The complexity of quantifier elimination and cylindrical algebraic decomposition

  • Authors:

  • Christopher W. Brown;James H. Davenport

  • Affiliations:

  • United States Naval Academy, Annapolis;University of Bath, Bath, England

  • Venue:
  • Proceedings of the 2007 international symposium on Symbolic and algebraic computation
  • Year:
  • 2007

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Abstract

This paper has two parts. In the first part we give a simple and constructive proof that quantifier elimination in real algebra is doubly exponential, even when there is only one free variable and all polynomials in the quantified input are linear. The general result is not new, but we hope the simple and explicit nature of the proof makes it interesting. The second part of the paper uses the construction of the first part to prove some results on the effects of projection order on CAD construction -- roughly that there are CAD construction problems for which one order produces a constant number of cells and another produces a doubly exponential number of cells, and that there are problems for which all orders produce a doubly exponential number of cells. The second of these results implies that there is a true singly vs. doubly exponential gap between the worst-case running times of several modern quantifier elimination algorithms and CAD-based quantifier elimination when the number of quantifier alternations is constant.