## On the Combinatorial and Algebraic Complexity of Quantifier Elimination (1996)

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Citations: | 197 - 29 self |

### BibTeX

@MISC{Basu96onthe,

author = {Saugata Basu and Richard Pollack and Marie-Francoise Roy},

title = {On the Combinatorial and Algebraic Complexity of Quantifier Elimination},

year = {1996}

}

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### Abstract

In this paper, a new algorithm for performing quantifier elimination from first order formulas over real closed fields is given. This algorithm improves the complexity of the asymptotically fastest algorithm for this problem, known to this date. A new feature of this algorithm is that the role of the algebraic part (the dependence on the degrees of the input polynomials) and the combinatorial part (the dependence on the number of polynomials) are separated. Another new feature is that the degrees of the polynomials in the equivalent quantifier-free formula that is output, are independent of the number of input polynomials. As special cases of this algorithm, new and improved algorithms for deciding a sentence in the first order theory over real closed fields, and also for solving the existential problem in the first order theory over real closed fields, are obtained.

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Citation Context ...i who proved that there is an algorithm deciding any sentence of the first order theory of the reals. The complexity of the decision procedure for the first order theory of the reals, given by Tarski =-=[29]-=- (see also ([28]) was not elementary recursive. The first algorithm, with a reasonable worst-case time bound was given by Collins ([10]). His algorithm had a worst case running time doubly exponential... |

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Citation Context ... of polynomials Q 1 ae D[ �� X 1 ], such that the set SIGN 1 (��x 1 ) remains invariant over each cell of Q 1 : We make use of the classical theory of subresultant coefficient sequences (see [=-=21], or [22]) and comput-=-e the subresultant coefficient sequences, of the following pairs of polynomials. For every tuple u = (f( �� X 1 ; ffi 1 ; t 1 ); g 0 ( �� X 1 ; ffi 1 ; t 1 ); : : : ; g k1 ( �� X 1 ; ffi 1... |

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Citation Context ... problem in a singly exponential number of arithmetic operations over any ring and the quantifier elimination problem in a number of operations, doubly exponential in the number of blocks. Renegar's (=-=[25]-=-) algorithms which solve the general decision problem in time (sd) \PiO(k i ) , and the quantifier elimination problem in time (sd) (l+1)\PiO(k i ) were the best to date. The degrees of the polynomial... |

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Citation Context ...contained in a ball of radius a b with a and b in Z of bit size (�� + log(m))d O(k) . Our techniques can be applied to decision problems in complex geometry improving slightly the results of [12] =-=and [20]-=-, since we achieve the same complexity with deterministic algorithms, rather than with probabilistic ones. Proposition 7 Given a set P m polynomials of degreesd in k variables with coefficients in D[i... |

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Citation Context ...ber a b with a and b of bit length �� \Theta d O(k) in every connected component of C. This result improves the bound in [19], where the dependence on k was doubly exponential as well as the bound=-= in [14]-=- which depended on s. 3.2 Real and Complex Decision Problems As applications of our techniques, we give improved algorithms for several problems in real and complex geometry. We can prove the followin... |

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Citation Context ...the drawback that they require that the coefficients of the input polynomials be integers rather than belonging to an arbitrary ordered ring contained in a real closed field. Heintz, Roy and Solerno (=-=[16]-=-) gave an algorithm to decide the existential problem in a singly exponential number of arithmetic operations over any ring and the quantifier elimination problem in a number of operations, doubly exp... |

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Citation Context ...own properties of subresultant coefficients, it can be easily shown that as �� x 1 varies over each cell of Q 1 , the set SIGN 1 (��x 1 ) remains invariant, as claimed. Next, we eliminate the =-=block X [2] . M-=-ore precisely, we compute a set of polynomials Q 2 ae D[ �� X 2 ]; with the property that the k 2 dimensional fibre over each point in one cell of Q 2 , meets exactly the same cells of Q 1 . This ... |

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Citation Context ...ingle polynomial equation whose degree is bounded by 2d. It is possible to compute points in every connected component of such an algebraic set usingsd O(k) arithmetic operations (this is implicit in =-=[9]-=-, [25]). We follow [3] and compute these points using the Cell Representatives Subroutine. We briefly outline the Cell Representatives Subroutine. Given a polynomial Q 2 R[X 1 ; : : : ; X k ] we defin... |

32 |
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Citation Context ...[15]) gave an algorithm to solve the decision problem for the existential theory of the reals. Their algorithm has a time complexity which is singly exponential in the number of variables. Grigor'ev (=-=[13]-=-), extended this result to the general decision problem and achieved doubly exponential complexity in the number of blocks. It should be noted that for a fixed value of !, this is only singly exponent... |

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Citation Context ...t]. We say that such points are definedsover D[ffl]. We denote by eval ffl the map from the valuation ring of Rhffli to R, obtained by evaluating the Puiseux series at ffl = 0. We refer the reader to =-=[3]-=- for a complete list of properties of Puiseux series that we utilize. Whenever we compute a point x = (x 1 ; : : : ; x k ) what we actually compute is : 1. A univariate polynomial f(t). 2. A root, say... |

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Citation Context ...irst algorithm, with a reasonable worst-case time bound was given by Collins ([10]). His algorithm had a worst case running time doubly exponential in the number of variables. Grigor'ev and Vorobjov (=-=[15]-=-) gave an algorithm to solve the decision problem for the existential theory of the reals. Their algorithm has a time complexity which is singly exponential in the number of variables. Grigor'ev ([13]... |

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Citation Context ...2 R k j P i (x) ? 0g is a non-empty sign condition, there exists a rational number a b with a and b of bit length �� \Theta d O(k) in every connected component of C. This result improves the bound=-= in [19]-=-, where the dependence on k was doubly exponential as well as the bound in [14] which depended on s. 3.2 Real and Complex Decision Problems As applications of our techniques, we give improved algorith... |

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Citation Context ...ing a set of polynomials Q 1 ae D[ �� X 1 ], such that the set SIGN 1 (��x 1 ) remains invariant over each cell of Q 1 : We make use of the classical theory of subresultant coefficient sequenc=-=es (see [21], or [22]) a-=-nd compute the subresultant coefficient sequences, of the following pairs of polynomials. For every tuple u = (f( �� X 1 ; ffi 1 ; t 1 ); g 0 ( �� X 1 ; ffi 1 ; t 1 ); : : : ; g k1 ( �� X ... |

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Citation Context ...pecialized to y and z respectively. The quantifier free formula can then be written as \Psi(Y ) =sy2T \Psi y (Y ): In order to construct the formulae \Psi y , we make use of the multivariate version (=-=[24]-=-) of the sign determination algorithm due to Ben Or, Kozen and Reif, in an inverse fashion. We call this the Inverse Sign Determination Subroutine. Roughly, the idea is the following. 2.3 The Inverse ... |

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Citation Context ...e Inverse Sign Determination Subroutine. Roughly, the idea is the following. 2.3 The Inverse Sign Determination Subroutine We assume that the reader is familiar with the Sign Determination Algorithm (=-=[27]-=-,[24], inspired by[5]). Given a multivariate polynomialh, and another system of polynomials, T , having only a finite number of real zeros, denoted by Z(T ), we define the Sturm query of h with respec... |

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Citation Context ...ints are contained in a ball of radius a b with a and b in Z of bit size (�� + log(m))d O(k) . Our techniques can be applied to decision problems in complex geometry improving slightly the results=-= of [12]-=- and [20], since we achieve the same complexity with deterministic algorithms, rather than with probabilistic ones. Proposition 7 Given a set P m polynomials of degreesd in k variables with coefficien... |

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