S. McCallum. An improved projection operator for cylindrical algebraic decomposition of three dimensional space. J. Symbolic Computation, 5:141--161, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....space into a finite set of regions where each polynomial s evaluation is sign invariant. The quantifier elimination procedure for real closed fields is obtained as a side effect of the CAD decomposition. Over the last 25 years, the CAD algorithm has been improved and made more efficient [28], 27] 19] One such efficient implementation is available via the tool QEPCAD [20] which is built over a symbolic algebra library called SACLIB [11] The tool QEPCAD can be used to perform quantifier elimination over the first order theory of real closed fields and, consequently, it can be ....

S. McCallum. An improved projection operator for cylindrical algebraic decomposition of three dimensional space. J. Symbolic Computation, 5:141--161, 1988.

....n dimensional space into a nite set of regions where each polynomial s evaluation is sign invariant. The quanti er elimination procedure for real closed elds is obtained as a side e ect of the CAD decomposition. Over the last 25 years, the CAD algorithm has been improved and made more ecient [29], 27] 19] One such ecient implementation is available via the tool QEPCAD [20] which is built over a symbolic algebra library called SACLIB [11] The tool QEPCAD can be used to perform quanti er elimination over the rst order theory of real closed elds and, consequently, it can be used as ....

S. McCallum. An improved projection operator for cylindrical algebraic decomposition of three dimensional space. J. Symbolic Computation, 5:141-161, 1988.

....of the abstract system requires logical reasoning in the theory of reals. The rst order theory of real closed elds is known to be decidable [23] and the rst practical algorithm, based on cylindrical algebraic decomposition, was given in [5] which has gone through several improvements [16, 14, 12]. We use the rst order theory of reals to represent sets of continuous states and use reasoning over this theory for creating abstract transition system. Preliminaries The signature of the rst order theory of reals consists of function symbols f ; g, constants R, and predicate symbols f= ....

S. McCallum. An improved projection operator for cylindrical algebraic decomposition of three dimensional space. J. Symbolic Computation, 5:141-161, 1988.

.... Hong, 1996; Weispfenning, 1997; Dorato et al. 1997; Hong, 1997) The first algorithm which is practically useful has been given in (Collins, 1975) with improvements described in (Arnon et al. 1988; Arnon and McCallum, 1982; Collins, 1994; Collins and Hong, 1991; Collins and Johnson, 1989; McCallum, 1988). Other algorithmic approaches are described in (Renegar, 1992; Weispfenning, 1998; Dolzmann and Sturm, 1997; Gonz alez Vega, 1998) Many applications, especially in mechanical engineering and in numerical analysis (see (Liska and Steinberg, 1993; G onzalez L opez and Recio, 1993; Jirstrand, ....

McCallum, S. (1988). An improved projection operator for cylindrical algebraic decomposition. Journal of Symbolic Computation, 5(1,2).

....2000 Abstract This technical report is a preliminary version of a paper on improved projection for Cylindrical Algebraic Decomposition. It is being made available for ISSAC 2000 because of its bearing on [Bro00] McCallum s projection operator for Cylindrical Algebraic Decomposition (CAD) [McC98, McC88, McC84] represented a huge step forward for the practical utility of the CAD algorithm. This report presents a simple theorem showing that the mathematics in McCallum s paper actually point to a better projection operator than he proposes a reduced McCallum projection. As with McCallum s ....

....important for the efficiency of CAD construction that the projection operator produce as small a set of polynomials as possible, while still ensuring the cylindrical arrangement of cells in the resulting decomposition. For most problems, the current best projection operator is due to McCallum [McC84, McC88, McC98]. In this paper it is shown that certain polynomials included in McCallum s projection are in fact unnecessary, and thus the paper provides a reduced projection factor set. Additionally, it is shown that in certain situations in the CAD based quantifier elimination method, still more polynomials ....

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S. McCallum. An improved projection operator for cylindrical algebraic decomposition. In B. Caviness and J. Johnson, editors, Quantifier Elimination and Cylindrical Algebraic Decomposition, Texts and Monographs in Symbolic Computation. Springer-Verlag, Vienna, 1998.

....2000 Abstract This technical report is a preliminary version of a paper on improved projection for Cylindrical Algebraic Decomposition. It is being made available for ISSAC 2000 because of its bearing on [Bro00] McCallum s projection operator for Cylindrical Algebraic Decomposition (CAD) [McC98, McC88, McC84] represented a huge step forward for the practical utility of the CAD algorithm. This report presents a simple theorem showing that the mathematics in McCallum s paper actually point to a better projection operator than he proposes a reduced McCallum projection. As with McCallum s ....

....important for the efficiency of CAD construction that the projection operator produce as small a set of polynomials as possible, while still ensuring the cylindrical arrangement of cells in the resulting decomposition. For most problems, the current best projection operator is due to McCallum [McC84, McC88, McC98]. In this paper it is shown that certain polynomials included in McCallum s projection are in fact unnecessary, and thus the paper provides a reduced projection factor set. Additionally, it is shown that in certain situations in the CAD based quantifier elimination method, still more polynomials ....

S. McCallum. An Improved Projection Operator for Cylindrical Algebraic Decomposition. PhD thesis, University of Wisconsin-Madison, 1984.

.... computer science, engineering and industrial computations can be reduced to the QE problem (see (Hong, 1996; Dorato et al. 1997) The rst algorithm which is practically useful has been given in (Collins, 1975) with improvements described in (Arnon et al. 1988; Collins and Hong, 1991; McCallum, 1988). Other algorithmic approaches are described in (Renegar, 1992; Weispfenning, 1998; Dolzmann and Sturm, 1997; Gonz alez Vega, 1998) Many applications, especially in mechanical engineering and in numerical analysis (see (Liska and Steinberg, 1993; Jirstrand, 1997) lead to QE problems with ....

McCallum, S. (1988). An improved projection operator for cylindrical algebraic decomposition. Journal of Symbolic Computation, 5(1,2).

....at some non trivial examples of the CAD produced during quantifier elimination by CAD. We have implemented the algorithm for constructing a minimal CAD outlined in the previous section and integrated it into QEPCAD, allowing us to compare the minimal CAD with the original CAD. Note: The McCallum [McC84] projection is used in all examples, with equational constraints where possible. 3.3.1 The complex product of an edge and a square Consider the complex segment L = fx i j x 2 [0; 2]g, and the complex square S = fx iy j x 2 [2; 4] y 2 [ Gamma1; 1]g. We use quantifier elimination to find the ....

S. McCallum. An Improved Projection Operator for Cylindrical Algebraic Decomposition. PhD thesis, University of Wisconsin-Madison, 1984. 13

....point on which the matrix is true, then the input sentence is true. Otherwise it is false. We will call this process Decision Stage. In [10] it is shown that the worst time complexity of this algorithm is dominated by L 3 (md) 2 O(n) Collins algorithm has gone through various improvements [1, 28, 18, 19, 27, 20, 11, 26, 6, 25]. All these improvements, though do not change the theoretical complexity, gave significant speed ups in practice, enabling mechanical solution of various nontrivial problems [3, 20, 21] 3 Algorithm of Grigor ev and Vorobjov Now we give an overview of the algorithm of Grigor ev and Vorobjov ....

S. McCallum. An Improved Projection Operator for Cylindrical Algebraic Decomposition. PhD thesis, University of Wisconsin-Madison, 1984.

....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 ....

S. McCallum. An improved projection operator for cylindrical algebraic decomposition. Journal of Symbolic Computation, 5(1,2), 1988.

....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 ....

S. McCallum. An Improved Projection Operator for Cylindrical Algebraic Decomposition. PhD thesis, University of Wisconsin-Madison, 1984.

....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 ....

S. McCallum. An Improved Projection Operator for Cylindrical Algebraic Decomposition. PhD thesis, University of Wisconsin-Madison, 1984.

....that it has a common real solution. The main idea underlying the method is to decompose a hyper real space into a finite number of regions, called cells, such that every polynomial occurring in the input formula has a constant sign in each cell. This method has gone through various improvements [11, 129, 92, 93, 123, 94, 48, 119, 28, 114], enabling mechanical solution of diverse nontrivial problems [13, 94, 95] Very recently several other methods with better theoretical complexities are proposed in the literature [78, 77, 143, 144] However the study by Hong [96] suggests that CAD is still the best one for practical purpose. ffl ....

S. McCallum. An Improved Projection Operator for Cylindrical Algebraic Decomposition. PhD thesis, University of Wisconsin-Madison, 1984. Technical Annex PE7195/ACCLAIM 92

....coefficients, we see immediately that P is absolutely positive 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 ....

McCallum, S. (1984). An Improved Projection Operator for Cylindrical Algebraic Decomposition. PhD thesis, University of Wisconsin-Madison.

....case of the second problem, known as quantifier elimination in the first order theory of real closed fields. Around 1930 Tarski [46] gave the first quantifier elimination algorithm for this theory. Following Tarski s work, various improvements and new algorithms have been devised and analyzed [45, 13, 27, 8, 14, 3, 40, 5, 7, 21, 23, 22, 12, 24, 41, 42, 43, 28, 29, 38, 30, 15, 37, 11, 36, 31, 32, 33]. See [4] for an extensive bibliography on this subject. Among them the following three algorithms are most well known: Collins [14] Grigor ev [23, 22] and Renegar [41, 42, 43] Among the three, Renegar s algorithm has the best theoretical time complexity. But as analyzed in [31] for inputs ....

S. McCallum. An Improved Projection Operator for Cylindrical Algebraic Decomposition. PhD thesis, University of Wisconsin-Madison, 1984.

....the number of times warning messages are printed when not needed. Neither McCallum s modi ed 1 lifting method nor our improvement of it have been implemented in QEPCAD the design of the system would make implementing such a feature quite dicult. 1 Introduction The McCallum Projection [11] (and the Improved McCallum Projection [3] represents a huge improvement over the original projection [6] for CAD construction, as the projection factor set it produces is much smaller. Unfortunately, it also involves a slightly more complicated idea of lifting. In particular, lifting becomes ....

....all the elements of S. If G is the GCD of the elements of S, then the roots of G are exactly the common roots of the elements of S i.e. G is a minimal delineating polynomial Thus, instead of choosing one of the non zero elements of DP and using it as a delineating polynomial (as suggested in [11]) we use the GCD of all non zero elements of DP as a delineating polynomial. 2.2 In practice One of the practical bene ts of computing a minimal delineating polynomial when faced with a projection factor P (x 1 ; x k 1 ) that vanishes identically over a point 2 R k is that one ....

McCallum, S. An improved projection operator for cylindrical algebraic decomposition. In Quantier Elimination and Cylindrical Algebraic Decomposition (1998), B. Caviness and J. Johnson, Eds., Texts and Monographs in Symbolic Computation, Springer-Verlag, Vienna.

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S. McCallum. An improved projection operator for cylindrical algebraic decomposition of three dimensional space. J. Symbolic Computation, 5:141--161, 1988.

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S. McCallum. An improved projection operator for cylindrical algebraic decomposition of three dimensional space. J. Symbolic Computation, 5:141--161, 1988. 28

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S. McCallum. An improved projection operator for cylindrical algebraic decomposition of three dimensional space. J. Symbolic Computation, 5:141--161, 1988.

No context found.

S. McCallum. An improved projection operator for cylindrical algebraic decomposition of three dimensional space. J. Symbolic Computation, 5:141--161, 1988.

No context found.

S. McCallum. An improved projection operator for cylindrical algebraic decomposition of three dimensional space. J. Symbolic Computation, 5:141--161, 1988.

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