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sav07_lecture_9 [2007/04/19 11:13] vkuncak |
sav07_lecture_9 [2007/05/11 17:26] (current) ghid.maatouk old revision restored |
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| \end{equation*} | \end{equation*} | ||
| where $a^1,\ldots,a^n$ are variables ranging over a lattice $A$ (one for each control-flow graph node), and $f_k$ are monotonic functions on $A$ (one for each edge in the control-flow graph). | where $a^1,\ldots,a^n$ are variables ranging over a lattice $A$ (one for each control-flow graph node), and $f_k$ are monotonic functions on $A$ (one for each edge in the control-flow graph). | ||
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| === Narrowing === | === Narrowing === | ||
| - | Once we have reached above the fixpoint (using widening or anything else), we can always apply the fixpoint transformer and improve the precision of the results. | + | Once we have reached above the fixpoint (using widening or anything else), we can always apply the fixpoint operator $G$ and improve the precision of the results. |
| + | |||
| + | If this is too slow, we can stop at any point or use a narrowing operator, the result is sound. | ||
| + | |||
| + | Instead of using the fixpoint operator $G$, we can use any operator that leaves our current value above fixpoint. | ||
| + | |||
| + | See Combination of Abstractions in ASTREE, Section 8.1, define $\Delta$ such that $\gamma(a) \cap \gamma(b) \subseteq \gamma(a \Delta b)$ | ||
| + | * example: take $\sqcap$ as $\Delta$, or even some improved version | ||
| === Reduced product === | === Reduced product === | ||
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| Reduced product: [[http://www.di.ens.fr/~cousot/COUSOTpapers/POPL79.shtml|Systematic Design of Program Analysis Frameworks]], Section 10.1. | Reduced product: [[http://www.di.ens.fr/~cousot/COUSOTpapers/POPL79.shtml|Systematic Design of Program Analysis Frameworks]], Section 10.1. | ||
| + | Key point: define $\sigma : A_1 \times A_2 \to A_1 \times A_2$ by | ||
| + | \begin{equation*} | ||
| + | \sigma(a_1,a_2) = \sqcap \{ (a_1',a_2') \mid \gamma(a_1') \cap \gamma(a_2') = \gamma(a_1) \cap \gamma(a_2) \} | ||
| + | \end{equation*} | ||
| + | The function $\sigma$ improves the precision of the element of the lattice without changing its meaning. Subsequent computations with these values then give more precise results. | ||