Differences

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--- msol_over_strings [2007/05/11 16:58]
ghid.maatouk
+++ msol_over_strings [2007/05/27 13:27]
vaibhav.rajan
@@ Line 1: / Line 1: @@
 ====== MSOL over Strings ======
@@ Line 26: / Line 27: @@
   [\![F_1 \lor F_2]\!]e &=& [\![F_1]\!]e\ \lor\ [\![F_2]\!]e \\ \
   [\![\lnot F]\!]e &=& \lnot ([\![F]\!]e) \\ \
-  [\![\exists v. F]\!]e &=& \exists S. S\ \mbox{is finite}\ \land\ S \subseteq N_0. [\![F]\!](e[v \mapsto S])
+  [\![\exists v. F]\!]e &=& \exists S. S\ \mbox{is finite}\ \land\ S \subseteq N_0 \land [\![F]\!](e[v \mapsto S])
 \end{eqnarray*}
@@ Line 41: / Line 44: @@
   * Intersection: $(A = B \cap C) = (A \subseteq B \land A \subseteq C) \land (\forall A_1. A_1 \subseteq B \land A_1 \subseteq C \rightarrow A_1 \subseteq A)$
   * Union: $(A = B \cup C) = (B \subseteq A \land C \subseteq A) \land (\forall A_1. B \subseteq A_1 \land C \subseteq A_1 \rightarrow A \subseteq A_1)$
-  * Set difference: $(A = B \setminus C) = (A \cup C = B \land A \cap C = \emptyset)$
+  * Set difference: $(A = B \setminus C) = (A \cup (B \cap C) = B \land A \cap C = \emptyset)$
     (or just use element-wise definitions with singletons)
   * If $k$ is a fixed constant, properties $\mbox{card}(A) \geq k$, $\mbox{card}(A)\leq k$, $\mbox{card}(A)=k$
@@ Line 54: / Line 57: @@
 \end{equation*}
+This does not give the smallest set containing both $u$ and $v$. The reflexive transitive closure, T is:
+\begin{equation*}
+(F \subseteq T )\land (\forall x. x \in S \rightarrow (x,x) \in T ) \land ((\exists k.(u,k) \in T \land (k,v) \in T) \rightarrow ((u,v) \in T))
+\end{equation*}
+The underlying smallest set S containing $u$ and $v$ is given by:
+\begin{equation*}
+S = \{x | \exists k. ((k, x) \in T \lor (x,k) \in T) \}
+\end{equation*}
 **Using transitive closure and successors:**
   * Constant zero: $(x=0) = One(x) \land \lnot (\exists y. One(y) \land s(y,x))$