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18
REDLOG Computer Algebra Meets Computer Logic
 ACM SIGSAM Bulletin
, 1996
"... . redlog is a package that extends the computer algebra system reduce to a computer logic system, i.e., a system that provides algorithms for the symbolic manipulation of firstorder formulas over some temporarily fixed language and theory. In contrast to theorem provers, the methods applied know a ..."
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Cited by 104 (30 self)
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. redlog is a package that extends the computer algebra system reduce to a computer logic system, i.e., a system that provides algorithms for the symbolic manipulation of firstorder formulas over some temporarily fixed language and theory. In contrast to theorem provers, the methods applied know about the underlying algebraic theory and make use of it. Though the focus is on simplification, parametric linear optimization, and quantifier elimination, redlog is designed as a generalpurpose system. We describe the functionality of redlog as it appears to the user, and explain the design issues and implementation techniques. ? The second author was supported by the dfg (Schwerpunktprogramm: Algorithmische Zahlentheorie und Algebra) 1 Introduction redlog stands for reduce logic system. It provides an extension of the computer algebra system (cas) reduce to a computer logic system (cls) implementing symbolic algorithms on firstorder formulas w.r.t. temporarily fixed firstorder languag...
Quantifier Elimination for Real Algebra  the Quadratic Case and Beyond
 AAECC
, 1993
"... . We present a new, "elementary" quantifier elimination method for various special cases of the general quantifier elimination problem for the firstorder theory of real numbers. These include the elimination of one existential quantifier 9x in front of quantifierfree formulas restricted by a non ..."
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Cited by 42 (4 self)
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. We present a new, "elementary" quantifier elimination method for various special cases of the general quantifier elimination problem for the firstorder theory of real numbers. These include the elimination of one existential quantifier 9x in front of quantifierfree formulas restricted by a nontrivial quadratic equation in x (the case considered also in [7]), and more generally in front of arbitrary quantifierfree formulas involving only polynomials that are quadratic in x. The method generalizes the linear quantifier elimination method by virtual substitution of test terms in [9]. It yields a quantifier elimination method for an arbitrary number of quantifiers in certain formulas involving only linear and quadratic occurences of the quantified variables. Moreover, for existential formulas ' of this kind it yields sample answers to the query represented by '. The method is implemented in reduce as part of the redlog package (see [4, 5]). Experiments show that the method is appl...
Real Quantifier Elimination in Practice
 Algorithmic Algebra and Number Theory
, 1998
"... We give a survey of three implemented real quantifier elimination methods: partial cylindrical algebraic decomposition, virtual substitution of test terms, and a combination of Gröbner basis computations with multivariate real root counting. We examine the scope of these implementations for applicat ..."
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Cited by 32 (6 self)
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We give a survey of three implemented real quantifier elimination methods: partial cylindrical algebraic decomposition, virtual substitution of test terms, and a combination of Gröbner basis computations with multivariate real root counting. We examine the scope of these implementations for applications in various fields of science, engineering, and economics.
A new approach for automatic theorem proving in real geometry
 Journal of Automated Reasoning
, 1998
"... Abstract. We present a new method for proving geometric theorems in the real plane or higher dimension. The method is derived from elimination set ideas for quantifier elimination in linear and quadratic formulas over the reals. In contrast to other approaches, our method can also prove theorems who ..."
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Cited by 30 (15 self)
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Abstract. We present a new method for proving geometric theorems in the real plane or higher dimension. The method is derived from elimination set ideas for quantifier elimination in linear and quadratic formulas over the reals. In contrast to other approaches, our method can also prove theorems whose complex analogues fail. Moreover, the problem formulation may involve order inequalities. After specification of independent variables, nondegeneracy conditions are generated automatically. Moreover, when trying to prove conjectures that – apart from nondegeneracy conditions – do not hold in the claimed generality, missing premises are found automatically. We demonstrate the applicability of our method to nontrivial examples. Key words: real quantifier elimination, real geometry, automatic theorem proving over the reals. 1.
Computational Geometry Problems in REDLOG
 AUTOMATED DEDUCTION IN GEOMETRY
, 1998
"... We solve algorithmic geometrical problems in real 3space or the real plane arising from applications in the area of cad, computer vision, and motion planning. The problems include parallel and central projection problems, shade and cast shadow problems, reconstruction of objects from images, offset ..."
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Cited by 19 (11 self)
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We solve algorithmic geometrical problems in real 3space or the real plane arising from applications in the area of cad, computer vision, and motion planning. The problems include parallel and central projection problems, shade and cast shadow problems, reconstruction of objects from images, offsets of objects, Voronoi diagrams of a finite families of objects, and collision of moving objects. Our tools are real elimination algorithms implemented in the reduce package redlog. In many cases the problems can be solved uniformly in unspecified parameters. The power of the method is illustrated by examples many of which have been outside the scope of real elimination methods so far.
Solving Parametric Polynomial Equations And Inequalities By Symbolic Algorithms
 COMPUTER ALGEBRA IN SCIENCE AND ENGINEERING
, 1995
"... The talk gives a survey on some symbolic algorithmic methods for solving systems of algebraic equations with special emphasis on parametric systems. Besides complex solutions I consider also real solutions of systems including inequalities. The techniques described include the Euclidean algorithm, ..."
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Cited by 13 (1 self)
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The talk gives a survey on some symbolic algorithmic methods for solving systems of algebraic equations with special emphasis on parametric systems. Besides complex solutions I consider also real solutions of systems including inequalities. The techniques described include the Euclidean algorithm, Gröbner bases, characteristic sets, univariate and multivariate SturmSylvester theorems, comprehensive Grobner bases and elimination methods for parametric optimization problems. Some examples illustrate the use of symbolic algorithms for the solution of parametric systems.
The Loop Parallelizer LooPo
 Proc. Sixth Workshop on Compilers for Parallel Computers, volume 21 of Konferenzen des Forschungszentrums Jülich
, 1996
"... . We report on a prototype for testing different methods of spacetime mapping loop nests. LooPo admits perfect or imperfect loop nests in a number of imperative languages, takes data dependences from the user or derives them itself from the source code, provides a choice of strategies for sched ..."
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Cited by 11 (2 self)
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. We report on a prototype for testing different methods of spacetime mapping loop nests. LooPo admits perfect or imperfect loop nests in a number of imperative languages, takes data dependences from the user or derives them itself from the source code, provides a choice of strategies for scheduling and allocating the loop nest's iterations, and produces synchronous or asynchronous parallel target code for sharedmemory or distributedmemory machines. 1 Why LooPo? LooPo is not meant to be yet another parallelizing compiler. It is a prototype system whose purpose is to assist us in the research on and evaluation of spacetime mapping methods for loop parallelization. To that end, it implements the complete path from executable source code to executable target code, with switches for choosing alternative methods. At present, we provide several inequality solving methods, several schedulers and several methods of code generation, one dependence analyzer (we are working on a second...
Semilinear motion planning in REDLOG
 AAECC
, 1999
"... We study a new type of motion planning problem in dimension 2 and 3 via linear and quadratic quantifier elimination. The object to be moved and the free space are both semilinear sets with no convexity assumptions. The admissible motions are finite continuous sequences of translations along prescrib ..."
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Cited by 7 (1 self)
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We study a new type of motion planning problem in dimension 2 and 3 via linear and quadratic quantifier elimination. The object to be moved and the free space are both semilinear sets with no convexity assumptions. The admissible motions are finite continuous sequences of translations along prescribed directions. When the number of translations is bounded in advance, then the corresponding path finding problem can be modelled and solved as a linear quantifier elimination problem. Moreover the problem to find a shortest or almost shortest admissible path can be modelled as a special quadratic quantifier elimination problem. We give upper complexity bounds on these problems, report experimental results using the elimination facilities of the redlog package of reduce, and indicate a possible application.
Deciding LinearTrigonometric Problems
 In ISSAC 2000
, 2000
"... In this paper, we present a decision procedure for certain lineartrigonometric problems for the reals and integers formalized in a suitable firstorder language. The inputs are restricted to formulas, where all but one of the quanti ed variables occur linearly and at most one occurs both linearly a ..."
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Cited by 5 (1 self)
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In this paper, we present a decision procedure for certain lineartrigonometric problems for the reals and integers formalized in a suitable firstorder language. The inputs are restricted to formulas, where all but one of the quanti ed variables occur linearly and at most one occurs both linearly and in a specific trigonometric function. Moreover we allow in addition the integerpart operation in formulas. Besides ordinary quantifiers, we allow also counting quantifiers. Furthermore we also determine the qualitative structure of the connected components of the satisfaction set of the mixed lineartrigonometric variable. We also consider the decision of these problems in subfields of the real algebraic numbers.