Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision Next revision Both sides next revision | ||
|
sav08:abstract_interpretation_recipe [2008/05/08 12:39] vkuncak |
sav08:abstract_interpretation_recipe [2008/05/08 12:41] vkuncak |
||
|---|---|---|---|
| Line 7: | Line 7: | ||
| * $E \subseteq V \times V$ are control-flow graph edges | * $E \subseteq V \times V$ are control-flow graph edges | ||
| * $r : E \to 2^{PS \times PS}$, so each $r(p_1,p_2) \subseteq PS \times PS$ is relation describing the meaning of command between $p_1$ and $p_2$ | * $r : E \to 2^{PS \times PS}$, so each $r(p_1,p_2) \subseteq PS \times PS$ is relation describing the meaning of command between $p_1$ and $p_2$ | ||
| + | We can define meaning of program in this form using [[Collecting Semantics]]. | ||
| ===== Summary of key steps ==== | ===== Summary of key steps ==== | ||
| Line 14: | Line 15: | ||
| * define lattice ordering $\sqsubseteq$ on $A$ such that $a_1 \le a_2 \rightarrow \gamma(a_1) \subseteq \gamma(a_2)$ | * define lattice ordering $\sqsubseteq$ on $A$ such that $a_1 \le a_2 \rightarrow \gamma(a_1) \subseteq \gamma(a_2)$ | ||
| * define $sp^\# : A \times R \to A$ that maps an abstract element and a CFG statement to new abstract element, such that $sp(\gamma(a),r) \subseteq \gamma ( sp^{\#} (a,r))$ | * define $sp^\# : A \times R \to A$ that maps an abstract element and a CFG statement to new abstract element, such that $sp(\gamma(a),r) \subseteq \gamma ( sp^{\#} (a,r))$ | ||
| - | (for example, by defining function $\alpha$ so that $(\alpha,\gamma)$ becomes a [[Galois Connection]]) | + | (for example, by defining function $\alpha$ so that $(\alpha,\gamma)$ becomes a [[Galois Connection on Lattices]]) |
| * extend $sp^{\#}$ to work on control-flow graphs, by defining $F^\# : (V \to A) \to (V \to A)$ as follows (below, $g^{\#} : V \to A$) | * extend $sp^{\#}$ to work on control-flow graphs, by defining $F^\# : (V \to A) \to (V \to A)$ as follows (below, $g^{\#} : V \to A$) | ||
| \[ | \[ | ||