On Approximate Majority and Probabilistic Time

Abstract.

We prove new results on the circuit complexity of approximate majority, which is the problem of computing the majority of a given bit string whose fraction of 1’s is bounded away from 1/2 (by a constant). We then apply these results to obtain new relationships between probabilistic time, BPTime (t), and alternating time, ∑_O(1)Time (t). Our main results are the following:

• 1. We prove that depth-3 circuits with bottom fan-in (log n)/2 that compute approximate majority on n bits must have size at least $2^{n^{0.1}}$. As a corollary we obtain that there is no black-box proof that BPTime (t) $\subseteq \sum_2$Time (o(t²)). This complements the (black-box) result that BPTime (t) $\subseteq \sum_2$Time (t² · poly log t) (Sipser and Gács, STOC ’83; Lautemann, IPL ’83).

• 2. We prove that approximate majority is computable by uniform polynomial-size circuits of depth 3. Prior to our work, the only known polynomial-size depth-3 circuits for approximate majority were non-uniform (Ajtai, Ann. Pure Appl. Logic ’83). We also prove that BPTime (t) $\subseteq \sum_3$Time (t · poly log t). This complements our results in (1).

• 3. We prove new lower bounds for solving QSAT₃ $\in \sum_3$Time (n · poly log n) on probabilistic computational models. In particular, we prove that solving QSAT₃ requires time n^1+Ω(1) on Turing machines with a random-access input tape and a sequential-access work tape that is initialized with random bits. No nontrivial lower bound was previously known on this model (for a function computable in linear space).