# 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} }

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

An Average-Case Depth Hierarchy Theorem for Boolean Circuits

- Mathematics, Computer ScienceJ. ACM
- 2017

The average-case depth hierarchy theorem implies that the polynomial hierarchy is infinite relative to a random oracle with probability 1, confirming a conjecture of Håstad (1986a), Cai (1986), and Babai (1987).

A fixed-depth size-hierarchy theorem for AC0[⊕] via the coin problem

- Mathematics, Computer ScienceSTOC
- 2019

It is shown that for any fixed d, the class Cd,k of functions that have uniform AC0[⊕] formulas of depth d and size nk form an infinite hierarchy.

Constant Depth Formula and Partial Function

- Computer Science
- 2020

This work makes progress, on two fronts, towards showing MCSP is intractable under worst-case assumptions, and formulates a notion of lower bound statements being (P/poly)-recognizable that is closely related to Razborov and Rudich’s definition of being ( P/ poly)-constructive.

Constant Depth Formula and Partial Function Versions of MCSP are Hard

- Computer Science2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)
- 2020

This work makes progress, on two fronts, towards showing MCSP is intractable under worst-case assumptions, and formulates a notion of lower bound statements being (P/poly)-recognizable that is closely related to Razborov and Rudich's definition of being ( P/ poly)-constructive.

Superpolynomial Lower Bounds Against Low-Depth Algebraic Circuits

- Computer Science, Mathematics2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)
- 2022

It is observed that the first super polynomial lower bounds against general algebraic circuits of all constant depths over all fields of characteristic 0 implies the first deterministic sub-exponential time algorithm for solving the Polynomial Identity Testing (PIT) problem for all small depth circuits using the known connection between algebraic hardness and randomness.

On the Probabilistic Degree of an n-variate Boolean Function

- Computer Science, MathematicsElectron. Colloquium Comput. Complex.
- 2021

This paper shows that if the probabilistic degree of OR is ( log n ) c, then the minimum possible probabilism degree of such an f is at least ( logn ) c / ( c + 1 ) − o ( 1 ) , and it is shown that this is tight up to (log n ) o (1 ) factors.

Circuits with composite moduli

- Computer Science
- 2016

The main result is that every ACC circuit of polynomial size and depth d can be reduced to a depth-2 circuit SYM◦AND of size 2(logn)O(d) , which improves exponentially the previously best-known construction by Yao-Beigel-Tarui, which has size blowup 2 2O( d) .

Finer separations between shallow arithmetic circuits

- Mathematics, Computer ScienceElectron. Colloquium Comput. Complex.
- 2016

The proofs are much shorter and simpler than the two known proofs of n^{Omega(sqrt(d))} lower bound for homogeneous depth-4 circuits, albeit the proofs only work when d = O(log^2(n), which shows that the parameters of depth reductions are optimal for algebraic branching programs.

Poly-logarithmic Frege depth lower bounds via an expander switching lemma

- Mathematics, Computer ScienceSTOC
- 2016

We show that any polynomial-size Frege refutation of a certain linear-size unsatisfiable 3-CNF formula over n variables must have depth Ω(√logn). This is an exponential improvement over the previous…

A Near-Optimal Depth-Hierarchy Theorem for Small-Depth Multilinear Circuits

- Computer Science, Mathematics2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)
- 2018

Two separating examples may be viewed as algebraic analogues of variants of the Graph Reachability problem studied by Chen, Oliveira, Servedio and Tan (STOC 2016), who used them to prove lower bounds for constant-depth Boolean circuits.

## References

SHOWING 1-10 OF 92 REFERENCES

A Satisfiability Algorithm for AC$^0$

- Computer Science, MathematicsSODA 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, MathematicsSODA
- 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

- MathematicsElectron. 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

- Computer Science, MathematicsSTOC
- 1987

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

- Mathematics48th 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, Mathematics2012 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

- Mathematics, Computer Science
- 1995

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

- MathematicsThe 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

- Computer Science
- 1987

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 ScienceICALP
- 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.