G. Collins. Quantifier Elimination for Real Closed Fields by Cylindric Algebraic Decomposition. In 2nd GI Conference on Automata Theory and Formal Languages, pages 134--183, 1975.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Cole s idea to multidimensional parametric searching [22] see also [213, 229] The running time was further improved by Agarwal et al. 18] to d . Later Toledo [265] extended these techniques to comparisons involving nonlinear polynomials, using Collins s cylindrical algebraic decomposition [79]. The total running time of his procedure is O(T s (T p log n) For the sake of completeness, we present these higher dimensional extensions in an Appendix. 3 Alternative Approaches to Parametric Searching Despite its power and versatility, the parametric searching technique has some ....

G. E. Collins, Quantifier elimination for real closed fields by cylindrical algebraic decomposition, in: Proc. 2nd GI Conference on Automata Theory and Formal Languages, Lecture Notes Comput. Sci., Vol. 33, Springer-Verlag, Berlin, West Germany, 1975, pp. 134--183.

....tenth problem, even the quantifier free fragment of nonlinear arithmetic over the integers or rationals is undecidable. However, the first order theory of nonlinear arithmetic over the reals and the complex 9 numbers is decidable. Tarski [Tar48] gave a decision procedure for this theory. Collins [Col75] gave an improved quantifier elimination procedure that is the basis for a popular package called QEPCAD [CH91] These procedures have been successfully used in proving theorems in algebraic geometry. Buchberger s Grobner basis method for testing membership in polynomial ideals has also been ....

G. E. Collins. Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In Second GI Conference on Automata Theory and Formal Languages, number 33 in Lecture Notes in Computer Science, pages 134--183, Berlin, 1975. Springer-Verlag.

....Define the Hilbert map for the G action by # : R (# 1 (m) # k (m) 12 Since G is compact, # separates G orbits [31] Therefore #(R )istheGorbit space R G. Because # is a polynomial mapping, #(R ) is a semialgebraic subset of R , by the Tarski Seidenberg theorem [32]. According to Mather s refinement of Schwarz s theorem [33] the mapping # # : C # (R : f is split surjective. An easy diagram chase shows that the space of smooth functions on R G is isomorphic as aFrechet space to the space of Whitney smooth functions on R . We know ....

Collins, G., Quantifier elimination for real closed fields by cylindrical algebraic decomposition, Lect. Notes in Comp. Sci., 33, (1974), 134-183.

....to the assertion (3al, an, b1, b n ] al 0) A. A (a n 0) A (R(a, b) 0) A (I(a, b) 0) 2. 16) in the first order theory of 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 ....

G.E. Collins, Quantifier elimination for real closed fields by cylindrical algebraic decomposition, in Automata Theory and Formal Languages (Lecture Notes in Computer Science -33), edited by H. Brakhage (Springer-Verlag, Berlin HeidelbergNew York Tokyo,

....are formulated. Stability and stabilizability margins of a n D system will be defined in Section 3. Section 4 presents a quantifier elimination (QE) formulation for computing both the stability and stabilizability margins, which can be solved based on the Cylindrical Algebraic Decompostion (CAD) [4, 5], a powerful computational algebraic method for solving algebraic problems. Section 5 provides a QE formulation for the construction of a stabilizing compensator that may approximately achieve the stabilizability margin. In Section 6, the robust stability and stabilization problems of ....

....v =0) 4. 8) can be descibed by algebraic conditions imposed on its argument # which can be obtained by eliminating the quantifiers n , such an operation is called Quantifier Elimination (QE) A general computational method based on the Cylindrical Algebraic Decomposition (CAD) [4] of algebraic sets has been developed for solving the QE problems. Example 3. Let g(z 1 ,z 2 ) z 1 z 2 4. Though it is easy to inspect that the stability margin is 1, in the following we show a method in the essentials of CAD for solving the QE problem. Writting g(z 1 ,z 2 ) as g(z 1 ,z 2 ) u(x ....

G. E. Collins, Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition, Lecture Notes in Computer Science, Vol.33 (1975), pp. 134-183.

....determine the set of zeros of the gradient of on U . Then is constant on each connected component and the maximum of these constant values yields max U n . To compute the zero set of the gradient of and its connected components, one may for instance use cylindrical decomposition (see [Col 75] Of course, other algorithms from effective real algebraic geometry can be used instead. The correctness of our algorithm is clear. The termination of the loop in step 2 follows from the fact that the new F i is asymptotically smaller then F j , so that either the i class of F j strictly ....

G.E. Collins. Quantifier elimination for real closed fields by cylindrical algebraic decomposition. Proc. 2-nd conf. on automata theory and formal languages, Springer lect. notes in comp. sc. 33 (p. 134-183).

.... assign Sigma : Sigma e [ffg; Sigma i ) Sigma : Sigma e ; Sigma i [ffg) and Sigma : Sigma e ; Sigma i [ f Gammaf g) A process is eliminated, whenever the new system Sigma is algebraically inconsistent; this can be tested for instance by using cylindrical decompositions [3]. Each process leads to an effective normal basis B and an expansion algorithm for f w.r.t. B relative to the solution set of Sigma. Processes in which an error occurs correspond to regions relative to which f is undefined at infinity. We notice that the parallel computation process can be ....

G.E. Collins. Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In Proc. 2-nd conf. on automata theory and formal languages, volume 33 of Lect. Notes in Comp. Science, pages 134--183. Springer, 1975.

....only solution points, but also that some input variables be universally quantified. To date, constraint systems with universally existentially quantified variables have mainly be handled by symbolic methods, among which one may single out Cylindrical Algebraic Decomposition (CAD) by Collins [11]. CAD is quite a powerful method since it permits handling more than one quantified variable for disjunctions conjunctions of constraints. However, it has strong requirements on the form of the constraints it processes since they are limited to polynomial constraints. As far as camera control is ....

G. E. Collins. Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In Proc. of the 2nd GI Conf. Automata Theory and Formal Languages, volume 33 of LNCS, pages 134--183, Kaiserslauten, 1975. Springer.

....of n real algebraic hypersurfaces in R k it su#ces to first triangulate the arrangement and then compute the Betti of the corresponding simplicial complex. However, currently the most e#cient way known to obtain such a triangulation is via the technique of cylindrical algebraic decomposition [7], and this produces O(n 2 k ) simplices in the worst case. More e#cient ways of decomposing such an arrangement into topological balls have been proposed. In [6] the authors provide a decomposition into O # (n 2k 3 ) cells (see [13] for a recent improvement of this result) However, this ....

G. Collins, Quantifier elimination for real closed fields by cylindric algebraic decomposition. In Second GI Conference on Automata Theory and Formal Languages. Lecture Notes in Computer Science, vol. 33, pp. 134-183, Springer-Verlag, Berlin (1975).

....value. By taking D 0 to be the active domain of a given finite database over the reals, we get the analog in the real case of the Aylamazyan et al. theorem. The reader familiar with Collins s method for quantifier elimination in real closed fields through cylindrical algebraic decomposition (cad) [3, 4, 11] will not be surprised by the above observation. Indeed, it follows more or less directly from an obvious adaptation of the cad construction. However, we give an alternative self contained proof from first principles which abstracts away the purely algorithmical aspects of the cad construction and ....

G.E. Collins. Quantifier elimination for real closed fields by cylindrical algebraic decomposition. Lecture Notes in Computer Science 33, 134-- 183 (1975).

....numbers, and whose formulas are constructed of inequalities of rational forms (these rational forms are 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 ....

....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 computational complexity. Collins [5] proved a useful Lemma as a byproduct of his decision procedure: Lemma 9. If the solution of a formula of length n in the existential theory of real closed fields is not the zero vector 0, then it is of modulus at least 2 Gamma2 cn , for some constant c 0. Now consider an optimum path S(d; ....

G. E. Collins. Quantifier elimination for real closed fields by cylindric algebraic decomposition. In Proc. Second GI Conference on Automata Theory and Formal Languages, volume 33 of Lecture Notes in Computer Science, 1975.

....Given an algebraic plane curve, techniques for desingularization based on quadratic transformations are given in [Wal50, Abh90, AB88] However, the resulting algorithm can be exponential in the degree of the curve. Algorithms based on Collins cylindrical algebraic decomposition (CAD) Col75, ACM84] have been used for evaluating all components of algebraic curves [Arn83, SS83] However, its worst case complexity is doubly exponential in the number of variables. For plane curves, improved polynomial 3 Figure 2: Two surfaces intersecting in a loop time algorithms based on CAD have ....

G.E. Collins. Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In Lecture Notes in Computer Science, number 33, Springer-Verlag, 1975.

....i can be constructed in NC ( Bovet, Crescenzi, 1994] chapter 12) thus, so can E 1 ; Em . It follows that PC i (S) can be constructed in NC too. 5. 1 Construction of P invariant CAD sequences The first PTIME (n fixed) algorithm for constructing a P invariant CAD sequence was obtained by Collins in 1975 ( Collins, 1975] This algorithm was not proven to parallelise, but, Renegar ( Renegar, 1991] pg. 289) states that a parallel version probably could be developed. Based on ideas from [Ben Or et al. 1986] Renegar ( Renegar, 1992] gives an NC algorithm for the decision problem in the ....

....in NC ( Bovet, Crescenzi, 1994] chapter 12) thus, so can E 1 ; Em . It follows that PC i (S) can be constructed in NC too. 5. 1 Construction of P invariant CAD sequences The first PTIME (n fixed) algorithm for constructing a P invariant CAD sequence was obtained by Collins in 1975 ([Collins, 1975]) This algorithm was not proven to parallelise, but, Renegar ( Renegar, 1991] pg. 289) states that a parallel version probably could be developed. Based on ideas from [Ben Or et al. 1986] Renegar ( Renegar, 1992] gives an NC algorithm for the decision problem in the first order theory of the ....

Collins, G. Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition. Second GI conference on Automata and Formal Languages. Lecture Notes in Computer Science, 33, 1975, pg. 134-183.

.... Gamma 3D using only a variant of the Collins cylindrical algebraic decomposition algorithm which does not require a change of coordinates ( 4] Benedikt et al. use semi algebraic triviality (section 9. 3 of [8] in addition to the standard Collins cylindrical algebraic decomposition algorithm ([10], 3] to obtain closure (semi algebraic triviality requires a change of coordinates) Therefore, we show how closure (and PTIME data complexity) of FO poly Pconn Gamma 3D can be proven without requiring a change of coordinates. Also, our proof techniques highlight more clearly the essential ....

....; EC j ;m j ) can be constructed in PTIME since C n can. Hence, the formula OE from the proof of Theorem 6 can be constructed. 5. 3 Construction of P invariant CAD Sequences The first PTIME (n fixed) algorithm for constructing a P invariant CAD sequence was obtained by Collins in 1975 ([10]) We briefly outline the exposition of the algorithm given in [3] stating a few of its properties (without proof) which we will use explicitly. The algorithm proceeds in three stages: projection, base, and extension (these terms are from [3] In the projection phase, a series of finite sets of ....

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Collins, G. Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition, in: Second GI Conference on Automata and Formal Languages 1975, Lecture Notes in Computer Science, Vol. 33, 134-183.

No context found.

G. Collins. Quantifier Elimination for Real Closed Fields by Cylindric Algebraic Decomposition. In 2nd GI Conference on Automata Theory and Formal Languages, pages 134--183, 1975.

No context found.

G. E. Collins "Quantifier elimination for real closed fields by cylindrical algebraic decomposition," Springer Lecture Notes in Computer Science, 33, 1975, 515--532.

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G. E. Collins. Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In Proc. 2nd GI Conf. on Automata Theory and Formal Languages, volume 6, pages 134--183. LNCS, Springer, Berlin, 1975. Reprinted with corrections in: B. F. Caviness and J. R. Johnson (eds.), Quantifier Elimination and Cylindrical Algebraic Decomposition, 85--121. Springer, 1998.

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G. Collins, 1975. Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition. In: Second GI Conference on Automata Theory and Formal Languages, LNCS, Vol. 33, Springer Berlin, pp. 134-183.

No context found.

G. E. Collins, Quantifier elimination for real closed fields by cylindrical algebraic decomposition Springer Lecture Notes in Computer Science 33, 515- 532, (1975).

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G. E. Collins, Quantifier elimination for real closed fields by cylindrical algebraic decomposition Springer Lecture Notes in Computer Science 33, 515-532.

No context found.

G. E. Collins, Quantifier elimination for real closed fields by cylindrical algebraic decomposition Springer Lecture Notes in Computer Science 33, 515-532.

No context found.

G. E. Collins, Quantifier elimination for real closed fields by cylindrical algebraic decomposition, Lect. Notes in Comp. Sci., 33, 515-532, Springer-Verlag (1975).

No context found.

G. E. Collins, Quantifier elimination for real closed fields by cylindrical algebraic decomposition. Springer Lecture Notes in Computer Science 33, 515-532.

No context found.

G.E. Collins. Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In Proceedings of the Second GI Conference on Automata Theory and Formal Languages, volume 33 of LNCS, pages 161--182. Springer, 1975.

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G. E. Collins. Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition. Proceedings of the 2nd GI Conference on Automata Theory and Formal Languages, 1975.

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