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sav08:abstract_interpretation_recipe [2008/05/08 12:40] vkuncak |
sav08:abstract_interpretation_recipe [2015/04/21 17:30] (current) |
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| * $E \subseteq V \times V$ 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 ==== | ||
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| (for example, by defining function $\alpha$ so that $(\alpha,\gamma)$ becomes a [[Galois Connection on Lattices]]) | (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, $g^{\#} : V \to A$) | ||
| - | \[ | + | \begin{equation*} |
| - | F^{\#}(g^{\#})(p_2) = g^{\#}(p_2) \sqcup \bigsqcup_{(p_1,p_2) \in E} sp^{\#}(g^{\#}(p_1),r(p_1,p_2)) | + | F^{\#}(g^{\#})(p') = g^{\#}(p') \sqcup \bigsqcup_{(p,p') \in E} sp^{\#}(g^{\#}(p),r(p,p')) |
| - | \] | + | \end{equation*} |
| which is analogous to [[Collecting Semantics]] | which is analogous to [[Collecting Semantics]] | ||
| * compute $g^{\#}_* = lfp(F^\#)$ (this is easier than computing collecting semantics because lattice $A^n$ is simpler than $C^n$): | * compute $g^{\#}_* = lfp(F^\#)$ (this is easier than computing collecting semantics because lattice $A^n$ is simpler than $C^n$): | ||
| - | \[ | + | \begin{equation*} |
| g^{\#}_* = \bigsqcup_{n \ge 0} (F^{\#})^{n}(\bot^{\#}) | g^{\#}_* = \bigsqcup_{n \ge 0} (F^{\#})^{n}(\bot^{\#}) | ||
| - | \] | + | \end{equation*} |
| where $\bot^{\#}(p) = \bot_A$ for all $p \in V$ | where $\bot^{\#}(p) = \bot_A$ for all $p \in V$ | ||