Real functions, contraction mappings, and P-completeness

Abstract

This paper examines the notions of feasible real function and of NC real function. We introduce a uniform framework for describing what it means for a continuous real function to be computed by a Boolean circuit family, and we provide techniques for constructing such functions. As an example we construct a continuous real function that is complete for P , thus showing that the question of wheter NC = P can be reduced to the question of whether the class of feasible real functions equals the class of NC real functions. A corollary of this result is that there exists a family of feasible-size-magnitude polynomials that are complete for P . Finally we look at contraction mappings and ask whether the fixed points of an NC real contraction mapping can be found in NC . We give evidence that this is not the case in general by exhibiting an NC real function which is a contraction mapping over disjoint intervals of the real line, and for which the problem of finding the fixed point of any given interval is complete for P . Thus methods for locating fixed points which are based simply on contraction mappings are not likely to parallelize well.