S. Basu, R. Pollack, and M.-F. Roy. On the combinatorial and algebraic complexity of quantier elimination. In Proc. IEEE Symp. Foundations of Computer Science, Sante Fe, New Mexico, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....by the open interval constraints a ij 2 ( 1# 1) 3. The decision problem for the existential theory of the reals is solvable in s k 1 d O(k) arithmetic operations where k denotes the number of variables, s is the number of polynomial (in)equalities, and d is the highest polynomial degree [2]. This shows that for fixed k, a polynomial time algorithm is possible. In particular, stable matrix in interval family becomes polynomial time solvable if an a priori bound is given on the size of the matrix. The problems discussed in Corollary 1 also become polynomial time solvable when ....

Basu S., Pollack R., Roy M.-F., On the combinatorial and algebraic complexity of quantifier elimination, preprint, January 1995.

.... the real numbers, which is decidable according to a classic result of Tarski [71] Indeed, by the method of cylindrical algebraic decomposition (CAD) 20] this computation can be performed in time c for suitable constants c and c ( doubly exponential time ) Moreover, some more recent algorithms [34, 36, 60, 5] require only a time c ( singly exponential time ) Unfortunately, this computation seems at present to be unfeasible in practice even for n = 4. 17 3 The local half plane property For any element e E, the deletion of e from P is the polynomial pe on ground set E e that is obtained from P by ....

S. Basu, R. Pollack and M.-F. Roy, On the combinatorial and algebraic complexity of quantifier elimination, J. ACM 43 (1996), 1002 1045.

....techniques and on reducing the multivariate problem to easier univariate problems. These new approaches rely on the work of several groups of researchers: Grigor ev and Vorobjov [Gri88, GV88] Canny [Can88a, Can90] Heintz et al. HRS90] Renegar [Ren91, Ren92a, Ren92b, Ren92c] and Basu et al. [BPR96]. A few representative applications of computational algebra conclude this chapter (Section 33.7) 33.1 FIRST ORDER THEORY OF REALS The decision problem for the rst order theory of reals is to determine if a Tarski sentence in the rst order theory of reals is true or false. The quanti er ....

....(L log L log log L) m) n i 1) O(n i ) only if (x ) holds. Such a quanti er free formula takes the form (x ) f i;j (x ) T 0 where f i;j 2 R[x ] is a multivariate polynomial with real coecients. Signi cantly improved bounds were given by Basu et al. [BPR96] and are summarized as follows: I (m) J i (m) The total degrees of the polynomials f i;j (x ) are bounded by Nonetheless, comparing the above bounds to the bounds obtained in semilinear geometry , it appears that the combinatorial part of the complexity of both the ....

S. Basu, R. Pollack, and M.-F. Roy. On the combinatorial and algebraic complexity of quanti er elimination. J. Assoc. Comput. Mach., 43:1002-1045, 1996.

....This is one of the consequences of Mnev s universality theorem below. An upper bound for the worst case complexity of the realizability problem is given by the following theorem. It follows from general complexity bounds for algorithmic problems about semialgebraic sets by Basu, Pollack, and Roy [BPR96] see also Chapter 29 of this Handbook) THEOREM 6.3.2 Complexity of the Best General Algorithm Known The realizability of a rank d oriented matroid on n points can be decided by solving a system of S = real polynomial equations and strict inequalities of degree at most D = d 1 in K = ....

....of the Best General Algorithm Known The realizability of a rank d oriented matroid on n points can be decided by solving a system of S = real polynomial equations and strict inequalities of degree at most D = d 1 in K = n 1) d 1) variables. Thus, with the algorithms of [BPR96] the number of bit operations needed to decide realizability is (in the Turing machine model of complexity) bounded by (S K) O(K) THE UNIVERSALITY THEOREM A basic observation is that all oriented matroids of rank 2 are realizable. In particular, up to change of orientations and permuting ....

S. Basu, R. Pollack, and M.-F. Roy. On the combinatorial and algebraic complexity of quantifier elimination. J. ACM 43:1002--1045, 1996.

....procedure [BCR98,Mis93, Bos82] provides an explicit algorithm for deciding if (2.1) holds, so the problem is decidable. There are also a few alternative approaches to effectively answer this question, also based in decision algebra; see [Bos82] for a survey of classical available techniques, and [BPR96] for more efficient recent developments. Regarding complexity, the general problem of testing global nonnegativity of a polynomial function is NP hard (when the degree is at least four) as easily follows from reduction from the matrix copositivity problem; see [MK87] and Section 7.5. Therefore, ....

S. Basu, R. Pollack, and M.-F. Roy. On the combinatorial and algebraic complexity of quantifier elimination. J. ACM, 43(6):1002--1045, 1996.

....consider arbitrary polygonal traps with k vertices. Such a trap can be represented by 2k parameters. Our approach to computing a k vertex trap that allows a given part to pass in only one orientation uses high dimensional arrangements and quanttrier elimination. Using recent results by Basu et al. [4, 5], we obtain our final result [12] which is given below. Theorem 12. In O( nk) k2) time we can design a polygonal trap with k ver tices with the feeding property for a polygonal part with n vertices, or report that no such trap exists. ....

S. Basu, R. Pollack, and M-F. Roy. On the combinatorial and algebraic complexity of quantifier elimination. Journal of the ACM, 43:1002-1045, 1996.

....a list of formulas in defining the components of . The following recursive translation map takes as input an L sentence , and if it terminates, it returns for the least such that Note that termination is not problematic for PML sentences. Applying the best available algorithm by Basu et al. [87] (which significantly improves the version of Collins cylindrical algebraic decomposition currently implemented in the computer algebra tool REDLOG [88] the number of arithmetical operations required to perform this QElim procedure is bounded by , when the body of the argument formula is ....

....of future research warrant special mention. First, given the demand for maximally powerful model checking tools, there is a pressing need for investigation of efficient quantifier elimination for the pre or postimages of polynomial flows applied to semialgebraic sets, starting with a study of [87]. Given the intrinsic limits on algorithmic model checking, further investigation is required of the theory and practice of approximating systems with complex nonlinear dynamics by systems with tractable semialgebraic (or simpler) flows. There is much more to be done in deductive methods for ....

S. Basu, R. Pollack, and M.-F. Roy, "On the combinatorial and algebraic complexity of quantifier elimination," J. ACM, vol. 43, pp. 1002--1045, 1996.

.... [19] proposed elementary algorithms (doubly exponential in the number of variables n) In recent years the problem had received a signi cant attention, had attracted a number of powerful mathematical techniques, and as a result some very ecient quanti er elimination algorithms were designed (see [12, 11, 16, 1]) Attempting to extend the complexity results from algebraic to real analytic case, we have rstly to restrict the class of real analytic functions to a nitely de ned subclass which would include as many as possible important analytic functions (for example, all algebraic functions, ....

....the family of projections of all cells onto a coordinate subspace should constitute a cylindrical decomposition of the projection of the set. In this way the algorithms, having doubly exponential complexity bounds, were obtained in [3, 19] for algebraic case (more ecient modern algorithms [12, 11, 16, 1] don t use cylindrical decomposition) The technique of cylindrical cell decomposition was applied to Pfaan case in the context of model theoretic study of ominimality (see [5, 18] The complexity estimates which can be extracted from these works are apparently non elementary. Recently Gabrielov ....

Basu, S., Pollack, R., Roy, M.-F., On the combinatorial and algebraic complexity of quantier elimination, Journal of the ACM, 43, 1996, 1002-1045.

.... references: Foundational results on the complexity of solving (or counting the roots of) polynomial systems over R can be found in [Roy96] and faster recent algorithms can be found in [Roj98a, MP98] More generally, there are algorithms known for quanti er elimination over any real closed eld [Ren92, Can93, BPR96]. Curiously, the best current complexity bounds for the problems over R just mentioned are essentially the same as those for the corresponding problems over C . Notable recent exceptions include [BGHM97] and [RY01] where the complexity bounds depend mainly on quantities relating only to the ....

Basu, Saugata; Pollack, Richard; Roy, Marie-Francoise, \On the Combinatorial and Algebraic Complexity of Quantier Elimination," J. ACM 43 (1996), no. 6, pp. 1002-1045.

.... [21] proposed elementary algorithms (doubly exponential in the number of variables n) In resent years the problem had received a signi cant attention, had attracted a number of powerful mathematical techniques, and as a result some very ecient quanti er elimination algorithms were designed (see [13, 12, 18, 1]) Attempting to extend the complexity results from algebraic to real analytic case, we have rstly to restrict the class of real analytic functions to a nitely de ned subclass which would include as many as possible important analytic functions (for example, all algebraic functions, ....

....1) i.e. representing this set as a disjoint union of geometrically simple cells each of which is homeomorphic to an open ball of some dimension. In this way the algorithms having doubly exponential complexity bounds were obtained in [3, 21] for algebraic case (more ecient modern algorithms [13, 12, 18, 1] don t use cylindrical decomposition) The technique of cylindrical cell decomposition was applied to Pfaan case in the context of model theoretic study of o minimality (see [6, 20] The complexity estimates which can be extracted from these works are apparently nonelementary. Recently Gabrielov ....

Basu, S., Pollack, R., Roy, M.-F., On the combinatorial and algebraic complexity of quantier elimination, Journal of the ACM, 43, 1996, 1002-1045.

..... Proposition 13. There is an algorithm finding the optimal k partition of a set of n points which runs in time F (n; d; k) Essential use is made here of the sample points algorithm of Basu, Pollack and Roy, which produces representative points in the cells defined by a set of polynomials ([5] x3.1.3 p. 1028) The reduction is omitted from this extended abstract. 3.5 Simplification of 2 2 clustering by dimension reduction The set T of n points to be clustered may lie in a high dimensional Euclidean space. We need never consider a space of dimension greater than n Gamma 1: input ....

S. Basu, R. Pollack, and M.-F. Roy. On the combinatorial and algebraic complexity of quantifier elimination. J. ACM, 43(6):1002--1045, 1996.

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S. Basu, R. Pollack, M.-F. Roy. On the combinatorial and algebraic complexity of Quantifier Elimination. In Proc. 35th Annual IEEE Sympos. on the Foundations of Computer Science, 632--641, (1994).

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S. Basu, R. Pollack, M.-F. Roy, On the combinatorial and algebraic complexity of Quantifier Elimination, Journal of the Association for Computing Machinery, 43 (1996), 1002--1045.

.... problem over the reals: Is therean algorithmdecidin theexistenw of realsolution to a set of polyn99F[ equation withinhF8z coe#cienF[ was solved with a yesanF8 by Tarski [46]an Seiden[Wk [44] Again theexistenw ofan algorithm raises complexity questiony This isan active field of research [4, 6, 12, 15, 39]. 4. Brief discussion. Hilbert s problems have playedan importan rolein the developmen of real algebraic geometry. The previousdiscussion illustrate thispoinz However, some veryimportan ideas were developed withoutan conoutFw6 to these problems. Morse, for example, related thechanW in topology to ....

....developed withoutan conoutFw6 to these problems. Morse, for example, related thechanW in topology to theexistenF an local behaviour of criticalpoinc of afunW68F [32] see Figure 4. 1) Morse theory plays a key rolein thequan:z6F[8w [33, 35, 47] 3) an algorithmic aspects of real algebraic geometry [4, 12]. Bibliographic Information (1) Virginia Ragsdale (1870 1945) She graduated from Guilford College in 1892. Shewon the first scholarship established byBryn Mawr College for a Guilfordwoman graduatin with the highest degree average. She took her Bachelor degreein 1896an was awarded theBryn ....

S. Basu, R. Pollack,an M.-F. Roy, On the combinatorial and algebraic complexity of quantifier elimination, J. Assoc. Comput. Mach. 43

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S. Basu, R. Pollack, and M.-F. Roy. On the combinatorial and algebraic complexity of quantier elimination. In Proc. IEEE Symp. Foundations of Computer Science, Sante Fe, New Mexico, 1994.

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S. Basu, R. Pollack, and M.-F. Roy. On the combinatorial and algebraic complexity of quantier elimination. Journal of ACM, 43(6):10021045, 1996.

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S. Basu, R. Pollack, and M. F. Roy. On the combinatorial and algebraic complexity of quanti er elimination. Journal of the ACM, 43(6):1002-1045, 1996.

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Saugata Basu, Richard Pollack, and Marie-Francoise Roy. On the combinatorial and algebraic complexity of quanti er elimination. J. ACM, 43(6):1002-1045, 1996.

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S. Basu, R. Pollack, and M.-F. Roy. On the combinatorial and algebraic complexity of quantifier elimination. In S. Goldwasser, editor, Proceedings fo the 35th Annual Symposium on Foundations of Computer Science, pages 632--641, Los Alamitos, CA, USA, 1994. IEEE Computer Society Press.

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S. Basu, R. Pollack, and M.-F. Roy. On the combinatorial and algebraic complexity of quantifier elimination. Journal of the ACM, 43(6):1002--1045, 1996.

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S. Basu, R. Pollack, and M.-F. Roy. On the combinatorial and algebraic complexity of quantifier elimination. In IEEE Symposium on Foundations of Computer Science, 1994.

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S. Basu, R. Pollack, and M.-F. Roy. On the combinatorial and algebraic complexity of quantifier elimination. Journal of the ACM, 43(6):1002--1045, 1996.

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S. Basu, R. Pollack, and M.-F. Roy. On the combinatorial and algebraic complexity of quantier elimination. In Proc. IEEE Symp. Foundations of Computer Science, Sante Fe, New Mexico, 1994.

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BASU, S. AND ROY, M.F. On the combinatorial and algebraic complexity of quantifier elimination. Journal of the ACM (JACM) 43, 6 (1996), 1002--1045.

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BASU, S. AND ROY, M.F. On the combinatorial and algebraic complexity of quantifier elimination. Journal of the ACM (JACM) 43, 6 (1996), 1002--1045.

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