On Probabilistic Time versus Alternating Time
Sipser and GÃƒÂ¡cs, and independently Lautemann, proved in '83 that probabilistic polynomial time is contained in the second level of the polynomial-time hierarchy, i.e. BPP is in Sigma_2 P. This is essentially the only non-trivial upper bound that we have on the power of probabilistic computation. More precisely, the Sipser-GÃƒÂ¡cs-Lautemann simulation shows that probabilistic time can be simulated deterministically, using two quantifiers, **with a quadratic blow-up in the running time**. That is, BPTime(t) is contained in Sigma_2 Time(t^2).
In this talk we discuss whether this quadratic blow-up in the running time is necessary. We show that the quadratic blow-up is indeed necessary for black-box simulations that use two quantifiers, such as those of Sipser, GÃƒÂ¡cs, and Lautemann. To obtain this result, we prove a new circuit lower bound for computing **approximate majority**, i.e. computing the majority of a given bit-string whose fraction of 1's is bounded away from 1/2 (by a constant): We show that small depth-3 circuits for approximate majority must have bottom fan-in Omega(log n).
On the positive side, we obtain that probabilistic time can be simulated deterministically, using three quantifiers, in quasilinear time. That is, BPTime(t) is contained in Sigma_3 Time(t polylog t). Along the way, we show that approximate majority can be computed by uniform polynomial-size depth-3 circuits. This is a uniform version of a striking result by Ajtai that gives *non-uniform* polynomial-size depth-3 circuits for approximate majority.
If time permits, we will discuss some applications of our results to proving lower bounds on randomized Turing machines.
Probabilistic time
alternating time
polynomial-time hierarchy
approximate majority
constant-depth circuit
1-0
Regular Paper
Emanuele
Viola
Emanuele Viola
10.4230/DagSemProc.06111.11
Creative Commons Attribution 4.0 International license
https://creativecommons.org/licenses/by/4.0/legalcode