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# Differences

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sav08:interval_analysis_and_widening [2008/05/20 20:34]
vkuncak
sav08:interval_analysis_and_widening [2015/04/21 17:30] (current)
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How to interpret a command How to interpret a command
-$+\begin{equation*} x = y \otimes z x = y \otimes z -$+\end{equation*}
for some operation $\otimes$? for some operation $\otimes$?

$m(x) =$ ++++| $m(x) =$ ++++|
-$+\begin{equation*} [\min \{v_y \otimes v_z \mid v_y \in m(y), v_z \in m(z) \}, [\min \{v_y \otimes v_z \mid v_y \in m(y), v_z \in m(z) \}, \max \{v_y \otimes v_z \mid v_y \in m(y), v_z \in m(z) \}] \max \{v_y \otimes v_z \mid v_y \in m(y), v_z \in m(z) \}] -$+\end{equation*}
++++ ++++

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In fixpoint computation,​ compose $H_i$ with function ​ In fixpoint computation,​ compose $H_i$ with function ​
-$+\begin{equation*} w([a,b]) = [\max \{x \in J \mid x \le a\}, w([a,b]) = [\max \{x \in J \mid x \le a\}, \min \{x \in J \mid b \le x\}] \min \{x \in J \mid b \le x\}] -$+\end{equation*}

Approaches: Approaches:
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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 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
-$+\begin{equation*} - x_* = F(x_*) \sqsubseteq F^{\#}(y_*) \sqsubseteq (W \circ F^{\#​})(y_*) \sqsubseteq y_* + x_* = F^{\#}(x_*) \sqsubseteq F^{\#}(y_*) \sqsubseteq (W \circ F^{\#​})(y_*) \sqsubseteq y_* -$+\end{equation*}