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On the complexity of numerical analysis
 IN PROC. 21ST ANN. IEEE CONF. ON COMPUTATIONAL COMPLEXITY (CCC ’06
, 2006
"... We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis: • The BlumShubSmale model of computation over the reals. • A problem we call the “Generic Task of Numerical Computation, ” which captures an aspect of doing numerical computation ..."
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Cited by 48 (7 self)
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We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis: • The BlumShubSmale 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 divisionfree straightline program producing an integer N, decide whether N> 0. • In the BlumShubSmale model, polynomial time computation over the reals (on discrete inputs) is polynomialtime 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 polynomialtime 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.
Randomized and Deterministic Algorithms for the Dimension of Algebraic Varieties
 In Proc. 38th IEEE Symposium on Foundations of Computer Science
, 1997
"... We prove old and new results on the complexity of computing the dimension of algebraic varieties. In particular, we show that this problem is NPcomplete in the BlumShubSmale model of computation over C , that it admits a s O(1) D O(n) deterministic algorithm, and that for systems with integer ..."
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Cited by 25 (9 self)
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We prove old and new results on the complexity of computing the dimension of algebraic varieties. In particular, we show that this problem is NPcomplete in the BlumShubSmale model of computation over C , that it admits a s O(1) D O(n) deterministic algorithm, and that for systems with integer coefficients it is in the ArthurMerlin class under the Generalized Riemann Hypothesis. The first two results are based on a general derandomization argument. 1 Introduction We wish to compute the dimension of an algebraic variety V ` C n defined by a system of algebraic equations f 1 (x) = 0; : : : ; f s (x) = 0 (1) where f i 2 C [X 1 ; : : : ; Xn ]. This can be formalized as a decision problem DIMC . An instance of DIMC is a system of this form together with an integer d n. An instance is accepted if the variety defined by the system has dimension at least d. We also consider for each fixed value of d the restriction DIM d C of DIMC . For instance, DIM 0 C is the problem of dec...
Counting complexity classes for numeric computations II: Algebraic and semialgebraic sets (Extended Abstract)
 J. COMPL
, 2004
"... We define counting #P classes #P ¡ and in the BlumShubSmale setting of computations over the real or complex numbers, respectively. The problems of counting the number of solutions of systems of polynomial inequalities over ¢ , or of systems of polynomial equalities over £ , respectively, turn ou ..."
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Cited by 21 (13 self)
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We define counting #P classes #P ¡ and in the BlumShubSmale setting of computations over the real or complex numbers, respectively. The problems of counting the number of solutions of systems of polynomial inequalities over ¢ , or of systems of polynomial equalities over £ , respectively, turn out to be natural complete problems in these classes. We investigate to what extent the new counting classes capture the complexity of computing basic topological invariants of semialgebraic sets (over ¢ ) and algebraic sets (over £). We prove that the problem to compute the (modified) Euler characteristic of semialgebraic sets is FP #P¤complete, and that the problem to compute the geometric degree of complex algebraic sets is FP #P¥complete. We also define new counting complexity classes GCR and GCC in the classical Turing model via taking Boolean parts of the classes above, and show that the problems to compute the Euler characteristic and the geometric degree of (semi)algebraic sets given by integer polynomials are complete in these classes. We complement the results in the Turing model by proving, for all k ¦ ∈ , the FPSPACEhardness of the problem of computing the kth Betti number of the set of real zeros of a given integer polynomial. This holds with respect to the singular homology as well as for the BorelMoore homology.
Saturation and Stability in the Theory of Computation over the Reals
, 1997
"... This paper was motivated by the following two questions which arise in the theory of complexity for computation over ordered rings in the now famous computational model introduced by Blum, Shub and Smale: (i) is the answer to the question P =? NP the same in every realclosed field ? (ii) if P 6= ..."
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Cited by 15 (10 self)
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This paper was motivated by the following two questions which arise in the theory of complexity for computation over ordered rings in the now famous computational model introduced by Blum, Shub and Smale: (i) is the answer to the question P =? NP the same in every realclosed field ? (ii) if P 6= NP for R, does there exist a problem of R which is NP but not NPcomplete ? Some unclassical complexity classes arise naturally in the study of these questions. They are still open, but we could obtain unconditional results of independent interest. Michaux introduced =const complexity classes in an effort to attack question (i). We show that AR =const = AR , answering a question of his. Here A is the class of real problems which are algorithmic in bounded time. We also prove the stronger result: PARR =const = PARR , where PAR stands for parallel polynomial time. In our terminology, we say that R is Asaturated and PARsaturated. We also prove, at the nonuniform level, the above results for...
Variations by complexity theorists on three themes of
 Computational Complexity
, 2005
"... This paper surveys some connections between geometry and complexity. A main role is played by some quantities —degree, Euler characteristic, Betti numbers — associated to algebraic or semialgebraic sets. This role is twofold. On the one hand, lower bounds on the deterministic time (sequential and pa ..."
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Cited by 12 (4 self)
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This paper surveys some connections between geometry and complexity. A main role is played by some quantities —degree, Euler characteristic, Betti numbers — associated to algebraic or semialgebraic sets. This role is twofold. On the one hand, lower bounds on the deterministic time (sequential and parallel) necessary to decide a set S are established as functions of these quantities associated to S. The optimality of some algorithms is obtained as a consequence. On the other hand, the computation of these quantities gives rise to problems which turn out to be hard (or complete) in different complexity classes. These two kind of results thus turn the quantities above into measures of complexity in two quite different ways. 1
The Real Dimension Problem is NP_RComplete
, 1998
"... We show that computing the dimension of a semialgebraic set of R^n is a NP_Rcomplete problem in the BlumShubSmale model of computation over the reals. Since this problem is easily seen to be NP_Rhard, the main ingredient of the proof is a NP_R algorithm for computing the dimension. ..."
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Cited by 11 (0 self)
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We show that computing the dimension of a semialgebraic set of R^n is a NP_Rcomplete problem in the BlumShubSmale model of computation over the reals. Since this problem is easily seen to be NP_Rhard, the main ingredient of the proof is a NP_R algorithm for computing the dimension.
Elimination of parameters in the polynomial hierarchy
 THEORET. COMP. SCI
, 1998
"... 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 ..."
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Cited by 8 (6 self)
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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 9737) with the same title, and in particular includes new results on interactive protocols and boolean parts.
Circuits versus Trees in Algebraic Complexity
 In Proc. STACS 2000
, 2000
"... . 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 ca ..."
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Cited by 5 (4 self)
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. 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 [2123, 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...
M.: Uncomputability Below the Real Halting Problem
 CiE 2006. LNCS
, 2006
"... 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 BlumShubSmale ..."
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Cited by 2 (1 self)
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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 BlumShubSmale (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