| 11. | The second condition means that this lower bound is optimal.
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| 12. | Proofs of circuit lower bounds are strongly connected to derandomization.
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| 13. | But 4 to 6 months is more plausible as a lower bound.
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| 14. | All bounded sets have a lower bound, by definition.
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| 15. | Proving tight upper and lower bounds remains an open question.
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| 16. | The proof is typical of lower bounds in mechanism design.
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| 17. | Again, no tight lower bound has been proven.
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| 18. | Adding back the lower bound gives us our ranges:
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| 19. | Lower bounds for multiplication are also known for some classes of branching programs.
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| 20. | Lower bounds again are defined by inverting the order.
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