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sav08:galois_connection_on_lattices [2008/05/07 10:14] vkuncak |
sav08:galois_connection_on_lattices [2008/05/08 12:33] vkuncak |
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| * $\alpha$ is a [[wk>surjective function]] | * $\alpha$ is a [[wk>surjective function]] | ||
| * $\gamma$ is an [[wk>injective function]] | * $\gamma$ is an [[wk>injective function]] | ||
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| ===== Using Galois Connection ===== | ===== Using Galois Connection ===== | ||
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| + | Terminology: | ||
| + | * $C$ - concrete domain | ||
| + | * $A$ - abstract domain | ||
| + | * $\gamma$ - concretization function | ||
| + | * $\alpha$ - abstraction function | ||
| + | * $sp$ - (the usual) strongest postcondition (or just "strongest post"), also called | ||
| + | * $sp^{\#}$ - abstract post, abstract transformer (transforms one abstract element into another) | ||
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| + | Abstraction function for sign analysis. | ||
| Define $sp^\#$ using $\alpha$ and $\gamma$: | Define $sp^\#$ using $\alpha$ and $\gamma$: | ||
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| ===== Best Abstract Transformer ===== | ===== Best Abstract Transformer ===== | ||
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| + | $sp^{\#}$ can in general be arbitrarily precise while preserving correctness. | ||
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| + | Given $\gamma$, we would like $sp^{\#}$ to be as precise as possible while being correct. | ||
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| + | The most precise $sp^{\#}$ is called "best abstract transformer". | ||
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| + | If $\alpha(\gamma(a)) = a$, then $sp^{\#}$ defined using $\alpha$ and $\gamma$ is the most precise one (given the abstract domain and particular blocks in the control-flow graph). | ||
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