Difference between revisions of "Quantifier Elimination"

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(New page: Quantifier elimination is implemented in the REDUCE package Redlog. To get an idea consider the following illustrating example over the real numbers: * Consider two b...)
 
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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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<p>
 
* 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>.
 +
<p>
 
* 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>.
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* 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>.
  

Revision as of 22:30, 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>.
<p>
  • 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]