An average-case depth hierarchy theorem for boolean circuits
We prove an average-case depth hierarchy theorem for Boolean circuits over the standard basis of AND, OR, and NOT gates. Our hierarchy theorem says that for every d ≥ 2, there is an explicit n-variable Boolean function f , computed by a linear-size depth-d formula,which is such that any depth-(d - 1) circuit that agrees with f on (1/2 + on (1)) fraction of all inputs must have size exp(nΩ(1/d) ). This answers an open question posed by Hastad in his Ph.D. thesis (Hastad 1986b). Our average-case depth hierarchy theorem implies that the polynomial hierarchy is infinite relative to a random oracle with probability 1, confirming a conjecture of Hastad (1986a), Cai (1986), and Babai (1987). We also use our result to show that there is no "approximate converse" to the results of Linial, Mansour, Nisan (Linial et al. 1993) and (Boppana 1997) on the total influence of bounded-depth circuits. A key ingredient in our proof is a notion of random projections which generalize random restrictions.
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- Computation Theory & Mathematics
- 46 Information and computing sciences
- 08 Information and Computing Sciences
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Published In
DOI
EISSN
ISSN
Publication Date
Volume
Issue
Related Subject Headings
- Computation Theory & Mathematics
- 46 Information and computing sciences
- 08 Information and Computing Sciences