ECCC-Report TR10-109https://eccc.weizmann.ac.il/report/2010/109Comments and Revisions published for TR10-109en-usSun, 11 Jul 2010 03:39:28 +0300
Paper TR10-109
| A Counterexample to the Generalized Linial-Nisan Conjecture |
Scott Aaronson
https://eccc.weizmann.ac.il/report/2010/109In earlier work, we gave an oracle separating the relational versions of BQP and the polynomial hierarchy, and showed that an oracle separating the decision versions would follow from what we called the Generalized Linial-Nisan (GLN) Conjecture: that "almost k-wise independent" distributions are indistinguishable from the uniform distribution by constant-depth circuits. The original Linial-Nisan Conjecture was recently proved by Braverman; we offered a $200 prize for the generalized version. In this paper, we save ourselves $200 by showing that the GLN Conjecture is false, at least for circuits of depth 3 and higher.
As a byproduct, our counterexample also implies that Pi2P is not in P^NP relative to a random oracle with probability 1. It has been conjectured since the 1980s that PH is infinite relative to a random oracle, but the best previous result was NP!=coNP relative to a random oracle.
Finally, our counterexample implies that the famous results of Linial, Mansour, and Nisan, on the structure of AC0 functions, cannot be improved in several interesting respects.Sun, 11 Jul 2010 03:39:28 +0300https://eccc.weizmann.ac.il/report/2010/109