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sav07_lecture_4_skeleton [2007/03/22 20:30] vkuncak |
sav07_lecture_4_skeleton [2007/03/22 20:43] vkuncak |
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| We use weakest preconditions, although you could also use strongest postconditions or any other variants of the conversion from programs to formulas. | We use weakest preconditions, although you could also use strongest postconditions or any other variants of the conversion from programs to formulas. | ||
| + | |||
| ===== More on wp ===== | ===== More on wp ===== | ||
| Line 64: | Line 65: | ||
| Benefit: if there is x_{n+1} that is not changed, we do not need to write its properties in the loop invariant. This can make loop invariant shorter. | Benefit: if there is x_{n+1} that is not changed, we do not need to write its properties in the loop invariant. This can make loop invariant shorter. | ||
| + | |||
| + | ==== References about weakest precondition (in construction) ==== | ||
| + | * Back, Wright: Refinement Calculus | ||
| + | * Dijkstra | ||
| + | * Hoare, He | ||
| ===== Modeling data structures ===== | ===== Modeling data structures ===== | ||
| Line 76: | Line 82: | ||
| Array bounds checking. | Array bounds checking. | ||
| + | |||
| ==== Semantics of references ==== | ==== Semantics of references ==== | ||
| Objects as references, null as an object. | Objects as references, null as an object. | ||
| + | |||
| + | Program with class declaration | ||
| + | |||
| + | <code java> | ||
| + | class Node { | ||
| + | Node left, right; | ||
| + | } | ||
| + | </code> | ||
| + | |||
| + | How can we represent fields? | ||
| + | |||
| + | Possible mathematical model: fields as functions from objects to objects. | ||
| + | |||
| + | left : Node => Node | ||
| + | right : Node => Node | ||
| + | |||
| + | What is the meaning of assignment? | ||
| + | |||
| + | x.f = y | ||
| + | |||
| + | <latex> | ||
| + | f[x \mapsto y](z) = \left\{ \begin{array}{lr} | ||
| + | y, & z=x \\ | ||
| + | f(z), & z \neq x | ||
| + | \end{array}\right. | ||
| + | </latex> | ||
| + | |||
| + | left, right - uninterpreted functions (can have any value, depending on the program, unlike arithmetic functions such as +,-,* that have fixed interpretation). | ||
| Null checks. | Null checks. | ||