# Quantifier Elimination

From redlog.eu

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,\quad 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,\quad 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]