Difference between revisions of "Quantifier Elimination"

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:<math>f(x,y)=x^2+xy+b,\quad g(x,y)=x+ay^2+b</math>.
 
:<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,\quad g(x,y)\leq0</math>.
  
 
The problem can be straightforwardly formulated as a first-order formula:
 
The problem can be straightforwardly formulated as a first-order formula:
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[[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>
 
<p>
  
 
[[image:qe-example.png|center]]
 
[[image:qe-example.png|center]]
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<br>
  
 
A comprehensive collection of [[Redlog]] examples can be found in the [[REMIS]] online database.
 
A comprehensive collection of [[Redlog]] examples can be found in the [[REMIS]] online database.

Revision as of 01:32, 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,\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>.

Qe-example.png


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

References

  • The Redlog Homepage [1]