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sav08:chaotic_iteration_in_abstract_interpretation [2008/05/20 13:16] vkuncak |
sav08:chaotic_iteration_in_abstract_interpretation [2008/05/20 19:34] vkuncak |
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| - | ====== Chaotic Iteration in Abstract Interpretation ====== | + | ====== Chaotic Iteration in Abstract Interpretation: How to compute the fixpoint? ====== |
| In [[Abstract Interpretation Recipe]], note that if the set of program points is $p'_1,\ldots,p'_n$, then we are solving the system of eqautions in $n$ variables $g_1,\ldots,g_n$ | In [[Abstract Interpretation Recipe]], note that if the set of program points is $p'_1,\ldots,p'_n$, then we are solving the system of eqautions in $n$ variables $g_1,\ldots,g_n$ | ||
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| \begin{array}{ll} | \begin{array}{ll} | ||
| g^{k+1}_i = H_i(g^k_1,\ldots,g^k_n) \\ | g^{k+1}_i = H_i(g^k_1,\ldots,g^k_n) \\ | ||
| - | g^{k+1}_j = g^k_j, \mbox{ for } j \neq i & \mbox{\bf chaotic iteration} | + | & \mbox{\bf chaotic iteration} \\ |
| + | g^{k+1}_j = g^k_j, \mbox{ for } j \neq i | ||
| \end{array} | \end{array} | ||
| \] | \] | ||
| - | here we require that the new value $H_i(g^k_1,\ldots,g^k_n)$ differs from the old one $g^k_i$, otherwise we select a different one. | + | here we require that the new value $H_i(g^k_1,\ldots,g^k_n)$ differs from the old one $g^k_i$. An iteration where at each step we select some equation $i$ (arbitrarily) is called //chaotic iteration//. It is abstract representation of different iteration strategies. |
| - | Then we pick a different $i$, as long as the result changes. This is //chaotic iteration//. | + | |
| Questions: | Questions: | ||
| Line 50: | Line 50: | ||
| Compare values $I$, $L_1$, $C_1$, $I_n$, $C_n$ in the lattice | Compare values $I$, $L_1$, $C_1$, $I_n$, $C_n$ in the lattice | ||
| - | * in general | + | * in general ++|$C_1 \sqsubseteq L_1$, generally $C_i \sqsubseteq L_i$ ++ |
| - | * when selecting equations by fixed permutation | + | * when selecting equations by fixed permutation ++| $L_1 \sqsubseteq C_n$, generally $L_i \sqsubseteq C_{ni}$ ++ |
| ==== Worklist Algorithm and Iteration Strategies ==== | ==== Worklist Algorithm and Iteration Strategies ==== | ||
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| Strongly connected component (SCC) of a directed graph: path between each two nodes of component. | Strongly connected component (SCC) of a directed graph: path between each two nodes of component. | ||
| * compute until fixpoint within each SCC | * compute until fixpoint within each SCC | ||
| + | |||
| + | If we generate control-flow graph from our simple language, what do strongly connected components correspond to? | ||
| ===== References ===== | ===== References ===== | ||