Thursday, April 9, 2009

Details for the IKW Theorem

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 LE such that L\notc\SIZE(nc), then BPPεDTIME(2nε).

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 C be a circuit on n bits of size nc, and let f:{0,1}n1/10c{0,1} be a function such that every size-n2c circuit has correlation at most 1/n2c. Then [some explicit PRG outputting n bits based on f's truth table and designs] 1/10-fools C."

2. The second point, raised by Pranjal, was about the uncareful (indeed, possibly incorrect) way I explained the derandomization of MA that comes out of assuming NEXPEXP. The wrong way to say it is to start with a fixed language in MA and then deduce that Arthur's nc-time randomized part can be replaced by some other NTIME(2na)-time algorithm (with advice). That is, of course, pointless, if ac, since Arthur could just enumerate all possible random strings. Rather, the key is to show that all of MA can be done by a nondeterministic (advice-taking) Arthur who uses fixed-exponential nondeterministic time.

In other words, the correct key lemma is:

NEXP\EXPc,MAio-[NTIME(2nc)/n].

This is nontrivial because the same c works for all languages in MA, regardless of their polytime complexity. And the proof of this uses the appeal to the length-specific IW Theorem described in Point 1 above.

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