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How do I prove Berman's theorem?

Submitted by: @import:stackexchange-cs··
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Problem

Berman's theorem states


If a unary language ( a language with all the strings of the type $1^i$, $ i > 0 $ ) is NP-Complete then P = NP.

I tried reducing SAT to a given unary language $L$ assuming it is NP-Complete. But I can't think of a way such that after applying the reduction so that SAT gets solved in polynomial time. How do I proceed further?

This is an exercise from Sanjeev Arora and Boaz Barak , but not homework.

Solution

Please refer this solution. Or see the paper by S. R. Mahaney, Sparse complete sets for NP: Solution of a conjecture by Berman and Hartmanis (Journal of Computer and System Sciences, 25:130-143, 1982).

Context

StackExchange Computer Science Q#53268, answer score: 4

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