# 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] |
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- | ====== Graphs as Interpretations ====== | ||

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- | 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 $\alpha(edge) = E$. | ||

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- | Example: $D = \{1,2,3,4\}$, $\alpha(edge) = \{ (1,2), (2,3), (1,3), (3,4) \}$. | ||

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- | For a class of graph properties we can write down a formula $F$ such that property holds for graph iff $F$ is true in the interpretation $I$ representing the graph. | ||

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- | **No self-loops:** | ||

- | \[ | ||

- | \forall x.\ \lnot edge(x,x) | ||

- | \] | ||

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- | **Undirected graph:** | ||

- | \[ | ||

- | \forall x.\ edge(x,y) \rightarrow edge(y,x) | ||

- | \] | ||

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- | **Tournament:** | ||

- | \[ | ||

- | \forall x, y.\ (edge(x,y) \lor edge(y,x)) \land \lnot (edge(x,y) \land edge(y,x)) | ||

- | \] | ||

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- | 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. | ||

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- | * [[http://citeseer.ist.psu.edu/benedikt95relational.html|Relational Expressive Power of Constraint Query Language]] | ||

- | * [[http://citeseer.ist.psu.edu/context/64580/0|H. Gaifman, On local and non-local properties, in Logic Colloquium '81, North Holland, 1982]] | ||

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- | Many more properties become expressible if we take as domain $D$ the set of all subsets of $V$ and allow set operations in our language. | ||