Differences
This shows you the differences between two versions of the page.
Both sides previous revision Previous revision Next revision | Previous revision | ||
sav08:graphs_as_interpretations [2008/03/19 16:16] vkuncak |
sav08:graphs_as_interpretations [2008/03/25 16:07] vkuncak |
||
---|---|---|---|
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 19: | Line 19: | ||
**Tournament:** | **Tournament:** | ||
\[ | \[ | ||
- | \forall x, y.\ (edge(x,y) \lor edge(y,x)) \land \lnot (edge(x,y) \land edge(y,x)) | + | (\forall x, y.\ x \neq y \rightarrow (edge(x,y) \lor edge(y,x)) \land \lnot (edge(x,y) \land edge(y,x))) \land (\forall x. \lnot edge(x,x)) |
\] | \] | ||
- | 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". | + | 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. |
- | + | ||
- | 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. | + | |
* [[http://citeseer.ist.psu.edu/benedikt95relational.html|Relational Expressive Power of Constraint Query Language]] | * [[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]] | * [[http://citeseer.ist.psu.edu/context/64580/0|H. Gaifman, On local and non-local properties, in Logic Colloquium '81, North Holland, 1982]] | ||
+ | |||
+ | 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. | ||