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sav08:mapping_fixpoints_under_lattice_morphisms [2008/05/07 10:31] vkuncak |
sav08:mapping_fixpoints_under_lattice_morphisms [2009/03/26 13:53] vkuncak |
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====== Mapping Fixpoints Under Lattice Morphisms ====== | ====== Mapping Fixpoints Under Lattice Morphisms ====== | ||
- | **Lemma:** Let $(X,\le)$ and $(Y,\sqsubseteq)$ be complete lattices, and $F : X \to X$, $\Gamma : X \to Y$, $F^\# : Y \to Y$ be complete morphisms (they distribute through arbitrary least upper bound) such that | + | **Definition:** Let $(X,\le)$ and $(Y,\sqsubseteq)$ be complete [[lattices]]. We call $F : X \to Y$ a **complete join-morphism** iff for each set $X_1 \subseteq X$ we have |
+ | \[ | ||
+ | F(\sqcup X_1) = \sqcup \{ F(a).\ a \in X_1 \} | ||
+ | \] | ||
+ | |||
+ | For example, $F(a_1 \sqcup a_2 \sqcup a_3) = F(a_1) \sqcup F(a_2) \sqcup F(a_3) | ||
+ | |||
+ | **Lemma:** Let $(X,\le)$ and $(Y,\sqsubseteq)$ be complete lattices, and $F : X \to X$, $\Gamma : X \to Y$, $F^\# : Y \to Y$ be complete join-morphisms such that | ||
\[ | \[ | ||
F(\Gamma(y)) \le \Gamma(F^\#(y)) | F(\Gamma(y)) \le \Gamma(F^\#(y)) | ||
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In other words, we can approximate $lfp(F)$ by computing $lfp(F^\#)$. | In other words, we can approximate $lfp(F)$ by computing $lfp(F^\#)$. | ||
- | **Definition:** Height of the lattice is the size of the longest strictly increasing sequence of elements $a_1 \sqsubset a_2 \sqsubset a_3 \ldots$. | ||
- | Lattice has finite height iff there exists no infinite strictly increasing sequence of elements in the lattice. | ||
- | |||
- | Observation: it is decidable to find least fixpoint of morphism $F^{\#}$ in a lattice with finite height. | ||
- | |||
- | Lattices with finite height | ||
- | * finite lattice | ||
- | * constant propagation lattice | ||