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sav08:interval_analysis_and_widening [2008/05/20 20:33]
vkuncak
sav08:interval_analysis_and_widening [2009/01/21 14:53]
vkuncak
Line 54: Line 54:
 Narrowing: after finding fixpoint of $(W \circ F)^{\#}$, apply $F^{\#}$ to improve precision. Narrowing: after finding fixpoint of $(W \circ F)^{\#}$, apply $F^{\#}$ to improve precision.
  
-Observation:​ if $F^{\#}$ and $W$ are $\omega$-continuous functions and $x \sqsubseteq W(x)$ for all $x$, then narrowing will improve the result, that is, if $x_* = lfp (F^{\#})$ and $y_* = lfp (W \circ F^{\#})$, then+Observation:​ if $F^{\#}$ and $W$ are $\omega$-continuous functions and $x \sqsubseteq W(x)$ for all $x$, then narrowing will improve the result, that is, if $x_* = lfp (F^{\#})$ and $y_* = lfp (W \circ F^{\#})$, then $x_* \sqsubseteq y_*$ and
 \[ \[
-    x_* = F(x_*) \sqsubseteq F^{\#}(y_*) \sqsubseteq (W \circ F^{\#​})(y_*) \sqsubseteq y_*+    x_* = F^{\#}(x_*) \sqsubseteq F^{\#}(y_*) \sqsubseteq (W \circ F^{\#​})(y_*) \sqsubseteq y_*
 \] \]