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--- sav08:definition_of_set_constraints [2008/05/22 11:50]
vkuncak
+++ sav08:definition_of_set_constraints [2008/05/22 11:55]
vkuncak
@@ Line 109: / Line 109: @@
 Let us simplify this expression:
 \[
-    f^{-2}(f(S,T)) =
+    f^{-1}(f(S,T)) =
 \]
-++++|
+++|
 \[
-   = f^{-2}(\{ f(s,t) \mid s \in S, t \in T \})
+   = f^{-1}(\{ f(s,t) \mid s \in S, t \in T \})
 \]
-++++
+++
+++|
-++++|
 \[
    = \{ s_1 \in S \mid \exists t_1 \in T \}
 \]
-++++
+++
+++|
-++++|
 \[
    = \left\{ \begin{array}{rl}
@@ Line 131: / Line 129: @@
      \right.
 \]
-++++
+++
-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}
+Consider conditional constraints of the form
+\[
+    T \neq \emptyset \rightarrow S_1 \subseteq S_2
+\]
+We can transform it into
+\[
+    f(S_1,T) \subseteq f(S_2,T)
+\]
+for some binary function symbol $f$.   Thus, we can introduce such conditional constraints without changing expressive power or decidability.