Weispfenning V., The Complexity of Linear Problems in Fields, Journal of Symbolic Computation, 5 (1-2), pp. 3-27 (1988).  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... quantified variables have mainly be handled by symbolic methods, among which one may single out Cylindrical Algebraic Decomposition (CAD) by Collins [14] Cylindrical Trigonometric Decomposition (CTD) by Pau and Schicho [30] and virtual substitution of parametric test points by Weispfenning [44, 16]. Some numerical methods based on a bisection scheme also exist that solve constraint systems with universally quantified variables by computing an inner approximation of the underlying relation [19, 26, 34] However, all these methods have strong requirements on the form of the constraints they ....

Volker WEISPFENNING. The complexity of linear problems in fields. Journal of Symbolic Computation, 5(1--2) :3--27, February--April ####.

.... sentences, the theory consisting of rational variables, rational constants, Gamma; 8; 9) are used in the tests of protocols, timing verification of hardware, and so on[1, 2] There has been a lot of work on the problem of finding precise time and space complexity of this algorithm[3, 4, 5, 6, 7]. Ferrante and Rackoff have proposed a decision algorithm for PRP sentences which runs in rfl d n ffid(2b 1) a time, where fl and ffi are constants, and r; n; d; a; b denote the maximum bit length of coefficients, the number of inequalities, the number of variables, the number of quantifier ....

V. Weispfenning : "The complexity of linear problems in fields," J. Symbolic Computation 5 pp.3-27, 1988.

....systems: SAC, now SAC II ALDES. Important improvements of the CAD method have been made by Collins and Hong [CH91] resulting in the partial CAD implemented in Hong s QEPCAD package [HCJE93] Weispfenning introduced an alternative approach to quantifier elimination first for linear formulas [Wei88]. This could be extended to arbitrary degrees [LW93, Wei96b, Wei94b] For degrees 1 and 2, the methods have been successfully implemented in the REDUCE package REDLOG written by the author and others [DS95a, DS96] Despite the bad theoretical complexity [Wei88, DH88] and the degree restrictions ....

.... elimination first for linear formulas [Wei88] This could be extended to arbitrary degrees [LW93, Wei96b, Wei94b] For degrees 1 and 2, the methods have been successfully implemented in the REDUCE package REDLOG written by the author and others [DS95a, DS96] Despite the bad theoretical complexity [Wei88, DH88] and the degree restrictions REDLOG turned out suitable for solving a number of practical problems [Wei94a, Wei96a] Meanwhile it is used commercially as part of an error diagnosis system for physical networks. We have now started a second implementation of the test term method in the C language ....

Volker Weispfenning. The complexity of linear problems in fields. Journal of Symbolic Computation, 5(1):3--27, February 1988.

....planning, term rewriting systems, algebraic topology, and computer graphics. In order to obtain a more eOEcient quantier elimination method for linear case, some other algorithms were elaborated. In [LJL91] some geometric techniques are used to perform quantier elimination in the linear case. In [Wei88] some work is done to determine the exact complexity of linear elimination procedures in dioeerent classes of elds. Wei83] describes quantier elimination for linear formulae, and [Wei96] for quadratic formulae. 1.1.2 Lazard s method The method of Lazard aims at computing the solutions of an ....

V. Weispfenning. The Complexity of Linear Problems in Fields. Journal of Symbolic Computation, 5(1&2):327, 1988.

....cases. Recently this implementation has been ported to the massively parallel IBM SP 2 machine (see Hong, et al. 24] and problems have shown a significant speedup. Recently, several new interesting approaches to QE for special problems have been developed: for linear problems, see Weispfenning [52], Loos and Weispfenning [34] for quadratic problems, see Weispfenning [53] and for cubic problems, see Weispfenning [55] These algorithms are able to treat problems where the quantified variables have low degree, while the non quantified variables can have arbitrary degree. The preliminary ....

V. Weispfenning, The complexity of linear problems in fields, J. Symb. Comp., 5 (1988), pp. 3--27.

....second order stable stabilizing compensator does exist. To find a particular compensator and to reduce the CPU time require to find a solution, the following supplementary techniques were used. Note here that all polynomials in (4. 23) are linear in A which allows us to apply Weispfenning s method (Weispfenning, 1988) to the elimination of 9A from (4.23) The result of the elimination has been simplified using REDLOG (Sturm, 1994) package to eliminate equations (we are interested only in solutions A 0 ; B 0 ; D 0 for which exists neighborhood of this point such that all point in the neighborhood are also ....

Weispfenning, V. (1988). The complexity of linear problems in fields. J. Symb. Comp., 5:3--27.

....engineering can be reduced to the problem of testing positiveness of polynomials. In 1930, Tarski [33, 34] showed that the problem is decidable. In fact, he gave a decision method for a more general problem than just testing positiveness. Since then, many improvements and new methods were proposed [7, 1, 26, 2, 3, 27, 13, 35, 30, 14, 15, 16, 9, 29, 17, 28, 6, 20, 19, 18, 25, 36, 8, 21, 22, 5, 12, 23]. However, these methods are computationally expensive due to their generality. Naturally one is interested in devising more efficient methods for the sub problem: testing positiveness. But then, this sub problem turns out to be still difficult. Thus, several authors (mainly from the field of term ....

V. Weispfenning. The complexity of linear problems in fields. Journal of Symbolic Computation, 5, 1988.

....This approach has been improved considerably by Collins and Hong resulting in partial cad, cf. CH91] which has been implemented by Hong et al. in the qepcad package. Besides the cad approach, the implementation of quantifier elimination procedures using test point ideas of Weispfenning, cf. [Wei88, LW93, Wei94, Wei97a], is under development since 1992. The latest implementation is part of the reduce package redlog developed by the author together with A. Dolzmann, cf. DS96, DS97a] redlog is freely available to the scientific community. 1 In general, the test point method can cope better than cad with input ....

....hydraulic networks. In Section 7 we finally summarize and evaluate our results. All computations have been performed on a sun Ultra 1 Model 140 workstation using 32 MB of memory. 2 The Test Point Method The elimination method we use dates back to a theoretical paper by Weispfenning in 1988, cf. [Wei88]. During the last five years a lot of theoretical work has been done to improve the method, cf. LW93, Wei94, DSW96, DS97b, Wei97a] After promising experimental implementations by Burhenne in 1990, cf. Bur90] and by the author in 1992, the method has been efficiently reimplemented within the ....

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Volker Weispfenning. The complexity of linear problems in fields. Journal of Symbolic Computation, 5(1--2):3--27, February--April 1988.

....order stable stabilizing compensator does exist. To find a particular compensator and to reduce the CPU time require to find a solution, the following supplementary techniques were used. Note here that all polynomials in (4.1. 19) are linear in A which allows us to apply Weispfenning s method [22] to the elimination of 9A from (4.1.19) The result of the elimination has been simplified using REDLOG [23] package to eliminate equations (we are interested only in solutions A 0 ; B 0 ; D 0 for which exists neighborhood of this point such that all point in the neighborhood are also solution and ....

V. Weispfenning, The Complexity of Linear problems in Fields, J. Symb. Comp., 5, (1988), 3--27.

....methods. Recently, implementations of quantifier elimination have been able to solve problems of interesting size in science, engineering, and also in economics, namely in operations research, cf. DSW97] and the references there. Quantifier elimination by virtual substitution of test terms [Wei88, LW93, Wei94, Wei97a] plays a crucial role for this success, in particular for problems involving numerous parameters. The purpose of our work is to study parallelization and distribution of the virtual substitution method. Here, distribution refers to parallelization on distributed memory hosts ....

....has gone through numerous improvements [McC88, Hon90, Hon92] resulting in partial cad [CH91] implemented in Hong s qepcad program based on saclib [BCE 93] a C version of sac 2. Around 1988 it has been shown that real quantifier elimination is inherently hard for some problem classes [DH88, Wei88] Thus the attention turned to special procedures for restricted problem classes, where the elimination procedures can be tuned to the structure of the problem. The focus was on considering formulas in which the occurrence of quantified variables is restricted to low degrees [Hon93b, Hon93c, ....

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Volker Weispfenning. The complexity of linear problems in fields. Journal of Symbolic Computation, 5(1--2):3--27, February--April 1988.

....[DS97] The prenex normal form computation minimizes the number of quantifier changes. See Chapter 5 [Normal Forms] page 25. Quantifier elimination computes quantifier free equivalents for given first order formulas. For ofsf and dvfsf we use a technique based on elimination set ideas [Wei88]. The ofsf implementation is restricted to at most quadratic occurrences of the quantified variables, but includes numerous heuristic strategies for coping with higher degrees. See [LW93] Wei97] for details on the method. The dvfsf implementation is restricted to formulas that are linear in the ....

....(b 0 and a b c = 0) and (d 0 and a c d 0) Chapter 3: Format and Handling of Formulas 10 3.4 DVFSF Operators Discretely valued fields are implemented as a one sorted language using the operators , and , which encode #, #= in the value group, respectively. For details see [Wei88], Stu98] or [DS99] Binary Infix operator equal Binary Infix operator neq Binary Infix operator div Binary Infix operator sdiv Binary Infix operator assoc Binary Infix operator nassoc The above operators may also be written as = and , respectively. Integer reciprocals in ....

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Volker Weispfenning. The complexity of linear problems in fields. Journal of Symbolic Computation, 5(1):3--27, February, 1988.

....equivalent in Z the length of 0 n grows doubly exponentially in n and hence in length( n ) This is even the case if integers are coded in unary notation in n and in binary notation in 0 n . Proof. Let be a new constant. By a slight extension of a result of Fischer Rabin [FiR74] see [W88], Lemma 5.5) there is a sequence of formulas f n (x)g in the language L 0 = f0; Gammag such that in any additive, abelian group G with a distinguished element of infinite order: n(x) holds in G iff x = i for some 0 i 2 2 n . Moreover the length of n (x) grows linearly in ....

....(x)g in the language L 0 = f0; Gammag such that in any additive, abelian group G with a distinguished element of infinite order: n(x) holds in G iff x = i for some 0 i 2 2 n . Moreover the length of n (x) grows linearly in n. In order to apply the topological technique of [W88] we have to pass to a non standard model of PA with modified topology: Let G : R Theta Z be the lexicographical product of the ordered additive abelian groups R and Z and put : 1; 0) 1 : 0; 1) Then by Presburger s axioms, G is a model of PA. For fixed n put (x) n(x) and let 0 ....

V. Weispfenning, The complexity of linear problems in fields, J. Symb. Comp. 5, 3-27 (1988).

....complexity of the general quantifier elimination problem for the reals stimulated research concerning more specialized procedures for subproblems given by input formulas of special forms. The case of input formulas in which all quantified variables occur only linearly has been handled in [13, 9]. Elimination of a single quantified variable that is restricted by a non trivial quadratic equation has been treated in [7] In both cases the specialized elementary methods perform significantly better than the general purpose method in [3, 6] in some test examples. Notice, however, that the ....

....perform significantly better than the general purpose method in [3, 6] in some test examples. Notice, however, that the theoretical worst case complexity of the quantifier elimination problem for the linear case is the same as for the general case up to undetermined multiplicative constants (see [12, 13]) In the present note we explore an extension of the ideas in [9] from the linear to the quadratic case. In the linear case (i.e. the case, where all quantified variables occur only linearly in the input formula) the elimination of a quantifier 9x was achieved by replacing 9x by a finite ....

V. Weispfenning, The complexity of linear problems in fields, J. Symb. Comp. 5 (1988), pp. 3-27.

....development. On one hand cad has gone through numerous improvements, cf. McC88] Hon90,Hon92] resulting in partial cad, cf. CH91] implemented in Hong s qepcad program. On the other hand it had been shown that real quantifier elimination is inherently hard for some problem classes, cf. DH88,Wei88] Thus the attention turned to special procedures for restricted problem classes, where the elimination procedures can be tuned to the structure of the problem. The focus was on considering formulas in which the occurrence of quantified variables is restricted to low degrees, cf. ....

....cf. DH88,Wei88] Thus the attention turned to special procedures for restricted problem classes, where the elimination procedures can be tuned to the structure of the problem. The focus was on considering formulas in which the occurrence of quantified variables is restricted to low degrees, cf. Wei88,LW93,Hon93a,Hon93b,GV93,Wei94b,Wei97a] This was initiated by the third author in 1988. In his virtual substitution method the number of parameters plays a minor role for the complexity. The worst case complexity of the method is doubly exponential only in the number of the quantifier blocks of ....

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Volker Weispfenning. The complexity of linear problems in fields. Journal of Symbolic Computation, 5(1--2):3--27, February--April 1988.

....types of applications: 1 Simulation of networks of components with piecewise linear (or piecewise quadratic) behaviour, 2 Not necessarily convex parametric linear programming. We present linear elimination algorithms for these problems that have been optimized and extended from earlier versions in (Weispfenning 1988, Loos Weispfenning 1993, Weispfenning 1996a, Burhenne 1990, Kappert 1995) to an extent that serious nonacademic application problems can be handled by these methods. In fact, since March 1995, reduce implementations of these methods are employed successfully in a commercial software system for ....

Weispfenning, V. (1988). The complexity of linear problems in fields, J. Symb. Comp. 5, 3-27.

....quadratic programming problems. The results indicate that the scope of our method goes significantly beyond that of the Fourier Motzkin method and includes problems of interesting size. The method is derived from a quantifier elimination method for linear problems in the reals developed in [Weispfenning 1] optimized in [Loos Weispfenning] and implemented in [Sturm] The basic idea is to test feasibility of a system of parametric linear constraints using a finite selection of parametric test points. In principle, this idea dates back to Th. Skolem in the 1920 s; in many variants and disguises ....

....n: x 2 i Gamma x i = 0 for 1 i n x n 1 = 1 x n i 1 = x n i x n i for 1 i n x 1 x n 1 : x n x 2n Gamma u = 0 Then S is feasible iff x i 2 f0; 1g for 1 i n and x n i = 2 i Gamma1 for 1 i n, and so iff u is an integer with 0 u 2 n Gamma 1. It has been shown in [Weispfenning 1] section 6, that any boolean combination of polynomial inequalities in u defining the set of all integers between 0 and 2 n Gamma 1 is necessarily of length exponential in n. 4 Linear optimization for parametric data In this section we generalize the results of section 2 for linear ....

V. Weispfenning, The complexity of linear problems in fields, J. Symb. Comp. 5 (1988), pp. 3-27.

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Weispfenning V., The Complexity of Linear Problems in Fields, Journal of Symbolic Computation, 5 (1-2), pp. 3-27 (1988).

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Volker Weispfenning. The complexity of linear problems in fields. Journal of Symbolic Computation, 5(1&2):3--27, February--April 1988.

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Weispfenning V., The Complexity of Linear Problems in Fields, Journal of Symbolic Computation, 5 (1-2), pp.3-27, (1988).

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Volker Weispfenning. The complexity of linear problems in fields. Journal of Symbolic Computation, 5(1&2):3--27, February--April 1988.

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Volker Weispfenning. The complexity of linear problems in fields. Journal of Symbolic Computation, 5(1&2):3--27, February--April 1988.

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V. Weispfenning. The complexity of linear problems in fields. Journal of Symbolic Computation, 5(1--2):3--27, 1988.

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Volker Weispfenning. The complexity of linear problems in fields. Journal of Symbolic Computation, 5(1&2):3--27, February--April 1988.

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V. Weispfenning. The complexity of linear problems in fields. Journal of Symbolic Computation, 5(1--2):3--27, 1988. 15

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Volker Weispfenning. The complexity of linear problems in fields. Journal of Symbolic Computation, 5(1&2):3--27, February--April 1988.

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