# An Average-Case Depth Hierarchy Theorem for Boolean Circuits

```@article{Rossman2015AnAD,
title={An Average-Case Depth Hierarchy Theorem for Boolean Circuits},
author={Benjamin Rossman and Rocco A. Servedio and Li-Yang Tan},
journal={2015 IEEE 56th Annual Symposium on Foundations of Computer Science},
year={2015},
pages={1030-1048}
}```
• Published 13 April 2015
• Mathematics, Computer Science
• 2015 IEEE 56th Annual Symposium on Foundations of Computer Science
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 [Has86b]. Our average-case depth…
48 Citations

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## References

SHOWING 1-10 OF 92 REFERENCES
A Satisfiability Algorithm for AC\$^0\$
• Computer Science, Mathematics
SODA 2012
• 2011
A zero-error randomized algorithm is given which takes an AC0 circuit as input and constructs a set of restrictions which partitions {0, 1}n so that under each restriction the value of the circuit is constant.
More Applications of the Polynomial Method to Algorithm Design
• Computer Science, Mathematics
SODA
• 2015
This paper extends the polynomial method to solve a number of problems in combinatorial pattern matching and Boolean algebra, considerably faster than previously known methods.
A Counterexample to the Generalized Linial-Nisan Conjecture
• S. Aaronson
• Mathematics
Electron. Colloquium Comput. Complex.
• 2010
The counterexample implies that the famous results of Linial, Mansour, and Nisan, on the structure of AC functions, cannot be improved in several interesting respects.
Algebraic methods in the theory of lower bounds for Boolean circuit complexity
It is proved that depth k circuits with gates NOT, OR and MODp where p is a prime require Exp(&Ogr;(n1/2k)) gates to calculate MODr functions for any r ≠ pm.
Polylogarithmic Independence Can Fool DNF Formulas
• L. Bazzi
• Mathematics
48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)
• 2007
It is shown that any k-wise independent probability measure on {0, 1}n can O(m2ldr 2ldr2-radick/10)-fool any boolean function computable by an rn-clauses DNF (or CNF) formula on n variables, and this resolves, asymptotically and up to a logm factor, the depth-2 circuits case of a conjecture due to Linial and Nisan (1990).
Approximating AC^0 by Small Height Decision Trees and a Deterministic Algorithm for #AC^0SAT
• Computer Science, Mathematics
2012 IEEE 27th Conference on Computational Complexity
• 2012
It is shown that any function in n variables computable by an unbounded fan-in circuit of AND, OR, and NOT gates that has size S and depth d can be approximated by a decision tree of height n - βn to within error exp(-βn), where β = β(S, d) = 2-O(d log4/5 S).
Bounded Arithmetic and Lower Bounds in Boolean Complexity
This work studies the question of provability of lower bounds on the complexity of explicitly given Boolean functions in weak fragments of Peano Arithmetic and gives a more constructive version of the proof of Hastad Switching Lemma.
Switching lemma for small restrictions and lower bounds for k-DNF resolution
• Mathematics
The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings.
• 2002
We prove a new switching lemma that works for restrictions that set only a small fraction of the variables and is applicable to DNFs with small conjunctions. We use this to prove lower bounds for the
Computational limitations of small-depth circuits
The techniques described in "Computational Limitations for Small Depth Circuits" can be used to demonstrate almost optimal lower bounds on the size of small depth circuits computing several different functions, such as parity and majority.
Approximation by DNF: Examples and Counterexamples
• Mathematics, Computer Science
ICALP
• 2007
For every constant 0 < e < 1/2 there is a DNF of size 2O√n that e-approximates the Majority function on n bits, and this is optimal up to the constant in the exponent.