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On the combinatorial and algebraic complexity of quantifier elimination

Authors:

Saugata Basu,

Richard Pollack,

Marie-Françoise RoyAuthors Info & Claims

Journal of the ACM (JACM), Volume 43, Issue 6

Pages 1002 - 1045

https://doi.org/10.1145/235809.235813

Published: 01 November 1996 Publication History

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Abstract

In this paper, a new algorithm for performing quantifier elimination from first order formulas over real closed fields in given. This algorithm improves the complexity of the asymptotically fastest algorithm for this problem, known to this data. A new feature of this algorithm is that the role of the algebraic part (the dependence on the degrees of the imput polynomials) and the combinatorial part (the dependence on the number of polynomials) are sparated. 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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Reviews

Reviewer: Robert Stewart Roos

The problem of quantifier elimination for first-order formulas over real closed fields has applications in areas such as control theory and control system design. Consequently, this long, dense paper deserves a wider readership than its title and abstract might at first suggest. It is largely self-contained, but the reader will need a thorough grounding in the theory of real closed fields to appreciate the details. The authors begin with a lengthy review of algebraic geometry and algorithms for solving multivariate polynomial systems, counting real solutions, and determining signs. They then present an algorithm for deciding the existential theory of real closed fields. For a Boolean formula with s polynomials, each having degree at most d , the algorithm's complexity is a product of two expressions, one exponential in d , the other exponential in s . (The complexity of the previous best algorithm for the decision problem did not achieve this separation in dependence on the combinatorial and algebraic aspects of the problem.) This is followed by an algorithm for the general decision problem for real closed fields. Finally, the authors present the quantifier elimination algorithm. Both of these algorithms have complexities that exhibit separation of the combinatorial and algebraic parts. The quantifier elimination algorithm improves the combinatorial complexity of the previously best known algorithm, due to Renegar. All three algorithms are well behaved; in particular, all three lend themselves readily to parallel implementation. There is a lengthy set of references. Clearly, a lot of care has been taken to present a difficult algorithm in a way that will appeal to the largest number of readers.

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Published In

Journal of the ACM Volume 43, Issue 6

Nov. 1996

174 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/235809

Issue’s Table of Contents

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 November 1996

Published in JACM Volume 43, Issue 6

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Abstract

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The complexity of quantifier elimination and cylindrical algebraic decomposition