J. Heintz, M.-F. Roy, and P. Solern. On the complexity of semi-algebraic sets. In Proceedings IFIP'89 San Francisco, North-Holland, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....improvement over Tarski s algorithm was due to Collins [Col75] it has a doubly exponential time complexity in the number of variables appearing in the sentence. Further improvements have been made by a number of researchers (Grigor ev Vorobjov [Gri88, GV88] Canny [Can88b, Can93] Heintz et al. [HRS89, HRS90], Renegar [Ren92a,b,c] and most recently by Basu et al. BPR98] In the following, we assume that our Tarski sentence is presented in its prenex form: where the Q i s form a sequence of alternating quanti ers (i.e. 8 or 9, with every pair of consecutive quanti ers distinct) with ....

J. Heintz, M.-F. Roy, and P. Solerno. On the complexity of semi-algebraic sets. In Proc. Internat. Fed. Info. Process. 89, pages 293-298. North-Holland, San Francisco, 1989.

....so, the problem size which can be solved with such algorithms is limited. In [14] Grigoriev and Vorobjov proposed an algorithm for deciding the emptiness of a semialgebraic set with a single exponential complexity in the number of variables. In this method as well as in most of its variants (see [20, 7, 15, 4, 5, 25]) the key idea is to apply deformations so that the projection critical points with respect to one coordinate de ne a nite set that meets every semi algebraic connected component of the deformed variety. In [4, 5, 25] the authors take, in Universit e de Paris VI, France y LORIA, ....

J. Heintz, M.-F. Roy, P. Solern o , On the Complexity of Semi-Algebraic Sets, Proc. IFIP 89, San Francisco. North-Holland 293-298 (1989).

....Collins algorithm is the fastest among them for inputs which can be decided in a reasonable amount of time. 1 Introduction Since Tarski [33] gave the first decision algorithm for the first order theory of the reals, many other algorithms with better theoretical complexities have been proposed [32, 9, 17, 5, 10, 4, 13, 15, 14, 7, 16, 29, 30, 31]. In this paper we compare the complexities of the following three algorithms on existential sentences: Collins [10] Grigor ev and Vorobjov s [15, 14] and Renegar s [29, 30, 31] 1 Let n be the number of variables in the input sentence, m the number of polynomials, d the degree 1 , and L ....

J. Heintz, M-F. Roy, and P. Solern'o. On the complexity of semialgebraic sets. In Proc. IFIP, pages 293--298, 1989.

....improvement over Tarski s algorithm is due to Collins [Col75] and has a doubly exponential time complexity in the number of variables appearing in the sentence. Further improvements have been made by a number of researchers (Grigor ev Vorobjov [Gri88, GV88] Canny [Can88b, Can93] Heintz et al. [HRS89, HRS90], Renegar [Ren92a,b,c] and most recently by Basu et al. BPR95] In the following, we assume that our Tarski sentence is presented as shown below, in its prenex form: Q 1 x [1] Q 2 x [2] Delta Delta Delta (Q x [ x [1] x [ where the Q i s form a sequence ....

J. Heintz, M.-F. Roy and P. Solern'o. On the Complexity of Semi-Algebraic Sets. In Proceedings IFIP 89 , pp. 293--298, San Francisco. North-Holland, 1989.

....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 ....

J. Heintz, M-F. Roy, and P. Solern'o. On the complexity of semialgebraic sets. In G. X. Ritter, editor, Proc. IFIP, pages 293--298. North-Holland, 1989.

....theorem proving and discovery, termination proof of term rewrite systems, constraint solving in logic programming, etc. During 1930 1950, Tarski [34] found the first quantifier elimination algorithm in this theory. Since then various improvements and new methods have been devised and analyzed [33, 8, 17, 5, 9, 1, 28, 2, 4, 13, 15, 14, 7, 16, 29, 30, 31, 18, 19, 26, 20, 11, 25, 6, 23, 22]. In this paper we investigate the parallelization of the algorithm which was originally devised by Collins [9] and improved by the author [20] In [32] Saunders, Lee, and Abdali report their work on parallelizing Collins original algorithm on a shared memory machine, achieving about 50 ....

J. Heintz, M-F. Roy, and P. Solern'o. On the complexity of semialgebraic sets. In Proc. IFIP, pages 293--298, 1989.

....from any B 0. y The question can be easily formulatedas a sentence in the first order theory of real closed fields. Thus, in principle, we can use any decision procedure for the theory (Tarski, 1951; Collins, 1975; Arnon, 1981; McCallum, 1984; Grigor ev, 1988; Canny, 1988; Weispfenning, 1988; Heintz et al. 1989; Hong, 1990; Collins and Hong, 1991; Renegar, 1992) to check the existence of bounds. However, since this is a very structured and special question, one can naturally find a special method which is more efficient than the general methods. In (Lankford, 1976) a special method is given (using ....

Heintz, J., Roy, M.-F., Solern'o, P. (1989). On the complexity of semialgebraic sets. In Ritter, G. X., editor, Proc. IFIP, pages 293--298. North-Holland.

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J. Heintz, M.-F. Roy, P. Solern' o On the Complexity of SemiAlgebraic Sets, Proc. IFIP 89, San Francisco. North- Holland 293-298 (1989). 60

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J. Heintz, M.-F. Roy, and P. Solern' o. On the complexity of semialgebraic sets. In Proc. IFIP San Francisco. North-Holland, pp. 293--298, 1989.

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J. Heintz, M.-F. Roy, and P. Solerno. On the complexity of semialgebraic sets. In G. Ritter, editor, Information Processing 89, Proceedings of the IFIP 11th World Computer Congress, San Francisco, USA, August 28 { September 1, 1989.

....to special equation systems with low value for the intrinsic parameter . On the other hand, even in worst case our algorithm improves upon the known d O(n) time procedures for the algorithmic problem under consideration, also in their most ecient versions [4] 5] see also [3] 12] 19] [39], 40] 41] 59] 60] 14] 13] However, this distinction does not become apparent when we measure complexities simply in terms of d and n (all mentioned algorithms have worst case complexities of type d O(n) but it becomes clearly visible when we use the B ezout number just ....

J. Heintz, M.{F. Roy, P. Solerno: On the complexity of semialgebraic sets, Proc. Information Processing 89 (IFIP 89) San Francisco 1989, G.X.Ritter, ed., North-Holland (1989) 293-298.

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J. Heintz, M.-F. Roy, P. Solern' o On the Complexity of SemiAlgebraic Sets, Proc. IFIP 89, San Francisco. North-Holland 293-298 (1989).

....is the same as determining f(1; k; k; s) is well known to be Theta( i s k j ) see [8] for example) This bound has played an important role in discrete and computational geometry for many years. For f(d; k; k; s) the best bound had been (sd) O(k) which was based on a result of Heintz [10]. Since the set of cells of sd hyperplanes is the same as the set of cells of s polynomials, each the product of d of the given linear polynomials, a lower bound of Omega Gamma i sd k j ) follows. This lower bound was recently shown to be an upper bound as well [14] For the case f(1; k; k ....

J. Heintz, M.-F. Roy, and P. Solern' o. On the complexity of semialgebraic sets. In Proc. IFIP San Francisco. North-Holland, pp. 293--298, 1989.

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J. Heintz, M.-F. Roy, P. Solern' o On the Complexity of Semi-Algebraic Sets, Proc. IFIP 89, San Francisco. North-Holland 293-298 (1989).

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J. Heintz, M.-F. Roy, and P. Solern. On the complexity of semi-algebraic sets. In Proceedings IFIP'89 San Francisco, North-Holland, 1989.

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J. Heintz, M.F. Roy, P. Solerno, On the complexity of semialgebraic sets, Proceedings IFIP 1989.

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J. Heintz, M.-F. Roy, P. Solerno, On the complexity of semialgebraic sets, Information Processing 89 (Proc. IFIP San Francisco), 293-298, North-Holland, New York, 1989. BIBLIOGRAPHY 68

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J. Heintz, M.F. Roy, P. Solerno, On the complexity of semialgebraic sets, Proceedings IFIP 1989.

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