Notes on Complexity Theory Last updated : December , 2011 Lecture 28

Recall that one motivation for studying non-uniform computation is the hope that it might be easier to prove lower bounds in that setting. (This is somewhat paradoxical, as non-uniform algorithms are more powerful than uniform algorithms; nevertheless, since circuits are more “combinatorial” in nature than uniform algorithms, there may still be justification for such hope.) The ultimate goal here would be to prove that NP 6⊂ P/poly, which would imply P 6 = NP. Unfortunately, after over two decades of attempts we are unable to prove anything close to this. Here, we show one example of a lower bound that we have been able to prove; we then discuss one “barrier” that partly explains why we have been unable to prove stronger bounds.