Difference between revisions of "Quantifier Elimination"

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:

$f(x,y)=x^2+xy+b,\quad g(x,y)=x+ay^2+b.$

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

$f(x,y)>0,\quad g(x,y)\leq0.$

The problem can be straightforwardly formulated as a first-order formula:

$\varphi=\forall x\exists y(x^2+xy+b>0 \land x+ay^2+b\leq0).$

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

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

References

• The Redlog Homepage [1]