| 11. | A full transitive closure is not needed; a commutative transitive closure and even weaker forms suffice.
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| 12. | More precisely, it is the transitive closure of the relation " is the mother of ".
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| 13. | Let \ Rightarrow ^ { * } be the reflexive transitive closure of the relation \ Rightarrow.
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| 14. | The Ancestral relation R ^ * is equal to the transitive closure R ^ + of R.
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| 15. | Therefore, the transitive reduction can be constructed in the same asymptotic time bounds as the transitive closure.
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| 16. | In computational complexity theory, the complexity class L is first-order logic with the commutative, transitive closure.
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| 17. | If is a DAG, its transitive closure is the graph with the most edges that represents the same reachability relation.
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| 18. | The "'trace "'is defined as the symmetric, reflexive and transitive closure of \ sim.
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| 19. | Finally, one takes the reflexive and transitive closure of " E ", which is then a monoid congruence.
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| 20. | First-order logic is too restrictive for some database applications, for instance because of its inability to express transitive closure.
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