TY - GEN

T1 - Learning arithmetic read-once formulas

AU - Bshouty, Nader H.

AU - Hancock, Thomas R.

AU - Hellerstein, Lisa

N1 - Funding Information:
*This research t supported by
Publisher Copyright:
© 1992 ACM.

PY - 1992/7/1

Y1 - 1992/7/1

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.

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U2 - 10.1145/129712.129747

DO - 10.1145/129712.129747

M3 - Conference contribution

AN - SCOPUS:0026981387

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 370

EP - 381

BT - Proceedings of the 24th Annual ACM Symposium on Theory of Computing, STOC 1992

PB - Association for Computing Machinery

T2 - 24th Annual ACM Symposium on Theory of Computing, STOC 1992

Y2 - 4 May 1992 through 6 May 1992

ER -