| 1. | Formally, is the reflexive-transitive closure of ?!.
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| 2. | To preserve transitivity, one must take the transitive closure.
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| 3. | Let \ vdash ^ * denote the transitive closure of \ vdash.
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| 4. | Next consider the reflexive, transitive closure of the " successor " relation.
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| 5. | The core is based on the Presburger Arithmetic and the transitive closure operation.
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| 6. | In particular, there is no transitive closure of set membership for such hypergraphs.
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| 7. | When transitive closure is added to second-order logic instead, we obtain PSPACE.
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| 8. | The relational example constitutes a relation algebra equipped with an operation of reflexive transitive closure.
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| 9. | Equivalently, a set is hereditarily finite if and only if its transitive closure is finite.
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| 10. | A full transitive closure is not needed; a commutative transitive closure and even weaker forms suffice.
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