## New Results on Quantifier Elimination Over Real Closed Fields and Applications to Constraint Databases (1999)

Venue: | Journal of the ACM |

Citations: | 32 - 4 self |

### BibTeX

@ARTICLE{Basu99newresults,

author = {Saugata Basu},

title = {New Results on Quantifier Elimination Over Real Closed Fields and Applications to Constraint Databases},

journal = {Journal of the ACM},

year = {1999},

volume = {46},

pages = {537--555}

}

### Years of Citing Articles

### OpenURL

### Abstract

In this paper we give a new algorithm for quantifier elimination in the first order theory of real closed fields that improves the complexity of the best known algorithm for this problem till now. Unlike previously known algorithms [3, 28, 22] the combinatorial part of the complexity (the part depending on the number of polynomials in the input) of this new algorithm is independent of the number of free variables. Moreover, under the assumption that each polynomial in the input depend only on a constant number of the free variables, the algebraic part of the complexity (the part depending on the degrees of the input polynomials) can also be made independent of the number of free variables. This new feature of our algorithm allows us to obtain a new algorithm for a variant of the quantifier elimination problem. We give an almost optimal algorithm for this new problem, which we call the uniform quantifier elimination problem. Using the uniform quantifier elimination algorithm, we give a...

### Citations

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Citation Context ... from a uniform sequence of formulas and obtain another uniform sequence of quantifier free formulas. 4 1.3 History The existence of an algorithm for quantifier elimination was first proved by Tarski =-=[31]-=- (see also [29]). However, the complexity of his algorithm is not elementary recursive. The first algorithm with a an elementary-recursive worst-case time bound was given by Collins [11]. His algorith... |

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Citation Context ... sequence of formulas and obtain another uniform sequence of quantifier free formulas. 4 1.3 History The existence of an algorithm for quantifier elimination was first proved by Tarski [31] (see also =-=[29]-=-). However, the complexity of his algorithm is not elementary recursive. The first algorithm with a an elementary-recursive worst-case time bound was given by Collins [11]. His algorithm had a worst c... |

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Citation Context ... in the number of variables. Heintz, Roy and Solerno [22] and Renegar [28] gave quantifier elimination algorithms which were doubly exponential only in the number of quantifier alternations. See also =-=[33]-=- for the lower bound proof. The best algorithm for this problem till now appeared in [3], which has a complexity of s (`+1)\Pi(k i +1) d (`+1)\PiO(k i ) : Moreover, the degrees of the polynomials appe... |

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Citation Context ...rithm for quantifier elimination in the first order theory of real closed fields that improves the complexity of the best known algorithm for this problem till now. Unlike previously known algorithms =-=[3, 28, 22]-=- the combinatorial part of the complexity (the part depending on the number of polynomials in the input) of this new algorithm is independent of the number of free variables. Moreover, under the assum... |

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Citation Context ... infinite collections of items to be stored in the database. They are a powerful generalization of Codd's relational model [32]. Different types of constraints have been considered by various authors =-=[27, 26, 17]-=- including dense linear order inequalities, real polynomial inequalities etc. Even though constraint databases permit infinite number of elements in the database, for the purposes of this paper it is ... |

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Citation Context ...ature). Recently however, there have been efforts to generalize computational geometry algorithms to problems where the constraints are low degree polynomials, not just linear ones (see, for example, =-=[1, 30, 24, 25, 5]-=- and also the survey by Chazelle [10]). From this point of view it makes sense to separate out the roles of the parameters s and d in the complexity analysis of algorithms in semi-algebraic geometry. ... |

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Citation Context ... Y 1 ; : : : ; Y ` . The degrees of the polynomials are bounded by d and their coefficients lie in a real closed field R: We are also given a first-order formula of the form (Q ! X [!] ) : : : (Q 1 X =-=[1]-=- )F (P 1 ; : : : ; P s ) (henceforth denoted \Phi(Y )) where Q i 2 f8; 9g; Q i 6= Q i+1 , Y = (Y 1 ; : : : ; Y ` ) is a block of ` free variables, X [i] is a block of k i variables with P 1i! k i = k;... |

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Citation Context ...rithm for quantifier elimination in the first order theory of real closed fields that improves the complexity of the best known algorithm for this problem till now. Unlike previously known algorithms =-=[3, 28, 22]-=- the combinatorial part of the complexity (the part depending on the number of polynomials in the input) of this new algorithm is independent of the number of free variables. Moreover, under the assum... |

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Citation Context ...ature). Recently however, there have been efforts to generalize computational geometry algorithms to problems where the constraints are low degree polynomials, not just linear ones (see, for example, =-=[1, 30, 24, 25, 5]-=- and also the survey by Chazelle [10]). From this point of view it makes sense to separate out the roles of the parameters s and d in the complexity analysis of algorithms in semi-algebraic geometry. ... |

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Citation Context ...y with natural domain semantics, outputs another query equivalent to it with active domain semantics. The equivalence of natural and active domain semantics over real closed fields was proved in [6], =-=[7]-=-. However, their proofs are essentially nonconstructive and one does not obtain an effective algorithm for converting a query with the natural domain semantics into an equivalent one with active domai... |

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Citation Context ...ples of queries that are not definable over finite models in a first order language with equality. Their undefinability has been shown using different proof techniques such as locality [16], 0/1 laws =-=[14, 15]-=-, EhrenfeuchtFraisse games [13]. These undefinability results also hold in the presence of an order relation [21]. Grumbach and Su [17] proved that parity is not definable over constraint databases wi... |

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Citation Context ...o generalize computational geometry algorithms to problems where the constraints are low degree polynomials, not just linear ones (see, for example, [1, 30, 24, 25, 5] and also the survey by Chazelle =-=[10]-=-). From this point of view it makes sense to separate out the roles of the parameters s and d in the complexity analysis of algorithms in semi-algebraic geometry. We call the part that depends on s th... |

10 | Complexity of computing semialgebraic descriptions of the connected components of a semialgebraic set - Basu, Pollack, et al. - 1998 |

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7 |
Ehrenfeucht an application of games to the completeness problem for formalized theories
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Citation Context ...ble over finite models in a first order language with equality. Their undefinability has been shown using different proof techniques such as locality [16], 0/1 laws [14, 15], EhrenfeuchtFraisse games =-=[13]-=-. These undefinability results also hold in the presence of an order relation [21]. Grumbach and Su [17] proved that parity is not definable over constraint databases with linear constraints. They als... |

3 | Uniform Quantifier Elimination and Constraint Query - Basu - 1997 |

2 |
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Citation Context ...A preliminary version of this paper appeared in the Proceedings of the 38th IEEE Symposium on the Foundations of Computer Science, 1997. recently, a constructive proof was also given independently in =-=[8]-=-. However, complexity issues were not considered and no algorithm with a reasonable complexity bound was known for this latter problem till now. We also point out interesting logical consequences of t... |

2 |
M.-.F Roy Thom's lemma, the coding of real algebraic numbers and the topology of semi-algebraic sets
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Citation Context ... description of these subroutines. For x 2 R and f 2 R[X] we write oe x;f for the sign vector (sign(f (0) (x)); sign(f (1) (x)); : : : ; sign(f (deg(f)) (x): It is a consequence of Thom's lemma ([9], =-=[12]-=-) that if f(x) = 0 then oe x;f distinguishes x from all the other roots of f and if oe x;f 6= oe y;f then these two sign vectors easily enable us to determine whether x ! y or y ! x. 12 A k-univariate... |