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 1978), the Karp–Lipton theorem is also evidence that the use of randomization does not lead to polynomial time algorithms for NP-complete problems.
Suppose that polynomial sized circuits for SAT not only exist, but also that they could be constructed by a polynomial time algorithm. Then this supposition implies that SAT itself could be solved by a polynomial time algorithm that constructs the circuit and then applies it. That is, efficiently constructable circuits for SAT would lead to a stronger collapse, P = NP.
The assumption of the Karp–Lipton theorem, that these circuits exist, is weaker. But it is still possible for an algorithm in the complexity class to guess a correct circuit for SAT. The complexity class describes problems of the form
To understand the Karp–Lipton proof in more detail, we consider the problem of testing whether a circuit c is a correct circuit for solving SAT instances of a given size, and show that this circuit testing problem belongs to . That is, there exists a polynomial time computable predicate V such that c is a correct circuit if and only if, for all polynomially-bounded z, V(c,z) is true.
The circuit c is a correct circuit for SAT if it satisfies two properties:
The first of these two properties is already in the form of problems in class . To verify the second property, we use the self-reducibility property of SAT.
Self-reducibility describes the phenomenon that, if we can quickly test whether a SAT instance is solvable, we can almost as quickly find an explicit solution to the instance. To find a solution to an instance s, choose one of the Boolean variables x that is input to s, and make two smaller instances s0 and s1 where si denotes the formula formed by replacing x with the constant i. Once these two smaller instances have been constructed, apply the test for solvability to each of them. If one of these two tests returns that the smaller instance is satisfiable, continue solving that instance until a complete solution has been derived.
To use self-reducibility to check the second property of a correct circuit for SAT, we rewrite it as follows:
Thus, we can test in whether c is a valid circuit for solving SAT.
see Random_self-reducibility for more information
The Karp–Lipton theorem can be restated as a result about Boolean formulas with polynomially-bounded quantifiers. Problems in are described by formulas of this type, with the syntax