P ≠ NC over the p-adic numbers

  • Authors:

  • Michael Maller;Jennifer Whitehead

  • Affiliations:

  • Department of Mathematics, Queens College, Flushing, New York;Department of Computer Science, Queens College, Flushing, New York

  • Venue:
  • Journal of Complexity
  • Year:
  • 2003

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Abstract

We show that in the Blum-Shub-Smale model of computation, over the p-adic numbers Qp, the class NCQp is strictly contained in the class PQp. That is, there exist sets of p-adic numbers which can be recognized in sequential polynomial time, but which cannot be recognized in polylogarithmic parallel time. We use the sets (x1,x2,...,xn) ∈ Qpn ⊂ Qp∞ such that x12n = x2. We also show that the inclusion PARQp ⊂ EXPQp is strict. These results extend previous work of Cucker, and of Blum, Cucker, Shub and Smale in the real case.