Two P-complete problems in the theory of the reals

Abstract

A classical subject in computational complexity what usually called algebraic complexity. This area deals with the complexity of algorithms that take their inputs on R n, where R a ring, understood as the number of ring operations the algorithm performs as a function of n and the main kind of results are both upper and lower bounds. A recent survey about this topic [7]. A very special case in algebraic complexity the one dealing with R = ]1%. Here, the assumption that all the elements in the have unit size, and that the cost of the operations also unitary, reflects the particular features of algorithms in numerical analysis. Very recently an article of L. Blum, M. Shub and S. Smale put into this scenery another approach to the complexity: the structural one (see [4]). To do so, they deviced a model of real Turing machines, over the which the basis of a theory of computability and a theory of complexity built. In particular analogs of the classes P and NP over the reals are introduced, and the problem of deciding whether a degree 4 real polynomial has a root shown to be NP-complete, thus, unlikely decidable in polynomial time. On the other hand, for a wide variety of problems, fast parallel algorithms have been designed during the last years. These algorithms use as a model of parallel machine a family of circuits with polynomial size and polylogarithmic depth, for which a logspace uniformity condition valid. However, this is of course not possible for arbitrary constants over an uncountable field (see [6]). For the Boolean model, there are many problems for which algorithms working in polynomial time can be given that solve them and no good fast parallel algorithm known that do the same. Some of these problems share the property that if a fast parallel algorithm found that solves one of them, then every problem in P can be decided by such a fast parallel algorithm. These problems, that are said to be P complete constitute a class whose elements, even when no proof known till now for P ~ NC, are usually considered as problems not admitting a solution in fast parallel time. In section 2 of this paper we propose the parallel real RAM as a model for defining a class of problems dealing with real numbers that are decidable in fast parallel time. This formulation avoids uniformity considerations and the resulting class trivially included in P. We use reductions in this class (in fact, using constant parallel time) to give our main result, the existence of P-complete problems for the real model. In section 3, two