### Abstract

A formula is read-once if each variable appears at most once in it. An arithmetic read-once formula is one in which the operators are addition, subtraction, multiplication, and division. We present polynomial time algorithms for exactly learning (or interpolating) arithmetic read-once formulas computing functions over a field. We present an algorithm that uses randomized membership queries (or substitutions) to identify such formulas over large finite fields and infinite fields. We also present a deterministic algorithm that uses equivalence queries as well as membership queries to identify arithmetic read-once formulas over small finite fields. We then non-constructively show the existence of deterministic membership query (interpolation) algorithms for arbitrary formulas over fields of characteristic 0 and for division-free formulas over large or infinite fields. Our algorithms assume we are able to efficiently perform arithmetic operations on field elements and compute square roots in the field. It is shown that the ability to compute square roots is necessary, in the sense that the problem of computing n-1 square roots in a field can be reduced to the problem of identifying an arithmetic formula over n variables in that field. Our equivalence queries are of a slightly non-standard form, in which counterexamples are required to not be inputs on which the formula evaluates to 0/0. This assumption is shown to be necessary for fields of size o(n/log n), for which it is shown that there is no polynomial time identification algorithm that uses just membership and standard equivalence queries.

Original language | English (US) |
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Title of host publication | Conference Proceedings of the Annual ACM Symposium on Theory of Computing |

Editors | Anon |

Publisher | Publ by ACM |

Pages | 370-381 |

Number of pages | 12 |

ISBN (Print) | 0897915119 |

State | Published - 1992 |

Event | Proceedings of the 24th Annual ACM Symposium on the Theory of Computing - Victoria, BC, Can Duration: May 4 1992 → May 6 1992 |

### Other

Other | Proceedings of the 24th Annual ACM Symposium on the Theory of Computing |
---|---|

City | Victoria, BC, Can |

Period | 5/4/92 → 5/6/92 |

### Fingerprint

### ASJC Scopus subject areas

- Software

### Cite this

*Conference Proceedings of the Annual ACM Symposium on Theory of Computing*(pp. 370-381). Publ by ACM.

**Learning arithmetic read-once formulas.** / Bshouty, Nader H.; Hancock, Thomas R.; Hellerstein, Lisa.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Conference Proceedings of the Annual ACM Symposium on Theory of Computing.*Publ by ACM, pp. 370-381, Proceedings of the 24th Annual ACM Symposium on the Theory of Computing, Victoria, BC, Can, 5/4/92.

}

TY - GEN

T1 - Learning arithmetic read-once formulas

AU - Bshouty, Nader H.

AU - Hancock, Thomas R.

AU - Hellerstein, Lisa

PY - 1992

Y1 - 1992

N2 - A formula is read-once if each variable appears at most once in it. An arithmetic read-once formula is one in which the operators are addition, subtraction, multiplication, and division. We present polynomial time algorithms for exactly learning (or interpolating) arithmetic read-once formulas computing functions over a field. We present an algorithm that uses randomized membership queries (or substitutions) to identify such formulas over large finite fields and infinite fields. We also present a deterministic algorithm that uses equivalence queries as well as membership queries to identify arithmetic read-once formulas over small finite fields. We then non-constructively show the existence of deterministic membership query (interpolation) algorithms for arbitrary formulas over fields of characteristic 0 and for division-free formulas over large or infinite fields. Our algorithms assume we are able to efficiently perform arithmetic operations on field elements and compute square roots in the field. It is shown that the ability to compute square roots is necessary, in the sense that the problem of computing n-1 square roots in a field can be reduced to the problem of identifying an arithmetic formula over n variables in that field. Our equivalence queries are of a slightly non-standard form, in which counterexamples are required to not be inputs on which the formula evaluates to 0/0. This assumption is shown to be necessary for fields of size o(n/log n), for which it is shown that there is no polynomial time identification algorithm that uses just membership and standard equivalence queries.

AB - A formula is read-once if each variable appears at most once in it. An arithmetic read-once formula is one in which the operators are addition, subtraction, multiplication, and division. We present polynomial time algorithms for exactly learning (or interpolating) arithmetic read-once formulas computing functions over a field. We present an algorithm that uses randomized membership queries (or substitutions) to identify such formulas over large finite fields and infinite fields. We also present a deterministic algorithm that uses equivalence queries as well as membership queries to identify arithmetic read-once formulas over small finite fields. We then non-constructively show the existence of deterministic membership query (interpolation) algorithms for arbitrary formulas over fields of characteristic 0 and for division-free formulas over large or infinite fields. Our algorithms assume we are able to efficiently perform arithmetic operations on field elements and compute square roots in the field. It is shown that the ability to compute square roots is necessary, in the sense that the problem of computing n-1 square roots in a field can be reduced to the problem of identifying an arithmetic formula over n variables in that field. Our equivalence queries are of a slightly non-standard form, in which counterexamples are required to not be inputs on which the formula evaluates to 0/0. This assumption is shown to be necessary for fields of size o(n/log n), for which it is shown that there is no polynomial time identification algorithm that uses just membership and standard equivalence queries.

UR - http://www.scopus.com/inward/record.url?scp=0026981387&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0026981387&partnerID=8YFLogxK

M3 - Conference contribution

SN - 0897915119

SP - 370

EP - 381

BT - Conference Proceedings of the Annual ACM Symposium on Theory of Computing

A2 - Anon, null

PB - Publ by ACM

ER -