Variant quantifier elimination

  • Authors:

  • Hoon Hong;Mohab Safey El Din

  • Affiliations:

  • Department of Mathematics, North Carolina State University, Raleigh NC 27695, USA;INRIA, Paris-Rocquencourt, SALSA Project, UPMC, Univ Paris 06, LIP6 CNRS, UMR 7606, LIP6 UFR Ingéniérie 919, LIP6 Passy-Kennedy Case 169, 4, Place Jussieu, F-75252 Paris, France

  • Venue:
  • Journal of Symbolic Computation
  • Year:
  • 2012

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Abstract

We describe an algorithm (VQE) for a variant of the real quantifier elimination problem (QE). The variant problem requires the input to satisfy a certain extra condition, and allows the output to be almost equivalent to the input. The motivation/rationale for studying such a variant QE problem is that many quantified formulas arising in applications do satisfy the extra conditions. Furthermore, in most applications, it is sufficient that the output formula is almost equivalent to the input formula. The main idea underlying the algorithm is to substitute the repeated projection step of CAD by a single projection without carrying out a parametric existential decision over the reals. We find that the algorithm can tackle important and challenging problems, such as numerical stability analysis of the widely-used MacCormack's scheme. The problem has been practically out of reach for standard QE algorithms in spite of many attempts to tackle it. However, the current implementation of VQE can solve it in about 12 hours. This paper extends the results reported at the conference ISSAC 2009.