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sav08:hoare_logic [2008/03/03 09:44] pedagand |
sav08:hoare_logic [2008/03/03 10:31] pedagand |
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| Note: weakest conditions (predicates) correspond to largest sets; strongest conditions (predicates) correspond to smallest sets that satisfy a given property (Graphically, a stronger condition $x > 0 \land y > 0$ denotes one quadrant in plane, whereas a weaker condition $x > 0$ denotes the entire half-plane.) | Note: weakest conditions (predicates) correspond to largest sets; strongest conditions (predicates) correspond to smallest sets that satisfy a given property (Graphically, a stronger condition $x > 0 \land y > 0$ denotes one quadrant in plane, whereas a weaker condition $x > 0$ denotes the entire half-plane.) | ||
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| Note the similarity with relation composition. | Note the similarity with relation composition. | ||
| - | FIXME Graphical illustration. | + | {{sav08:sp.png?400x250|}} |
| ==== Lemma: Characterization of sp ==== | ==== Lemma: Characterization of sp ==== | ||
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| - $\{P\} r \{ sp(P,r) \}$ | - $\{P\} r \{ sp(P,r) \}$ | ||
| - $\forall Q \subseteq S.\ \{P\} r \{Q\} \rightarrow sp(P,r) \subseteq Q$ | - $\forall Q \subseteq S.\ \{P\} r \{Q\} \rightarrow sp(P,r) \subseteq Q$ | ||
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| ===== Weakest Precondition - wp ===== | ===== Weakest Precondition - wp ===== | ||
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| Note that this is in general not the same as $sp(Q,r^{-1})$ when relation is non-deterministic. | Note that this is in general not the same as $sp(Q,r^{-1})$ when relation is non-deterministic. | ||
| - | FIXME Graphical illustration. | + | {{sav08:wp.png?400x250|}} |
| ==== Lemma: Characterization of wp ==== | ==== Lemma: Characterization of wp ==== | ||