Are lower bounds easier over the reals?

Abstract

We show that proving lower bounds in algebraic models of computation may not be easier than in the standard Turing machine model. For instance, a superpolynomial lower bound on the size of an algebraic circuit solving the real knapsack problem (or on the running time of a real Turing machine) would imply a separation of P from PSPACE. A more general result relates parallel complexity classes in boolean and real models of computation. We also propose a few problems in algebraic complexity and topological complexity.