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sav08:hoare_logic [2009/02/25 14:35] vkuncak |
sav08:hoare_logic [2009/03/04 11:03] vkuncak |
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| Hoare logic is a way of inserting annotations into code to make proofs about program behavior simpler. | Hoare logic is a way of inserting annotations into code to make proofs about program behavior simpler. | ||
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| ===== Example Proof ===== | ===== Example Proof ===== | ||
| - | <code> | + | <code java> |
| - | {0 <= y} | + | //{0 <= y} |
| i = y; | i = y; | ||
| - | {0 <= y & i = y} | + | //{0 <= y & i = y} |
| r = 0; | r = 0; | ||
| - | {0 <= y & i = y & r = 0} | + | //{0 <= y & i = y & r = 0} |
| - | while {r = (y-i)*x & 0 <= i} | + | while //{r = (y-i)*x & 0 <= i} |
| (i > 0) ( | (i > 0) ( | ||
| - | {r = (y-i)*x & 0 < i} | + | //{r = (y-i)*x & 0 < i} |
| r = r + x; | r = r + x; | ||
| - | {r = (y-i+1)*x & 0 < i} | + | //{r = (y-i+1)*x & 0 < i} |
| i = i - 1 | i = i - 1 | ||
| - | {r = (y-i)*x & 0 <= i} | + | //{r = (y-i)*x & 0 <= i} |
| ) | ) | ||
| - | {r = x * y} | + | //{r = x * y} |
| </code> | </code> | ||
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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. | + | This is simply relation image of a set. (See [[Sets and relations#Relation Image]].) |
| {{sav08:sp.png?400x250|}} | {{sav08:sp.png?400x250|}} | ||