The proof of the IKW Theorem I gave in class was a bit sketchy; let me discuss two points about it, raised by Or and Pranjal.
1. The first point, raised by Or, was about appealing to the Impagliazzo-Wigderson Theorem (or implicitly, the Nisan-Wigderson Theorem). Normally when one thinks about that theorem one thinks of the statement, "If there is a language such that , then .
However, this is not what the IKW Theorem appeals to; you should think of that as a corollary of the IW/NW Theorems. The Theorems themselves are input-length-specific; e.g., for Nisan-Wigderson, the statement we're using is of the form "Let be a circuit on bits of size , and let be a function such that every size- circuit has correlation at most . Then [some explicit PRG outputting bits based on 's truth table and designs] -fools ."
2. The second point, raised by Pranjal, was about the uncareful (indeed, possibly incorrect) way I explained the derandomization of that comes out of assuming . The wrong way to say it is to start with a fixed language in and then deduce that Arthur's -time randomized part can be replaced by some other -time algorithm (with advice). That is, of course, pointless, if , since Arthur could just enumerate all possible random strings. Rather, the key is to show that all of can be done by a nondeterministic (advice-taking) Arthur who uses fixed-exponential nondeterministic time.
In other words, the correct key lemma is:
.
This is nontrivial because the same works for all languages in , regardless of their polytime complexity. And the proof of this uses the appeal to the length-specific IW Theorem described in Point 1 above.
Thursday, April 9, 2009
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