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sav08:hoare_logic [2008/02/29 19:20] vkuncak |
sav08:hoare_logic [2008/03/03 10:31] pedagand |
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| {r = x * y} | {r = x * y} | ||
| </code> | </code> | ||
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| ===== Hoare Triple for Sets and Relations ===== | ===== Hoare Triple for Sets and Relations ===== | ||
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| means | means | ||
| \[ | \[ | ||
| - | \forall s,s' \in S. s \in S \land (s,s') \in r \rightarrow s' \in Q | + | \forall s,s' \in S. s \in P \land (s,s') \in r \rightarrow s' \in Q |
| \] | \] | ||
| We call $P$ precondition and $Q$ postcondition. | We call $P$ precondition and $Q$ postcondition. | ||
| 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 ==== | ||