We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis: • The Blum-Shub-Smale model of computation over the reals. • A problem we call the “Generic Task of Numerical Computation, ” which captures an aspect of doing numerical computation in floating point, similar to the “long exponent model ” that has been studied in the numerical computing community. We show that both of these approaches hinge on the question of understanding the complexity of the following problem, which we call PosSLP: Given a division-free straight-line program producing an integer N, decide whether N> 0. • In the Blum-Shub-Smale model, polynomial time computation over the reals (on discrete inputs) is polynomial-time equivalent to PosSLP, when there are only algebraic constants. We conjecture that using transcendental constants provides no additional power, beyond nonuniform reductions to PosSLP, and we present some preliminary results supporting this conjecture. • The Generic Task of Numerical Computation is also polynomial-time equivalent to PosSLP. We prove that PosSLP lies in the counting hierarchy. Combining this with work of Tiwari, we obtain that the Euclidean Traveling Salesman Problem lies in the counting hierarchy – the previous best upper bound for this important problem (in terms of classical complexity classes) being PSPACE. In the course of developing the context for our results on arithmetic circuits, we present some new observations on the complexity of ACIT: the Arithmetic Circuit Identity Testing problem. In particular, we show that if n! is not ultimately easy, then ACIT has subexponential complexity.

Diagonalization Theorem Let a quasi-ordered set (W;) be fixed. Elements of the set W of sequences in W will be called families in the sequel. We may formally define a quasi-order p (the abstract p-projection) on the set W of families as in ( 3 on the page before). Two families f and g are said to be p-equivalent iff f p g and g p f . We call the equivalence classes p-degrees and denote by D p the poset of all p-degrees with the partial order induced by p . f ! p g shall mean that f p g but not g p f . The join f [g of two families f ; g 2W is defined as f [g := ( f 0 ; g 0 ; f 1 ; g 1 ; f 2 ; g 2 ; : : :) : (4) It is easy to see that the join of two p-degrees is well-defined and that it is the smallest upper bound of these p-degrees in D p . The poset D p of p-degrees is thus a join-semilattice.

Blum, Cucker, Shub and Smale have shown that the problem "P = NP?" has the same answer in all algebraically closed elds of characteristic 0. We generalize this result to the polynomial hierarchy: if it collapses over an algebraically closed eld of characteristic 0, then it must collapse at the same level over all algebraically closed fields of characteristic 0. The main ingredient of their proof was a theorem on the elimination of parameters, which we also extend to the polynomial hierarchy. Similar but somewhat weaker results hold in positive characteristic. The present paper updates a report (LIP Research Report 97-37) with the same title, and in particular includes new results on interactive protocols and boolean parts.

In the BSS model of real number computations we prove a concrete and explicit semi-decidable language to be undecidable yet not reducible from (and thus strictly easier than) the real Halting Language. This solution to Post’s Problem over the reals significantly differs from its classical, discrete variant where advanced diagonalization techniques are only known to yield the existence of such intermediate Turing degrees. Then we strengthen the above result and show as well the existence of an uncountable number of incomparable semi-decidable Turing degrees below the real Halting Problem in the BSS model. Again, our proof will give concrete such problems representing these different degrees. Finally we show the corresponding result for the linear BSS model, that is over (R, +, −,<)rather than (R, +, −, ×, ÷,<).

We show that computing the dimension of a semi-algebraic set of R^n is a NP_R-complete problem in the Blum-Shub-Smale model of computation over the reals. Since this problem is easily seen to be NP_R-hard, the main ingredient of the proof is a NP_R algorithm for computing the dimension.

. This survey is devoted to some aspects of the \P = NP ?" problem over the real numbers and more general algebraic structures. We argue that given a structure M , it is important to nd out whether NPM problems can be solved by polynomial depth computation trees, and if so whether these trees can be eciently simulated by circuits. Point location, a problem of computational geometry, comes into play in the study of these questions for several structures of interest. 1 Introduction In algebraic complexity one measures the complexity of an algorithm by the number of basic operations performed during a computation. The basic operations are usually arithmetic operations and comparisons, but sometimes transcendental functions are also allowed [21-23, 26]. Even when the set of basic operations has been xed, the complexity of a problem depends on the particular model of computation considered. The two main categories of interest for this paper are circuits and trees. In section 2 and...

We introduce a new class VPSPACE of families of polynomials. Roughly speaking, a family of polynomials is in VPSPACE if its coefficients can be computed in polynomial space. Our main theorem is that if (uniform, constant-free) VPSPACE families can be evaluated efficiently then the class PARR of decision problems that can be solved in parallel polynomial time over the real numbers collapses to PR. As a result, one must first be able to show that there are VPSPACE families which are hard to evaluate in order to separate PR from NPR, or even from PARR.

Abstract. Most of the existing work in real number computation theory concentrates on complexity issues rather than computability aspects. Though some natural problems like deciding membership in the Mandelbrot set or in the set of rational numbers are known to be undecidable in the Blum-Shub-Smale (BSS) model of computation over the reals, there has not been much work on different degrees of undecidability. A typical question into this direction is the real version of Post’s classical problem: Are there some explicit undecidable problems below the real Halting Problem? In this paper we study three different topics related to such questions: First an extension of a positive answer to Post’s problem to the linear setting. We then analyze how additional real constants increase the power of a BSS machine. And finally a real variant of the classical word problem for groups is presented which we establish reducible to and from (that is, complete for) the BSS Halting problem. 1

. We show that deciding whether an algebraic variety has an irreducible component of codimension at least d is an NPC -complete problem for every fixed d (and is in the Arthur-Merlin class if we assume a bit model of computation). However, when d is not fixed but is instead part of the input, we show that the problem is not likely to be in NPC or in coNPC . These results are generalized to arbitrary constructible sets. We also study the complexity of a few other related problems. 1. Introduction It was shown in [14] that computing the dimension of algebraic varieties is NP C - complete in the Blum-Shub-Smale model of computation, and that in the bit model this problem is in AM (the Arthur-Merlin complexity class) assuming the Generalized Riemann Hypothesis (GRH). The dimension of a variety is the dimension of its largest irreducible component, and the dimensions of smaller components may also be of interest (see for instance [18]). In this paper we investigate the complexity of compu...

Computing the maximum bichromatic discrepancy is an interesting theoretical problem with important applications in computational learning theory, computational geometry and computer graphics. In this paper we give algorithms to compute the maximum bichromatic discrepancy for simple geometric ranges, including rectangles and halfspaces. In addition, we give extensions to other discrepancy problems. 1 Introduction The main theme of this paper is to present efficient algorithms that solve the problem of computing the maximum bichromatic discrepancy for axis oriented rectangles. This problem arises naturally in different areas of computer science, such as computational learning theory, computational geometry and computer graphics ([Ma], [DG]), and has applications in all these areas. In computational learning theory, the problem of agnostic PAC-learning with simple geometric hypotheses can be reduced to the problem of computing the maximum bichromatic discrepancy for simple geometric ra...