# Difference between revisions of "Quantifier Elimination"

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To get an idea consider the following illustrating example over the real numbers: | To get an idea consider the following illustrating example over the real numbers: | ||

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* Consider two bivariate polynomials with parameters a and b over the reals: <math>f(x,y)=x^2+xy+b</math> and <math>g(x,y)=x+ay^2+b</math>. | * Consider two bivariate polynomials with parameters a and b over the reals: <math>f(x,y)=x^2+xy+b</math> and <math>g(x,y)=x+ay^2+b</math>. | ||

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* We are interested in necessary and sufficient conditions on the parameters a and b such that the following holds: For each real point x there exists some real point y such that <math>f(x,y)>0</math> and <math>g(x,y)\leq0</math>. | * We are interested in necessary and sufficient conditions on the parameters a and b such that the following holds: For each real point x there exists some real point y such that <math>f(x,y)>0</math> and <math>g(x,y)\leq0</math>. | ||

## Revision as of 22:31, 23 April 2009

Quantifier elimination is implemented in the REDUCE package Redlog.

To get an idea consider the following illustrating example over the real numbers:

- Consider two bivariate polynomials with parameters a and b over the reals: <math>f(x,y)=x^2+xy+b</math> and <math>g(x,y)=x+ay^2+b</math>.

- We are interested in necessary and sufficient conditions on the parameters a and b such that the following holds: For each real point x there exists some real point y such that <math>f(x,y)>0</math> and <math>g(x,y)\leq0</math>.

- The problem can be straightforwardly formulated as a first-order formula: <math>\varphi=\forall x\exists y(x^2+xy+b>0 \land x+ay^2+b\leq0)</math>.

- REDLOG's quantifier elimintion delivers within half a second necessary and sufficient conditions, which are obviously equivalent to <math> a<0\land b>0</math>.

A comprehensive collection of Redlog examples can be found in the REMIS online database.

# References

- The Redlog Homepage [1]