A deterministic algorithm for sparse multivariate polynomial interpolation
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
On the decidability of sparse univariate polynomial interpolation
STOC '90 Proceedings of the twenty-second annual ACM symposium on Theory of computing
Interpolation and approximation of sparse multivariate polynomials over GF(2)
SIAM Journal on Computing
Learning read-once formulas over fields and extended bases
COLT '91 Proceedings of the fourth annual workshop on Computational learning theory
Learning read-once formulas with queries
Journal of the ACM (JACM)
Fast Probabilistic Algorithms for Verification of Polynomial Identities
Journal of the ACM (JACM)
Testing polynomials which are easy to compute (Extended Abstract)
STOC '80 Proceedings of the twelfth annual ACM symposium on Theory of computing
Fast Parallel Algorithms for Sparse Multivariate Polynomial Interpolationover Finite Fields
Fast Parallel Algorithms for Sparse Multivariate Polynomial Interpolationover Finite Fields
Computational Complexity of Learning Read-Once Formulas over Different Bases
Computational Complexity of Learning Read-Once Formulas over Different Bases
Computational learning theory: survey and selected bibliography
STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
Learning Boolean read-once formulas with arbitrary symmetric and constant fan-in gates
COLT '92 Proceedings of the fifth annual workshop on Computational learning theory
Asking questions to minimize errors
COLT '93 Proceedings of the sixth annual conference on Computational learning theory
On learning arithmetic read-once formulas with exponentiation (extended abstract)
COLT '94 Proceedings of the seventh annual conference on Computational learning theory
Learning from a consistently ignorant teacher
COLT '94 Proceedings of the seventh annual conference on Computational learning theory
On the learnability of Zn-DNF formulas (extended abstract)
COLT '95 Proceedings of the eighth annual conference on Computational learning theory
Learning from examples with unspecified attribute values (extended abstract)
COLT '97 Proceedings of the tenth annual conference on Computational learning theory
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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 algorithm 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.