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469
A Scheme for Integrating Concrete Domains into Concept Languages
, 1991
"... A drawback which concept languages based on klone have is that all the terminological knowledge has to be defined on an abstract logical level. In many applications, one would like to be able to refer to concrete domains and predicates on these domains when defining concepts. Examples for such conc ..."
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Cited by 280 (21 self)
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A drawback which concept languages based on klone have is that all the terminological knowledge has to be defined on an abstract logical level. In many applications, one would like to be able to refer to concrete domains and predicates on these domains when defining concepts. Examples for such concrete domains are the integers, the real numbers, or also nonarithmetic domains, and predicates could be equality, inequality, or more complex predicates. In the present paper we shall propose a scheme for integrating such concrete domains into concept languages rather than describing a particular extension by some specific concrete domain. We shall define a terminological and an assertional language, and consider the important inference problems such as subsumption, instantiation, and consistency. The formal semantics as well as the reasoning algorithms are given on the scheme level. In contrast to existing klone based systems, these algorithms will be not only sound but also complete. The...
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 235 (30 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.
A proof of the Kepler conjecture
 Math. Intelligencer
, 1994
"... This section describes the structure of the proof of ..."
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Cited by 200 (13 self)
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This section describes the structure of the proof of
COMPUTATION OF EQUILIBRIA in Finite Games
, 1996
"... We review the current state of the art of methods for numerical computation of Nash equilibria for nitenperson games. Classical path following methods, such as the LemkeHowson algorithm for two person games, and Scarftype fixed point algorithms for nperson games provide globally convergent metho ..."
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Cited by 143 (1 self)
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We review the current state of the art of methods for numerical computation of Nash equilibria for nitenperson games. Classical path following methods, such as the LemkeHowson algorithm for two person games, and Scarftype fixed point algorithms for nperson games provide globally convergent methods for finding a sample equilibrium. For large problems, methods which are not globally convergent, such as sequential linear complementarity methods may be preferred on the grounds of speed. None of these methods are capable of characterizing the entire set of Nash equilibria. More computationally intensive methods, which derive from the theory of semialgebraic sets are required for finding all equilibria. These methods can also be applied to compute various equilibrium refinements.
Motion Planning in the Presence of Moving Obstacles
, 1985
"... This paper investigates the computational complexity of planning the motion of a body B in 2D or 3D space, so as to avoid collision with moving obstacles of known, easily computed, trajectories. Dynamic movement problems are of fundamental importance to robotics, but their computational compl ..."
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Cited by 126 (10 self)
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This paper investigates the computational complexity of planning the motion of a body B in 2D or 3D space, so as to avoid collision with moving obstacles of known, easily computed, trajectories. Dynamic movement problems are of fundamental importance to robotics, but their computational complexity has not previously been investigated. We provide evidence that the 3D dynamic movement problem is intractable even if B has only a constant number of degrees of freedom of movement. In particular, we prove the problem is PSPACEhard if B is given a velocity modulus bound on its movements and is NP hard even if B has no velocity modulus bound, where in both cases B has 6 degrees of freedom. To prove these results we use a unique method of simulation of a Turing machine which uses time to encode configurations (whereas previous lower bound proofs in robotic motion planning used the system position to encode configurations and so required unbounded number of degrees of freedom)...
Efficient algorithms for geometric optimization
 ACM Comput. Surv
, 1998
"... We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, pruneandsearch techniques for linear progra ..."
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Cited by 117 (12 self)
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We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, pruneandsearch techniques for linear programming and related problems, and LPtype problems and their efficient solution. We then describe a variety of applications of these and other techniques to numerous problems in geometric optimization, including facility location, proximity problems, statistical estimators and metrology, placement and intersection of polygons and polyhedra, and ray shooting and other querytype problems.
Linear invariant generation using nonlinear constraint solving
 IN COMPUTER AIDED VERIFICATION
, 2003
"... We present a new method for the generation of linear invariants which reduces the problem to a nonlinear constraint solving problem. Our method, based on Farkas' Lemma, synthesizes linear invariants by extracting nonlinear constraints on the coefficients of a target invariant from a program. ..."
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Cited by 108 (14 self)
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We present a new method for the generation of linear invariants which reduces the problem to a nonlinear constraint solving problem. Our method, based on Farkas' Lemma, synthesizes linear invariants by extracting nonlinear constraints on the coefficients of a target invariant from a program. These constraints guarantee that the linear invariant is inductive. We then apply existing techniques, including specialized quantifier elimination methods over the reals, to solve these nonlinear constraints. Our method has the advantage of being complete for inductive invariants. To our knowledge, this is the first sound and complete technique for generating inductive invariants of this form. We illustrate the practicality of our method on several examples, including cases in which traditional methods based on abstract interpretation with widening fail to generate sufficiently strong invariants.
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 105 (11 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...
Applications of parametric searching in geometric optimization
 J. Algorithms
, 1994
"... z Sivan Toledo x ..."
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Towards Exact Geometric Computation
, 1994
"... Exact computation is assumed in most algorithms in computational geometry. In practice, implementors perform computation in some fixedprecision model, usually the machine floatingpoint arithmetic. Such implementations have many wellknown problems, here informally called "robustness issues&quo ..."
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Cited by 94 (6 self)
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Exact computation is assumed in most algorithms in computational geometry. In practice, implementors perform computation in some fixedprecision model, usually the machine floatingpoint arithmetic. Such implementations have many wellknown problems, here informally called "robustness issues". To reconcile theory and practice, authors have suggested that theoretical algorithms ought to be redesigned to become robust under fixedprecision arithmetic. We suggest that in many cases, implementors should make robustness a nonissue by computing exactly. The advantages of exact computation are too many to ignore. Many of the presumed difficulties of exact computation are partly surmountable and partly inherent with the robustness goal. This paper formulates the theoretical framework for exact computation based on algebraic numbers. We then examine the practical support needed to make the exact approach a viable alternative. It turns out that the exact computation paradigm encomp...