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msol_over_strings [2008/05/14 19:21] vkuncak |
msol_over_strings [2008/05/14 19:23] vkuncak |
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| \end{equation*} | \end{equation*} | ||
| - | This does not give the smallest set containing both $u$ and $v$. The reflexive transitive closure, T is: | ||
| - | \begin{equation*} | ||
| - | (F \subseteq T )\land (\forall x. x \in S \rightarrow (x,x) \in T ) \land ((\exists k.(u,k) \in T \land (k,v) \in T) \rightarrow ((u,v) \in T)) | ||
| - | \end{equation*} | ||
| - | The underlying smallest set S containing $u$ and $v$ is given by: | ||
| - | \begin{equation*} | ||
| - | S = \{x | \exists k. ((k, x) \in T \lor (x,k) \in T) \} | ||
| - | \end{equation*} | ||
| **Using transitive closure and successors:** | **Using transitive closure and successors:** | ||
| * Constant zero: $(x=0) = One(x) \land \lnot (\exists y. One(y) \land s(y,x))$ | * Constant zero: $(x=0) = One(x) \land \lnot (\exists y. One(y) \land s(y,x))$ | ||