Interpolating Arithmetic Read-Once Formulas in Parallel

A formula is read-once if each variable appears at most once in it. An arithmetic read-once formula is one in which the operations are addition, subtraction, multiplication, and division (and constants are allowed). We present a randomized (Las Vegas) parallel algorithm for the exact interpolation of arithmetic read-once formulas over suuciently large elds. More speciically, for n-variable read-once formulas, and elds of size at least 3(n 2 + 3n ? 2), our algorithm runs in O(log 2 n) parallel steps using O(n 4) processors (where the eld operations are charged unit cost). This complements other results which imply that other classes of read-once formulas cannot be interpolated|or even learned with membership and equivalence queries|in poly-logarithmic-time with polynomially many processors (even though they can be learned sequentially in polynomial-time). These classes include boolean read-once formulas and arithmetic read-once formulas over elds of size o(n= log n) (for n variable read-once formulas).