@MISC{_1non-uniform, author = {}, title = {1 Non-Uniform Complexity}, year = {} }

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Abstract

We have seen that there exist “very hard ” languages (i.e., languages that require circuits of size (1 − ε)2n /n). If we can show that there exists a language in N P that is even “moderately hard” (i.e., requires circuits of super-polynomial size) then we will have proved P ̸ = N P. (In some sense, it would be even nicer to show some concrete language in N P that requires circuits of super-polynomial size. But mere existence of such a language is enough.) Here we show that for every c there is a language in Σ2 ∩ Π2 that is not in size(nc). Note that this does not prove Σ2 ∩ Π2 ̸ ⊆ P /poly since, for every c, the language we obtain is different. (Indeed, using the time hierarchy theorem, we have that for every c there is a language in P that is not in time(nc).) What is particularly interesting here is that (1) we prove a non-uniform lower bound and (2) the proof is, in some sense, rather simple. Theorem 1 For every c, there is a language in Σ4 ∩ Π4 that is not in size(n c). Proof Fix some c. For each n, let Cn be the lexicographically first circuit on n inputs such that (the function computed by) Cn cannot be computed by any circuit of size at most n c. By the non-uniform hierarchy theorem (see [1]), there exists such a Cn of size at most n c+1 (for n large