Karp–Lipton theorem

In complexity theory, the Karp–Lipton theorem states that if the Boolean satisfiability problem (SAT) can be solved by Boolean circuits with a polynomial number of logic gates, then

That is, if we assume that NP, the class of nondeterministic polynomial time problems, can be contained in the non-uniform polynomial time complexity class P/poly, then this assumption implies the collapse of the polynomial hierarchy at its second level. Such a collapse is believed unlikely, so the theorem is generally viewed by complexity theorists as evidence for the nonexistence of polynomial size circuits for SAT or for other NP-complete problems. A proof that such circuits do not exist would imply that P ≠ NP. As P/poly contains all problems solvable in randomized polynomial time (Adleman's theorem), the Karp–Lipton theorem is also evidence that the use of randomization does not lead to polynomial time algorithms for NP-complete problems.

The Karp–Lipton theorem is named after Richard M. Karp and Richard J. Lipton, who first proved it in 1980. (Their original proof collapsed PH to ${\displaystyle \Sigma _{3}}$, but Michael Sipser improved it to ${\displaystyle \Sigma _{2}}$.)

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