Lab for Automated Reasoning and Analysis LARA

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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:​** **No self-loops:​**
-\[+\begin{equation*}
     \forall x.\ \lnot edge(x,x)     \forall x.\ \lnot edge(x,x)
-\]+\end{equation*}
  
 **Undirected graph:** **Undirected graph:**
-\[+\begin{equation*}
    ​\forall x.\ edge(x,y) \rightarrow edge(y,x)    ​\forall x.\ edge(x,y) \rightarrow edge(y,x)
-\]+\end{equation*}
  
 **Tournament:​** **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. 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.
 
sav08/graphs_as_interpretations.txt · Last modified: 2015/04/21 17:30 (external edit)
 
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