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sav08:definition_of_set_constraints [2008/05/22 11:51] vkuncak |
sav08:definition_of_set_constraints [2008/05/22 11:55] vkuncak |
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Let us simplify this expression: | Let us simplify this expression: | ||
\[ | \[ | ||
- | f^{-2}(f(S,T)) = | + | f^{-1}(f(S,T)) = |
\] | \] | ||
++| | ++| | ||
\[ | \[ | ||
- | = f^{-2}(\{ f(s,t) \mid s \in S, t \in T \}) | + | = f^{-1}(\{ f(s,t) \mid s \in S, t \in T \}) |
\] | \] | ||
++ | ++ | ||
Line 131: | Line 131: | ||
++ | ++ | ||
- | An important property we would like to have in this semantic is a very intuitive one : \[ [[f^{-1}(f(S_1, S_2))]] = [[S_1]] \] | + | Consider conditional constraints of the form |
- | This property is in fact not conserved as we can easily see using a very simple counter-example : | + | \[ |
- | \[ [[f^{-1}(f(S_1, \emptyset))]] = [[f^{-1}(\lbrace f(t_1, t_2) | t_1 \in [[S_1]] \wedge t_2 \in \emptyset \rbrace)]] = [[f^{-1}(\emptyset)]] = \emptyset \] | + | T \neq \emptyset \rightarrow S_1 \subseteq S_2 |
- | + | \] | |
- | The correct interpretation of this property is : | + | We can transform it into |
- | \begin{displaymath} [[f^{-1}(f(S_1, S_2))]] = \left\{ \begin{array}{ll} | + | \[ |
- | \emptyset & if [[S_2]] = \emptyset & | + | f(S_1,T) \subseteq f(S_2,T) |
- | [[S_1]] & otherwise \\ | + | \] |
- | \end{array} \right \end{displaymath} | + | for some binary function symbol $f$. Thus, we can introduce such conditional constraints without changing expressive power or decidability. |