1, 1, 0, 0, 1, 0, ..., 1) is allowable or not. This expert can be a human expert or an unknown system whose rules have to be inferred. Further, we assume that a sampling of m observations has resulted in M1 instances which the expert has classified as allowable and M2 = m - M1 instances which are not allowable. We call these instances positive and negative examples, respectively. The objective of this research is to infer a set of logical rules for the entire system based upon the m, and possibly, additional sample observations. The proposed algorithm in this paper is based on an highly efficient branch-and-bound formulation. This algorithm configures a sequence of logical clauses in conjunctive normal form (CNF), that when are taken together, accept all the positive examples and reject all the negative examples. Some computational results indicate that the proposed approach can process problems that involve hundreds of positive and negative examples in a few CPU seconds and with small memory requirements.

AB - Consider a logical system with N entities which assume binary values of either TRUE (1) or FALSE (0). There are 2N vectors, each with N components, of this type. Even when a modest value of N, e.g. N = 50, the number of such vectors exceeds one quadrillion. We assume that an 'expert' exists which can ascertain whether a particular vector (observation) such as (1, 1, 0, 0, 1, 0, ..., 1) is allowable or not. This expert can be a human expert or an unknown system whose rules have to be inferred. Further, we assume that a sampling of m observations has resulted in M1 instances which the expert has classified as allowable and M2 = m - M1 instances which are not allowable. We call these instances positive and negative examples, respectively. The objective of this research is to infer a set of logical rules for the entire system based upon the m, and possibly, additional sample observations. The proposed algorithm in this paper is based on an highly efficient branch-and-bound formulation. This algorithm configures a sequence of logical clauses in conjunctive normal form (CNF), that when are taken together, accept all the positive examples and reject all the negative examples. Some computational results indicate that the proposed approach can process problems that involve hundreds of positive and negative examples in a few CPU seconds and with small memory requirements.

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U2 - 10.1016/0305-0548(94)90051-5

DO - 10.1016/0305-0548(94)90051-5

M3 - Article

VL - 21

SP - 185

EP - 197

JO - Surveys in Operations Research and Management Science

JF - Surveys in Operations Research and Management Science

SN - 0305-0548

IS - 2

ER -