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 sav08:graphs_as_interpretations [2008/03/19 16:37]vkuncak sav08:graphs_as_interpretations [2015/04/21 17:30] (current) Both sides previous revision Previous revision 2008/03/25 16:07 vkuncak 2008/03/23 13:50 maysam English Grammer2008/03/19 16:37 vkuncak 2008/03/19 16:16 vkuncak 2008/03/19 15:44 vkuncak 2008/03/19 15:43 vkuncak 2008/03/19 15:38 vkuncak 2008/03/19 15:38 vkuncak 2008/03/19 15:36 vkuncak created Next revision Previous revision 2008/03/25 16:07 vkuncak 2008/03/23 13:50 maysam English Grammer2008/03/19 16:37 vkuncak 2008/03/19 16:16 vkuncak 2008/03/19 15:44 vkuncak 2008/03/19 15:43 vkuncak 2008/03/19 15:38 vkuncak 2008/03/19 15:38 vkuncak 2008/03/19 15:36 vkuncak created Line 1: Line 1: ====== Graphs as Interpretations ====== ====== Graphs as Interpretations ====== - Directed graph is is given by a set of vertices $V$ and a set of edges $E \subseteq V \times V$.  Graph is therefore specified by an [[First-Order Logic Semantics|interpretation]] $I = (V,\alpha)$ in languge ${\cal L} = \{edge\}$ with $\alpha(edge) = E$. + Directed graph is given by a set of vertices $V$ and a set of edges $E \subseteq V \times V$.  Graph is therefore specified by an [[First-Order Logic Semantics|interpretation]] $I = (V,\alpha)$ in languge ${\cal L} = \{edge\}$ with $\alpha(edge) = E$. Example: $D = \{1,​2,​3,​4\}$,​ $\alpha(edge) = \{ (1,2), (2,3), (1,3), (3,4) \}$. Example: $D = \{1,​2,​3,​4\}$,​ $\alpha(edge) = \{ (1,2), (2,3), (1,3), (3,4) \}$. 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.