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26
On the Combinatorial and Algebraic Complexity of Quantifier Elimination
, 1996
"... In this paper, a new algorithm for performing quantifier elimination from first order formulas over real closed fields is given. This algorithm improves the complexity of the asymptotically fastest algorithm for this problem, known to this date. A new feature of this algorithm is that the role of th ..."
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Cited by 201 (29 self)
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In this paper, a new algorithm for performing quantifier elimination from first order formulas over real closed fields is given. This algorithm improves the complexity of the asymptotically fastest algorithm for this problem, known to this date. A new feature of this algorithm is that the role of the algebraic part (the dependence on the degrees of the input polynomials) and the combinatorial part (the dependence on the number of polynomials) are separated. Another new feature is that the degrees of the polynomials in the equivalent quantifierfree formula that is output, are independent of the number of input polynomials. As special cases of this algorithm, new and improved algorithms for deciding a sentence in the first order theory over real closed fields, and also for solving the existential problem in the first order theory over real closed fields, are obtained.
The Exact Computation Paradigm
, 1994
"... We describe a paradigm for numerical computing, based on exact computation. This emerging paradigm has many advantages compared to the standard paradigm which is based on fixedprecision. We first survey the literature on multiprecision number packages, a prerequisite for exact computation. Next ..."
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Cited by 95 (10 self)
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We describe a paradigm for numerical computing, based on exact computation. This emerging paradigm has many advantages compared to the standard paradigm which is based on fixedprecision. We first survey the literature on multiprecision number packages, a prerequisite for exact computation. Next we survey some recent applications of this paradigm. Finally, we outline some basic theory and techniques in this paradigm. 1 This paper will appear as a chapter in the 2nd edition of Computing in Euclidean Geometry, edited by D.Z. Du and F.K. Hwang, published by World Scientific Press, 1994. 1 1 Two Numerical Computing Paradigms Computation has always been intimately associated with numbers: computability theory was early on formulated as a theory of computable numbers, the first computers have been number crunchers and the original massproduced computers were pocket calculators. Although one's first exposure to computers today is likely to be some nonnumerical application, numeri...
A New Algorithm to Find a Point in Every Cell Defined by a Family of Polynomials
 B. Caviness and J. Johnson Eds., SpringerVerlag
, 1995
"... We consider s polynomials P 1 ; : : : ; P s in k ! s variables with coefficients in an ordered domain A contained in a real closed field R, each of degree at most d. We present a new algorithm which computes a point in each connected component of each nonempty sign condition over P 1 ; : : : ; P s ..."
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Cited by 40 (8 self)
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We consider s polynomials P 1 ; : : : ; P s in k ! s variables with coefficients in an ordered domain A contained in a real closed field R, each of degree at most d. We present a new algorithm which computes a point in each connected component of each nonempty sign condition over P 1 ; : : : ; P s . The output is the set of points together with the sign condition at each point. The algorithm uses s(s=k) k d O(k) arithmetic operations in A. The algorithm is nearly optimal in the sense that the size of the output can be as large as s(O(sd=k)) k . Previous algorithms of Canny and Renegar used (sd) O(k) operations [5, 7, 8, 15]. We use either these algorithms in the case s = 1 as a subroutine in our algorithm. As a bonus, our algorithm yields an independent proof of the bound on the number of connected components in all nonempty sign conditions ([14]) and also yields an independent proof of a theorem of Warren 1 Courant Institute of Mathematical Sciences, New York University, N...
Computing roadmaps of semialgebraic sets on a variety
 Journal of the AMS
, 1997
"... Let R be a real closed field, Z(Q) a real algebraic variety defined as the zero set of a polynomial Q ∈ R[X1,...,Xk]andSasemialgebraic subset of Z(Q), defined by a Boolean formula with atoms of the form P < 0,P> 0,P =0withP∈P, where P is a finite subset of R[X1,...,Xk]. A semialgebraic set C is se ..."
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Cited by 37 (13 self)
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Let R be a real closed field, Z(Q) a real algebraic variety defined as the zero set of a polynomial Q ∈ R[X1,...,Xk]andSasemialgebraic subset of Z(Q), defined by a Boolean formula with atoms of the form P < 0,P> 0,P =0withP∈P, where P is a finite subset of R[X1,...,Xk]. A semialgebraic set C is semialgebraically connected if it is nonempty and is not
Real Algebraic Numbers: Complexity Analysis and Experimentation
 RELIABLE IMPLEMENTATIONS OF REAL NUMBER ALGORITHMS: THEORY AND PRACTICE, LNCS (TO APPEAR
, 2006
"... We present algorithmic, complexity and implementation results concerning real root isolation of a polynomial of degree d, with integer coefficients of bit size ≤ τ, using Sturm (Habicht) sequences and the Bernstein subdivision solver. In particular, we unify and simplify the analysis of both metho ..."
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Cited by 30 (17 self)
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We present algorithmic, complexity and implementation results concerning real root isolation of a polynomial of degree d, with integer coefficients of bit size ≤ τ, using Sturm (Habicht) sequences and the Bernstein subdivision solver. In particular, we unify and simplify the analysis of both methods and we give an asymptotic complexity bound of eOB(d 4 τ 2). This matches the best known bounds for binary subdivision solvers. Moreover, we generalize this to cover the non squarefree polynomials and show that within the same complexity we can also compute the multiplicities of the roots. We also consider algorithms for sign evaluation, comparison of real algebraic numbers and simultaneous inequalities, and we improve the known bounds at least by a factor of d. Finally, we present our C++ implementation in synaps and some preliminary experiments on various data sets.
A Weak Version of the Blum, Shub & Smale Model
, 1994
"... We propose a weak version of the BlumShubSmale model of computation over the real numbers. In this weak model only a "moderate" usage of multiplications and divisions is allowed. The class of boolean languages recognizable in polynomial time is shown to be the complexity class P/poly. The main ..."
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Cited by 29 (6 self)
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We propose a weak version of the BlumShubSmale model of computation over the real numbers. In this weak model only a "moderate" usage of multiplications and divisions is allowed. The class of boolean languages recognizable in polynomial time is shown to be the complexity class P/poly. The main tool is a result on the existence of small rational points in semialgebraic sets which is of independent interest. As an application, we generalize recent results of Siegelmann & Sontag on recurrent neural networks, and of Maass on feedforward nets. A preliminary version of this paper was presented at the 1993 IEEE Symposium on Foundations of Computer Science. Additional results include: \Pi an efficient simulation of orderfree real Turing machines by probabilistic Turing machines in the full BlumShubSmale model; \Pi an efficient simulation of arithmetic circuits over the integers by boolean circuits; \Pi the strict inclusion of the real polynomial hierarchy in weak exponentia...
On the power of real Turing machines over binary inputs
 SIAM Journal on Computing
, 1997
"... this paper is to prove that BP(PAR IR ) = PSPACE/poly where PAR IR is the class of sets computed in parallel polynomial time by (ordinary) real Turing machines. As a consequence we obtain the existence of binary sets that do not belong to the Boolean part of PAR IR (an extension of the result in [20 ..."
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Cited by 24 (3 self)
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this paper is to prove that BP(PAR IR ) = PSPACE/poly where PAR IR is the class of sets computed in parallel polynomial time by (ordinary) real Turing machines. As a consequence we obtain the existence of binary sets that do not belong to the Boolean part of PAR IR (an extension of the result in [20] since PH IR ` PAR IR ) and a separation of complexity classes in the real setting.
Sample Complexity for Learning Recurrent Perceptron Mappings
 IEEE Trans. Inform. Theory
, 1996
"... Recurrent perceptron classifiers generalize the classical perceptron model. They take into account those correlations and dependences among input coordinates which arise from linear digital filtering. This paper provides tight bounds on sample complexity associated to the fitting of such models to e ..."
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Cited by 23 (10 self)
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Recurrent perceptron classifiers generalize the classical perceptron model. They take into account those correlations and dependences among input coordinates which arise from linear digital filtering. This paper provides tight bounds on sample complexity associated to the fitting of such models to experimental data. Keywords: perceptrons, recurrent models, neural networks, learning, VapnikChervonenkis dimension 1 Introduction One of the most popular approaches to binary pattern classification, underlying many statistical techniques, is based on perceptrons or linear discriminants ; see for instance the classical reference [9]. In this context, one is interested in classifying kdimensional input patterns v = (v 1 ; : : : ; v k ) into two disjoint classes A + and A \Gamma . A perceptron P which classifies vectors into A + and A \Gamma is characterized by a vector (of "weights") ~c 2 R k , and operates as follows. One forms the inner product ~c:v = c 1 v 1 + : : : c k v k . I...
On real Turing machines that toss coins
, 1995
"... In this paper we consider real counterparts of classical probabilistic complexity classes in the framework of real Turing machines as introduced by Blum, Shub, and Smale [2]. We give an extension of the wellknown "BPP ` P=poly" result from discrete complexity theory to a very general setting in the ..."
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Cited by 14 (8 self)
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In this paper we consider real counterparts of classical probabilistic complexity classes in the framework of real Turing machines as introduced by Blum, Shub, and Smale [2]. We give an extension of the wellknown "BPP ` P=poly" result from discrete complexity theory to a very general setting in the real number model. This result holds for real inputs, real outputs, and random elements drawn from an arbitrary probability distribution over IR m . Then we turn to the study of Boolean parts, that is, classes of languages of zeroone vectors accepted by real machines. In particular we show that the classes BPP , PP , PH, and PSPACE are not enlarged by allowing the use of real constants and arithmetic at unit cost provided we restrict branching to equality tests. Introduction 3 1 Introduction This paper deals with probabilistic complexity classes in the real number model of computation. We consider both uniform and nonuniform models. The classical nonuniform models are straightline ...
On Computing a Set of Points meeting every Semialgebraically Connected Component of a Family of Polynomials on a Variety
, 1997
"... We consider a family of s polynomials, P = fP 1 ; : : : ; P s g; in k variables with coefficients in a real closed field R, each of degree at most d, and an algebraic variety V of real dimension k 0 which is defined as the zero set of a polynomial Q of degree at most d. The number of semialgebr ..."
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Cited by 12 (5 self)
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We consider a family of s polynomials, P = fP 1 ; : : : ; P s g; in k variables with coefficients in a real closed field R, each of degree at most d, and an algebraic variety V of real dimension k 0 which is defined as the zero set of a polynomial Q of degree at most d. The number of semialgebraically connected components of all nonempty sign condition on P over V is bounded by \Gamma O(s) k 0 \Delta d O(k) (see [4]). In this paper we present a new algorithm to compute a set of points meeting every semialgebraically connected component of each non empty sign condition of P over V . Its complexity is \Gamma O(s) k 0 \Delta sd O(k) . This interpolates a sequence of results between the algorithm of BenOr, Kozen and Reif [5] which is the case k 0 = 0, in one variable, and the algorithm of BasuPollackRoy [1] which is the case k 0 = k. It Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, U.S.A.. Supported in part by NSF grants CCR...