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sav08:galois_connection_on_lattices [2008/05/07 08:52] vkuncak created |
sav08:galois_connection_on_lattices [2008/05/07 10:18] vkuncak |
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| ==== Galois Connection ==== | ==== Galois Connection ==== | ||
| - | Galois connection is defined by two monotonic functions $\alpha : C \to A$ and $\gamma : A \to C$ between [[partial order]]s $\leq$ on $C$ and $\sqsubseteq$ on $A$, such that | + | Galois connection is defined by two monotonic functions $\alpha : C \to A$ and $\gamma : A \to C$ between [[:partial order]]s $\leq$ on $C$ and $\sqsubseteq$ on $A$, such that |
| \begin{equation*} | \begin{equation*} | ||
| \alpha(c) \sqsubseteq a\ \iff\ c \leq \gamma(a) \qquad (*) | \alpha(c) \sqsubseteq a\ \iff\ c \leq \gamma(a) \qquad (*) | ||
| Line 7: | Line 7: | ||
| for all $c$ and $a$ (intuitively, the condition means that $c$ is approximated by $a$). | for all $c$ and $a$ (intuitively, the condition means that $c$ is approximated by $a$). | ||
| - | **Lemma:** The condition $(*)$ is equivalent to the conjunction of these two conditions: | + | **Lemma:** The condition $(*)$ holds iff the conjunction of these two conditions: |
| \begin{eqnarray*} | \begin{eqnarray*} | ||
| c &\leq& \gamma(\alpha(c)) \\ | c &\leq& \gamma(\alpha(c)) \\ | ||
| \alpha(\gamma(a)) &\sqsubseteq& a | \alpha(\gamma(a)) &\sqsubseteq& a | ||
| \end{eqnarray*} | \end{eqnarray*} | ||
| - | hold for all $c$ and $a$. | + | holds for all $c$ and $a$. |
| **Lemma:** Let $\alpha$ and $\gamma$ satisfy the condition of Galois connection. Then the following three conditions are equivalent: | **Lemma:** Let $\alpha$ and $\gamma$ satisfy the condition of Galois connection. Then the following three conditions are equivalent: | ||
| - | |||
| * $\alpha(\gamma(a)) = a$ for all $a$ | * $\alpha(\gamma(a)) = a$ for all $a$ | ||
| * $\alpha$ is a [[wk>surjective function]] | * $\alpha$ is a [[wk>surjective function]] | ||
| * $\gamma$ is an [[wk>injective function]] | * $\gamma$ is an [[wk>injective function]] | ||
| + | |||
| + | ===== Using Galois Connection ===== | ||
| + | |||
| + | 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) | ||
| + | |||
| + | Define $sp^\#$ using $\alpha$ and $\gamma$: | ||
| + | ++++| | ||
| + | \[ | ||
| + | sp^\#(a,r) = \alpha(sp(\gamma(a),r) | ||
| + | \] | ||
| + | ++++ | ||
| + | |||
| + | ===== Best Abstract Transformer ===== | ||
| + | |||
| + | $sp^{\#}$ can in general be arbitrarily precise while preserving correctness. | ||
| + | |||
| + | Given $\gamma$, we would like $sp^{\#}$ to be as precise as possible while being correct. | ||
| + | |||
| + | The most precise $sp^{\#}$ is called "best abstract transformer". | ||