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:galois_connection_on_lattices [2008/05/07 10:03] vkuncak |
sav08:galois_connection_on_lattices [2008/05/07 10:22] vkuncak |
||
---|---|---|---|
Line 1: | Line 1: | ||
==== 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 18: | Line 18: | ||
* $\alpha$ 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) | ||
+ | |||
+ | Abstraction function for sign analysis. | ||
+ | |||
+ | 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". | ||