Differences

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--- sav08:graphs_as_interpretations [2008/03/23 13:50]
maysam English Grammer
+++ sav08:graphs_as_interpretations [2015/04/21 17:30] (current)
@@ Line 8: / Line 8: @@
 **No self-loops:**
-\[
+\begin{equation*}
     \forall x.\ \lnot edge(x,x)
-\]
+\end{equation*}
 **Undirected graph:**
-\[
+\begin{equation*}
    \forall x.\ edge(x,y) \rightarrow edge(y,x)
-\]
+\end{equation*}
 **Tournament:**
-\[
+\begin{equation*}
-   \forall x, y.\ (edge(x,y) \lor edge(y,x)) \land \lnot (edge(x,y) \land edge(y,x))
+   (\forall x, y.\ x \neq y \rightarrow (edge(x,y) \lor edge(y,x)) \land \lnot (edge(x,y) \land edge(y,x))) \land (\forall x. \lnot edge(x,x))
-\]
+\end{equation*}
 Note: there is no formula $F$ in this language ${\cal L} = \{edge\}$ that characterizes property "graph has no cycles".  All properties expressed in first-order logic on graphs are "local". Intuitively, formula with $k$ universal quantifiers says that if we pick any set of $k$ vertices in the graph, then they (and their close neighbors) can induce only one of the finitely many specified subgraphs.