B. F. Caviness, J. R. Johnson (eds.). Quantifier Elimination and Cylindri - cal Algebraic Decomposition, Texts and Monographs in Symbolic Computation of the Research Institute for Symbolic Computation (B. Buchberger, G.E. Collins, eds.), Springer, Wien-New York, 431 pages.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... for general polynomials Q by reducing them to the special case of multiaffine polynomials P (albeit in a larger number of variables) It is worth remarking that the half plane property is algorithmically decidable, using quantifier elimination methods for the theory of real closed fields [15]. Indeed, let P 0 be a polynomial in complex variables Xl, x, Setting x = a ib and separating out real and imaginary parts R(al, Cn, bl, bn) He P( ak ] ibk ) 2.15a) I(al, Cn, bl, bn) Im P( a k ] ibk ) 2.15b) the half plane property for P is immediately seen to be ....

B.F. Caviness and J.R. Johnson, editors, Quantifier Elimination and Cylindrical Algebraic Decomposition (Springer-Verlag, Wien New York, 1998).

....the exact value of such an element. However, it is often quite easy to find a small interval that provably contains a solution. For example, the expression x 2 is negative for x = 0 and positive for x = 4. Hence, by elementary analysis (Boltzmann intermediate value theorem) the interval [0, 4] contains a solution of x 2 = 0, and 2 = 0 is true. Note that the approach fails if we use the larger interval [ 4, 4] where the function is positive also for x = 4. By applying first order constraint satisfaction [29] one can find intervals that enclose the solutions of the occurring ....

....solution. For example, the expression x 2 is negative for x = 0 and positive for x = 4. Hence, by elementary analysis (Boltzmann intermediate value theorem) the interval [0, 4] contains a solution of x 2 = 0, and 2 = 0 is true. Note that the approach fails if we use the larger interval [ 4, 4], where the function is positive also for x = 4. By applying first order constraint satisfaction [29] one can find intervals that enclose the solutions of the occurring equalities tightly. But, up to now it was unclear, how to extend the approach sketched above to general first order constraint ....

B. F. Caviness and J. R. Johnson, editors. Quantifier Elimination and Cylindrical Algebraic Decomposition. Springer, 1998.

....) one wishes to find a quantifier free formula ( x) equivalent to 9y (y; x) In many cases, the complexity of algorithms to find such a is of the form O(N f(n) where N is the size of the formula, and f is some function. For example, if one uses cylindrical algebraic decomposition [3] for the real field, f is O(2 n ) In general, even if better algorithms are available, the complexity of constraint processing often increases with dimension to such an extent that it becomes unmanageable for large datasets (see, e.g. 10] Assume now that x is split into two disjoint tuples ....

....9y is equivalent to (2) k i=1 (9y ff i (y; u) fi i ( v) For a number of operations this is a significant improvement, as the exponent becomes lower. For example, in addition to quantifier elimination, data often has to be represented in a nice format (essentially, as union of cells [3]) and algorithms for doing this also benefit from reduction in the dimension [9, 10] Even though such a notion of independence may seem to be too much of a restriction, from the practical point of view it is sometimes necessary to insist on it, as the cost of general quantifier elimination and ....

[Article contains additional citation context not shown here]

B.F. Caviness and J.R. Johnson, Eds. Quantifier Elimination and Cylindrical Algebraic Decomposition. Springer Verlag, 1998.

....can find a cell decomposition C of R n such that on each cell C i , none of the functions f j changes its sign. Furthermore, this decomposition can be found in time O( kd) h(n) where d is the maximal degree of a polynomial among f j s, and h is some function (typically, h(n) O(2 n ) [10, 9]. It is important to notice that for a fixed dimension, the cell decomposition algorithm is thus in PTIME (in fact, in NC [7] 3 Topological properties and closure The goal of this section is to show that adding topological properties to languages like FO Lin and FO Poly results in closed query ....

B.F. Caviness and J.R. Johnson, Eds. Quantifier Elimination and Cylindrical Algebraic Decomposition. Springer Verlag, 1998.

....can find a cell decomposition C of R n such that on each cell C i , none of the functions f j changes its sign. Furthermore, this decomposition can be found in time O( kd) h(n) where d is the maximal degree of a polynomial among f j s, and h is some function (typically, h(n) O(2 n ) [11, 9]. It is important to notice that for a fixed dimension, the cell decomposition algorithm is thus in PTIME (in fact, in NC [6] 3 Topological properties and closure The goal of this section is to show that adding topological properties to languages like FO Lin and FO Poly results in closed ....

B.F. Caviness and J.R. Johnson, Eds. Quantifier Elimination and Cylindrical Algebraic Decomposition. Springer Verlag, 1998.

....SURVEYS All results not given an explicit reference may be traced in these surveys. ffl [Mis93] A text book for algorithmic algebra covering Grobner bases, characteristic sets, resultants and real algebra. Chapter 8 gives many details of the classical results in computational real algebra. ffl [CJ95]: An anthology of key papers in computational real algebra and real algebraic geometry. Contains reprints of the following papers cited in this chapter: BPR95, Col75, Ren91, Tar51] Computational Real Algebraic Geometry 19 ffl [AB88] A special issue of the Journal of Symbolic Computation on ....

B.F. Caviness and J.R. Johnson (Eds.). Quantifier Elimination and Cylindrical Algebraic Decomposition, Springer-Verlag, Wien, 1995.

....sentences ( and (y) from the Introduction are examples of L sentences. Note that, since semi algebraic sets are first order definable in R, the question of A j= S) given (S) and a definition of a semi algebraic set A, is effectively decidable, because the first order theory of R is decidable [7, 8, 19, 23]. Homeomorphism invariance and equivalence. We call two subsets A and B of R 2 homeomorphic if there is a homeomorphism h of R 2 such that h(A) B. A sentence (S) is called invariant under homeomorphisms (abbreviated as H invariant) if for any two homeomorphic semialgebraic sets A and B, ....

....B c are both empty or both not empty. This is a test obviously expressible in first order logic and therefore decidable for semi algebraic sets by Tarski s theorem [23] If c is another kind of cone, then both A c and B c are finite and symbolic algorithms for the first order theory of the reals [1, 7, 19, 8] can effectively enumerate them. The for loop always terminates since in each closed semialgebraic set only a finite number of cones can appear (see (i) of Property 2) In order to formulate the second corollary, we call two semi algebraic sets I equivalent under if they cannot be distinguished ....

[Article contains additional citation context not shown here]

B.F. Caviness, J.R. Johnson, editors. Quantifier Elimination and Cylindrical Algebraic Decomposition. Springer, Wien-New York, 1998.

....a so called semi algebraic Challenges of Symbolic Computation 7 set, namely those specializations of the quantified expression that are theorems, i.e. true formulas. The algorithmic aspects of QE have been studied extensively by G. Collins, D. Grigoriev, J. Renegar, H. Hong, and others (see (Caviness and Johnson 1998)) In general, the process is computationally hard, although several of the examples in Section 1 have efficient algorithms. Collins has suggested another minimax problem as a challenge benchmark problem for QE software with the intent to demonstrate the abilities of the different algorithms and ....

Caviness, B. F., Johnson, J. R., editors (1998). Quantifier Elimination and Cylindrical Algebraic Decomposition.

No context found.

B. F. Caviness, J. R. Johnson (eds.). Quantifier Elimination and Cylindri - cal Algebraic Decomposition, Texts and Monographs in Symbolic Computation of the Research Institute for Symbolic Computation (B. Buchberger, G.E. Collins, eds.), Springer, Wien-New York, 431 pages.

No context found.

B. F. Caviness and J. R. Johnson, editors. Quantifier Elimination and Cylindrical Algebraic Decomposition. Springer, Wien, 1998.

No context found.

Bob F. Caviness and Jeremy R. Johnson, editors. Quantifier Elimination and Cylindrical Algebraic Decomposition, Texts and Monographs in Symbolic Computation, Linz, 1993. Springer, Wien, New York, 1998.

No context found.

B. F. Caviness and J. R. Johnson, editors. Quantifier Elimination and Cylindrical Algebraic Decomposition. Springer, Wien, 1998.

No context found.

B. F. Caviness and J. R. Johnson, editors. Quantifier Elimination and Cylindrical Algebraic Decomposition. Springer, Wien, 1998.

No context found.

B. F. Caviness and J. R. Johnson, editors. Quantifier Elimination and Cylindrical Algebraic Decomposition. Springer, 1998.

No context found.

B. F. Caviness and J. R. Johnson, editors. Quantifier Elimination and Cylindrical Algebraic Decomposition. Springer, Wien, 1998.

No context found.

B.F. Caviness and Jeremy R. Johnson, editors. Quantifier Elimination and Cylindrical Algebraic Decomposition. Texts and Monographs in Symbolic Computation. Springer-Verlag, 1998.

No context found.

B.F. Caviness and J.R. Johnson (eds.) Quantifier Elimination and Cylindrical Algebraic Decomposition Springer-Verlag, Wien New York, 1998.

No context found.

B.F. Caviness and J.R. Johnson (eds.) Quantifier Elimination and Cylindrical Algebraic Decomposition Springer-Verlag, Wien New York, 1998.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC