J. Canny. Some Algebraic and Geometric Computations in PSPACE. In 20th ACM Symposium on Theory of Computing, pages 460--467, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....corresponding classical language. Theorem 5. Satisfiability of a formula in space. If the formula is quantifier free, then its satisfiability can be decided in polynomial space. The proof of this result makes use of results of Ben Or, Kozen and Reif [BKR] for the full language, and of Canny [Canny88] for the quantifier free case. 6 Conclusion A topic that has been of some interest in the quantum mechanics literature is the extent to which it is possible to eliminate the use of complex numbers, and to reason about quantum probabilities purely as real numbers. This requires the ....

J. F. Canny. Some algebraic and geometric computations in PSPACE. In Proc. 20th ACM Symp. on Theory of Computing, 460-467, 1988.

....rst algorithm with substantial 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 ....

....problem is to construct another quanti er free formula, x ) such that (x ) holds if and TABLE 33.1.1 Selected time complexity results. GENERAL OR EXISTENTIAL TIME COMPLEXITY SOURCE O( n i ) Col75] O(n 2 ) GV92] O( n i ) 4 2 [Gri88] 1 o(1) n 1) O(n [Can88b, Can93] General (L log L log log L) md) 2 O( n i [Ren92a,b,c] Existential (L log L log log L)m (m=n) General (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 ....

J.F. Canny. Some algebraic and geometric computations in PSPACE. In Proc. 20th Annu. ACM Sympos. Theory Comput., pages 460-467, 1988.

....the computation of any reversible Turing Machine on an input string to an instance of the mover s problem. Hopcroft et al. HJW84] improved on this result by proving that the mover s problem for 2D linkages is PSPACE hard. Later the generalized mover s problem was proved to be in PSPACE by Canny [Can88]. Canny and Reif [CR87] also proved that computing the shortest path for a point robot moving amidst polyhedral obstacles is NP hard. Asano et al. AKY96] introduced the problem of computing the d 1 optimal motion for a 2D rod (de ned by a directed line segment) amidst polygonal obstacles and ....

J. Canny. Some algebraic and geometric computations in PSPACE. In Proceedings of the 20th Annual ACM Symposium on Theory of Computing, pages 460-467, May 1988.

....research on this problem, but there is considerable previous research on related movement problems. Reif [12] provided the rst PSPACE hardness result for a robotic motion planning problem, and Schwartz and Sharir [16] gave motion planning algorithms using the theory of real closed elds (Canny [3]) Reif and Sharir [13] gave algorithms and computational complexity lower bound results for robotic motion with moving obstacles (also see Wilfong [19] Canny and Reif [4] showed the 3D minimal cost path problem with polygonal obstacles is NP hard, and Reif and Storer [15] applied the theory of ....

....are arithmetic expressions involving these real variables and xed rational constants which may be added and multiplied together) and the usual Boolean logical connectives AND, OR, NOT. Collins [5] gave a decision procedure for the existential theory of real closed elds that was improved by Canny [3] to run in polynomial space: Lemma 4 Given a formula of the existential theory of real closed elds of length n, the formula can be decided in n space and 2 time, and the existentially quanti ed variables can be determined, up to exponential bit precision, within this computational ....

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J. Canny. Some algebraic and geometric computations in PSPACE. In Richard Cole, editor, Proceedings of the 20th Annual ACM Symposium on the Theory of Computing, pages 460-467, Chicago, IL, May 1988. ACM Press.

....parsimonious: since all primitive tests are polynomial sign evaluations, the question of whether the current test is a logical consequence of previous tests can be phrased as a statement of the existential theory of the reals. This theory is at least NP hard and is decidable in polynomial space [3]. Unfortunately, the full power of the theory seems to be necessary for some problems. An example is the line arrangement problem: given a set of lines (specified by real coordinates (a; b; c) so that ax by = c) compute the combinatorial structure of the resulting arrangement in the plane. ....

John Canny. Some Algebraic and Geometric Computations in PSPACE. 20th Annual Symposium on the Theory of Computing (Chicago, Illinois), pages 460--467. Association for Computing Machinery, May 1988.

.... Theta; x n ; 0; 1; i.e. polynomial inequations over the reals; C; Gamma; Theta; x n ; 0; 1; i.e. polynomial equations over the complex numbers. The proof is based on the known results on the complexity of algorithms for the corresponding algebraic structures [25, 112, 69, 75]. If we allow non ground queries, DEXPTIME completeness should be replaced by NEXPTIME completeness. 10. Expressive power of logic programmingwith complex values The expressive power of datalog queries is defined in terms of input and output databases, i.e. finite sets of tuples. To extend the ....

J. Canny. Some algebraic and geometric computations in PSPACE. In Proc. 20th Annual ACM STOC, pp. 460--467, Chicago, Illinois, 1988.

....research on this problem, but there is considerable previous research on related movement problems. Reif [12] provided the first PSPACE hardness result for a robotic motion planning problem, and Schwartz and Sharir [17] gave motion planning algorithms using the theory of real closed fields (Canny [3]) Reif and Sharir [13] gave algorithms and computational complexity lower bound results for robotic motion with moving obstacles (also see Wilfong [19] Canny and Reif [4] showed the 3D minimal cost path problem with polygonal obstacles is NP hard, and Reif and Storer [14] applied the theory of ....

....arithmetic expressions involving these real variables and fixed rational constants which may be added and multiplied together) and the usual Boolean logical connectives AND, OR, NOT. Collins [5] gave a decision procedure for the existential theory of real closed fields that was improved by Canny [3] to run in polynomial space: Lemma 8. Given a formula of the existential theory of real closed fields of length n, the formula can be decided in n O(1) space and 2 O(n) time, and the existentially quantified variables can be determined, up to exponential bit precision, within this ....

[Article contains additional citation context not shown here]

J. Canny. Some algebraic and geometric computations in PSPACE. In Proceedings of the 20th Annual ACM Symposium on the Theory of Computing, 1988.

....point arithmetic. Though its combinatorial complexity remains the same, but its algebraic complexity has been improved significantly which is very important towards its implementation. Several numerical examples are also presented. 1 Introduction The silhouette algorithm developed by Canny[1, 2, 4] is a general motion planning algorithm whose complexity is known to be the best within the general and complete algorithms. But no significant implementation of the algorithm has reported so far within our knowledge. We have revisited the silhouette algorithm to improve its algebraic complexity. ....

....the algorithm to find the topology of the curves, and the issue of the intersection points is described in detail with the new algorithm for finding the ordinal numbers of the curves. We conclude the paper in Section 4. 2 Silhouette algorithm We review the top level of the silhouette algorithm [1, 2, 4] before breaking it down and re design each step of the algorithm. Algorithm 2.1 1. for i = 1; d 0 1 do for each subset R of CFs s.t. jRj = i do (a) M r2R ( j CS = 0 in r) b) Check if the silhouette curve C = 6(a 0 j M ) of the manifold M under a generic linear map a 0 is empty. c) If ....

J.Canny, Some algebraic and geometric computation in PSPACE, ACM Symp. on Theory of Computing, pp.460--467, 1988.

....reversible Turing Machine on an input string to an instance of the generalized mover s problem. Hopcroft, Joseph and Whitesides [8] improved this result by proving that the mover s problem for 2 D linkages is PSPACE hard. Later the generalized mover s problem was proved to be in PSPACE by Canny [1]. In [9] Hopcroft, Schwartz and Sharir proved that motion planning for multiple independent rectangular boxes sliding inside a rectangular box is PSPACE hard. Reif and Sharir [12] introduced the 3 D mover s problem in the presence of moving obstacles and showed that this problem is PSPACE hard ....

J. Canny. Some algebraic and geometric computations in PSPACE. In Richard Cole, editor, Proceedings of the 20th Annual ACM Symposium on the Theory of Computing, pages 460--467, Chicago, IL, May 1988. ACM Press.

....E mail: reif cs.duke.edu. 1 input string to an instance of the generalized mover s problem. Hopcroft, Joseph and Whitesides [15] improved this result by proving that the mover s problem for 2 D linkages is PSPACE hard. Later the generalized mover s problem was proved to be in PSPACE by Canny [5]. Most of the works on the algorithmic approach of this problem were based on the decomposition method [25, 26, 27, 28] which was first suggested by Reif. Using this method, the free space is partitioned into simple connected components (cells) The relationship between two components is ....

J. Canny. Some algebraic and geometric computations in PSPACE. In Richard Cole, editor, Proceedings of the 20th Annual ACM Symposium on the Theory of Computing, pages 460--467, Chicago, IL, May 1988. ACM Press.

....Machine on an input string to an instance of the generalized mover s problem. Hopcroft, Joseph and Whitesides [4] improved this result by proving that J. Reif and Z. Sun the mover s problem for 2 D linkages is PSPACE hard. Later the generalized mover s problem was proved to be in PSPACE by Canny [1]. Reif and Sharir [6] introduced the 3 D mover s problem in the presence of moving obstacles and showed that this problem is PSPACE hard even in the case where the object to be moved is a disc with bounded velocity. 1.2 Frictional Mechanical Systems and the Frictional Mover s Problem All the ....

J. Canny. Some algebraic and geometric computations in PSPACE. In Richard Cole, editor, Proceedings of the 20th Annual ACM Symposium on the Theory of Computing, pages 460--467, May 1988.

....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. Canny. Some algebraic and geometric computations in PSPACE. In Proceedings of the 20th annual ACM symposium on the theory of computing, pages 460--467, 1988.

....indeed su#cient, to derive a single adversary input for any problem, rather than explicitly 3 constructing a di#erent input for every algorithm. Infinitesimals have been used extensively in geometric perturbation techniques [EM90, EC91, Yap90] in algorithms dealing with real semialgebraic sets [Can88, Can93] and in other lower bound arguments [GKMS97, GV96] Second, we allow our adversary inputs to be degenerate. That is, both the original adversary input and the modified input contain r tuples in the zeroset of #. Although it appears that such inputs cannot be used in an adversary ....

....field is an ordered field, no proper algebraic extension of which is also an ordered field. The real closure # K of an ordered field K is the smallest real closed field that contains it. We refer the interested reader to [BCR87, HRR91] for further details and more formal definitions, and to [Can88, Can93] for previous algorithmic applications of real closed fields. A formula in the first order theory of the reals, or more simply, a firstorder formula, is a quantified Boolean combination of polynomial equations and inequalities. An elementary formula is a first order formula with no free ....

J. Canny. Some algebraic and geometric computations in PSPACE. In Proceedings of the Twentieth Annual ACM Symposium on the Theory of Computing, pages 460--467. ACM Press, Los Alamitos, Calif., 1988.

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J. Canny. Some Algebraic and Geometric Computations in PSPACE. In 20th ACM Symposium on Theory of Computing, pages 460--467, 1988.

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J. Canny. Some algebraic and geometric computations in pspace. In Proceedings of 20th ACM STOC, pages 460-467, 1988.

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John Canny. Some Algebraic and Geometric Computations in PSPACE. 20th Annual Symposium on the Theory of Computing (Chicago, Illinois), pages 460--467. Association for Computing Machinery, May 1988.

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J. Canny. Some Algebraic and Geometric Computations in PSPACE. In Proc. of 20th Annual Symp. on the Theory of Computing, pages 460--467. ACM, 1988.

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J. Canny. Some algebraic and geometric computations in pspace. In Proceedings of the 20th Symposium on the Theory of Computation, pages 460--467, 1988. Glencora Borradaile and Pascal Van Hentenryck

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Canny J.: Some algebraic and geometric computations in PSPACE. Proc. 20-th STOC (1988) 460-467.

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J. Canny, Some algebraic and geometric computations in PSPACE, in Proceedings of the 20th Annual ACM Symposium on Theory of Computing, 1988, pp. 460--467.

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Canny, J. (1988), \Some Algebraic and Geometric Computations in PSPACE," ##### ######### ## ###### ## #########,pp.460-467.

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J. Canny. Some algebraic and geometric computations in PSPACE. In Proceedings of the 20th Annual ACM Symposium on Theory of Computing, pages 460--469, Chicago, Illinois, May 1988.

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John Canny. Some algebraic and geometric computations in PSPACE. Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing (STOC), 20:460-467, 1988.

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J. Canny, Some algebraic and geometric computations in PSPACE, in Proceedings of the 20th Annual ACM Symposium on Theory of Computing, Chicago, IL, 1988, pp. 460--467.

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Canny, J., Some algebraic and geometric computations in PSPACE, Proc. 20th ACM Symp. on Theory of Computing, 1988, pp. 460--467.

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