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sav08:definition_of_set_constraints [2008/05/22 11:50] vkuncak |
sav08:definition_of_set_constraints [2008/05/22 11:51] vkuncak |
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| f^{-2}(f(S,T)) = | f^{-2}(f(S,T)) = | ||
| \] | \] | ||
| - | ++++| | + | ++| |
| \[ | \[ | ||
| = f^{-2}(\{ f(s,t) \mid s \in S, t \in T \}) | = f^{-2}(\{ f(s,t) \mid s \in S, t \in T \}) | ||
| \] | \] | ||
| - | ++++ | + | ++ |
| - | + | ++| | |
| - | ++++| | + | |
| \[ | \[ | ||
| = \{ s_1 \in S \mid \exists t_1 \in T \} | = \{ s_1 \in S \mid \exists t_1 \in T \} | ||
| \] | \] | ||
| - | ++++ | + | ++ |
| - | + | ++| | |
| - | ++++| | + | |
| \[ | \[ | ||
| = \left\{ \begin{array}{rl} | = \left\{ \begin{array}{rl} | ||
| Line 131: | Line 129: | ||
| \right. | \right. | ||
| \] | \] | ||
| - | ++++ | + | ++ |
| - | + | ||
| - | An important property we would like to have in this semantic is a very intuitive one : \[ [[f^{-1}(f(S_1, S_2))]] = [[S_1]] \] | + | |
| - | This property is in fact not conserved as we can easily see using a very simple counter-example : | + | |
| - | \[ [[f^{-1}(f(S_1, \emptyset))]] = [[f^{-1}(\lbrace f(t_1, t_2) | t_1 \in [[S_1]] \wedge t_2 \in \emptyset \rbrace)]] = [[f^{-1}(\emptyset)]] = \emptyset \] | + | |
| - | + | ||
| - | The correct interpretation of this property is : | + | |
| - | \begin{displaymath} [[f^{-1}(f(S_1, S_2))]] = \left\{ \begin{array}{ll} | + | |
| - | \emptyset & if [[S_2]] = \emptyset & | + | |
| - | [[S_1]] & otherwise \\ | + | |
| - | \end{array} \right \end{displaymath} | + | |