## [My Computational Complexity Web Log] Foundations of Complexity Lesson 16: Ladner's Theorem

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• Previous Lesson In the 1950 s, Friedberg and Muchnik independently showed that there were sets that were computably enumerable, not computable and not
Message 1 of 1 , Mar 24, 2003
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In the 1950's, Friedberg and Muchnik independently showed that there were sets that were computably enumerable, not computable and not complete. Does a similar result hold for complexity theory?

Suppose P≠NP. We have problems that are in P and problems that are NP-complete and we know these sets are disjoint. Is there anything else in NP? In 1975, Ladner showed the answer is yes.

Theorem (Ladner) If P≠NP then there is a set A in NP such that A is not in P and A is not NP-complete.

I wrote up two proofs of this result, one based on Ladner's proof and one based on a proof of Impagliazzo. The write-up is taken mostly from a paper by Rod Downey and myself.

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Posted by Lance Fortnow to My Computational Complexity Web Log at 3/24/2003 5:26:58 PM