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61
Complexity and Expressive Power of Logic Programming
, 1997
"... This paper surveys various complexity results on different forms of logic programming. The main focus is on decidable forms of logic programming, in particular, propositional logic programming and datalog, but we also mention general logic programming with function symbols. Next to classical results ..."
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Cited by 299 (56 self)
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This paper surveys various complexity results on different forms of logic programming. The main focus is on decidable forms of logic programming, in particular, propositional logic programming and datalog, but we also mention general logic programming with function symbols. Next to classical results on plain logic programming (pure Horn clause programs), more recent results on various important extensions of logic programming are surveyed. These include logic programming with different forms of negation, disjunctive logic programming, logic programming with equality, and constraint logic programming. The complexity of the unification problem is also addressed.
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 200 (28 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.
Matrices in Elimination Theory
, 1997
"... The last decade has witnessed the rebirth of resultant methods as a powerful computational tool for variable elimination and polynomial system solving. In particular, the advent of sparse elimination theory and toric varieties has provided ways to exploit the structure of polynomials encountered in ..."
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Cited by 47 (17 self)
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The last decade has witnessed the rebirth of resultant methods as a powerful computational tool for variable elimination and polynomial system solving. In particular, the advent of sparse elimination theory and toric varieties has provided ways to exploit the structure of polynomials encountered in a number of scientific and engineering applications. On the other hand, the Bezoutian reveals itself as an important tool in many areas connected to elimination theory and has its own merits, leading to new developments in effective algebraic geometry. This survey unifies the existing work on resultants, with emphasis on constructing matrices that generalize the classic matrices named after Sylvester, Bézout and Macaulay. The properties of the different matrix formulations are presented, including some complexity issues, with an emphasis on variable elimination theory. We compare toric resultant matrices to Macaulay's matrix and further conjecture the generalization of Macaulay's exact ratio...
An expectation/maximization nuclear vector replacement algorithm for automated NMR resonance assignments
 J. Biomol. NMR
"... Highthroughput NMR structural biology can play an important role in structural genomics. We report an automated procedure for highthroughput NMR resonance assignment for a protein of known structure, or of a homologous structure. These assignments are a prerequisite for probing protein–protein int ..."
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Cited by 37 (12 self)
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Highthroughput NMR structural biology can play an important role in structural genomics. We report an automated procedure for highthroughput NMR resonance assignment for a protein of known structure, or of a homologous structure. These assignments are a prerequisite for probing protein–protein interactions, protein–ligand binding, and dynamics by NMR. Assignments are also the starting point for structure determination and refinement. A new algorithm, called Nuclear Vector Replacement (NVR) is introduced to compute assignments that optimally correlate experimentally measured NH residual dipolar couplings (RDCs) to a given a priori wholeprotein 3D structural model. The algorithm requires only uniform 15 Nlabeling of the protein and processes unassigned H N 15 N HSQC spectra, H N 15 N RDCs, and sparse H NH N NOE’s (dNNs), all of which can be acquired in a fraction of the time needed to record the traditional suite of experiments used to perform resonance assignments. NVR runs in minutes and efficiently assigns the (H N, 15 N) backbone resonances as well as the dNNs of the 3D 15 NNOESY spectrum, in O(n 3) time. The algorithm is demonstrated on NMR data from a 76residue protein, human ubiquitin, matched to four structures, including
A singly exponential stratification scheme for real semialgebraic varieties and its applications
, 1991
"... ..."
Polar varieties and efficient real elimination
 MATH. Z
, 2001
"... Let S0 be a smooth and compact real variety given by a reduced regular sequence of polynomials f1,..., fp. This paper is devoted to the algorithmic problem of finding efficiently a representative point for each connected component of S0. For this purpose we exhibit explicit polynomial equations th ..."
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Cited by 30 (13 self)
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Let S0 be a smooth and compact real variety given by a reduced regular sequence of polynomials f1,..., fp. This paper is devoted to the algorithmic problem of finding efficiently a representative point for each connected component of S0. For this purpose we exhibit explicit polynomial equations that describe the generic polar varieties of S0. This leads to a procedure which solves our algorithmic problem in time that is polynomial in the (extrinsic) description length of the input equations f1,..., fp and in a suitably introduced, intrinsic geometric parameter, called the degree of the real interpretation of the given equation system f1,..., fp.
Decomposition plans for geometric constraint systems
 J. Symbolic Computation
, 2001
"... A central issue in dealing with geometric constraint systems for CAD/CAM/CAE is the generation of an optimal decomposition plan that not only aids efficient solution, but also captures design intent and supports conceptual design. Though complex, this issue has evolved and crystallized over the past ..."
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Cited by 30 (0 self)
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A central issue in dealing with geometric constraint systems for CAD/CAM/CAE is the generation of an optimal decomposition plan that not only aids efficient solution, but also captures design intent and supports conceptual design. Though complex, this issue has evolved and crystallized over the past few years, permitting us to take the next important step: in this paper, we formalize, motivate and explain the decomposition–recombination (DR)planning problem as well as several performance measures by which DRplanning algorithms can be analyzed and compared. These measures include: generality, validity, completeness, Church–Rosser property, complexity, best and worstchoice approximation factors, (strict) solvability preservation, ability to deal with underconstrained systems, and ability to incorporate conceptual design decompositions specified by the designer. The problem and several of the performance measures are formally defined here for the first time—they closely reflect specific requirements of CAD/CAM applications. The clear formulation of the problem and performance measures allow us to precisely analyze and compare existing DRplanners that use two wellknown types of decomposition methods: SR (constraint shape recognition) and MM (generalized maximum matching) on constraint graphs. This analysis additionally serves to illustrate and provide intuitive substance to the newly formalized measures. In Part II of this article, we use the new performance measures to guide the development of a new DRplanning algorithm which excels with respect to these performance measures. c ○ 2001 Academic Press 1.
Generalized polar varieties: Geometry and algorithms
, 2004
"... Let V be a closed algebraic subvariety of the n–dimensional projective space over the complex or real numbers and suppose that V is non–empty and equidimensional. The classic notion of a polar variety of V associated with a given linear subvariety of the ambient space of V was generalized and motiva ..."
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Cited by 26 (8 self)
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Let V be a closed algebraic subvariety of the n–dimensional projective space over the complex or real numbers and suppose that V is non–empty and equidimensional. The classic notion of a polar variety of V associated with a given linear subvariety of the ambient space of V was generalized and motivated in [2]. As particular instances of this notion of a generalized polar variety one reobtains the classic one and an alternative type of a polar varietiy, called dual. As main result of the present paper we show that for a generic choice of their parameters the generalized polar varieties of V are empty or equidimensional and smooth in any regular point of V. In the case that the variety V is affine and smooth and has a complete intersection ideal of definition, we are able, for a generic parameter choice, to describe locally the generalized polar varieties of V by explicit equations. Finally, we indicate how this description may be used in order to design in
Computational Real Algebraic Geometry
 HANDBOOK OF DISCRETE AND COMPUTATIONAL GEOMETRY, CHAPTER 29
, 1997
"... Computational real algebraic geometry studies various algorithmic questions dealing with the real solutions of a system of equalities, inequalities, and inequations of polynomials over the real numbers. This emerging field is largely motivated by the power and elegance with which it solves a broad ..."
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Cited by 25 (5 self)
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Computational real algebraic geometry studies various algorithmic questions dealing with the real solutions of a system of equalities, inequalities, and inequations of polynomials over the real numbers. This emerging field is largely motivated by the power and elegance with which it solves a broad and general class of problems arising in robotics, vision, computer aided design, geometric theorem proving, etc. The algorithmic problems that arise in this context are formulated as decision problems for the firstorder theory of reals and the related problems of quantifier elimination (Section 1). The associated geometric structures are then examined via an exploration of the semialgebraic sets (Section 2). Algorithmic problems for semialgebraic sets are considered next. In particular, there is a discussion of real algebraic numbers and their representation which relies on such classical theorems as Stu
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 25 (4 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.