Cook's versus Valiant's hypothesis
Valiant developed a nonuniform algebraic analogue of the theory of NP-completeness for computations with polynomials over a field ii in (Valiant, Proceedings of the 11th ACM STOC, 1979, pp. 249-261; Logic Algorithmic: An International Symposium held in honor of Ernst Specker, Vol. 30, 1982, pp. 365-380). This theory centers around his hypothesis VPk not equal VNPk, the analogue of Cook's hypothesis P not equal NP. We identify the Boolean parts of Valiant's algebraic complexity classes VPk and VNPk as familiar nonuniform complexity classes. As a consequence, we obtain rather strong evidence for Valiant's hypothesis: if it were wrong, then the nonuniform versions of NC and PH would be equal. In particular, the polynomial hierarchy would collapse to the second level. We show this for fields of characteristic zero and finite fields; in the first case we assume a generalized Riemann hypothesis. The crucial step in our proof is the elimination of constants in k, which relies on a recent method proposed by Koiran [16]. (C) 2000 Elsevier Science B.V. All rights reserved.
np-completeness algebraic complexity elimination of constants reduction module primes