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 n1+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).
Similar content being viewed by others
Author information
Authors and Affiliations
Corresponding author
Additional information
Manuscript received 4 June 2007
Rights and permissions
About this article
Cite this article
Viola, E. On Approximate Majority and Probabilistic Time. comput. complex. 18, 337 (2009). https://doi.org/10.1007/s00037-009-0267-3
Published:
DOI: https://doi.org/10.1007/s00037-009-0267-3
Keywords.
- Approximate majority
- probabilistic time
- constant-depth circuit
- alternating time
- polynomial-time hierarchy
- quasilinear time
- lower bound
- time-space tradeoff
- pseudorandom generator
- black-box