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:
  
* 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,\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</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>.
  
* 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>.
+
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>.
+
[[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>.
 +
 
 +
<p>
  
 
[[image:qe-example.png|center]]
 
[[image:qe-example.png|center]]

Revision as of 01:28, 24 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,\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</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>.

Qe-example.png

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

References

  • The Redlog Homepage [1]