An average-case depth hierarchy theorem for Boolean circuits (1504.03398v1)

Abstract: We prove an average-case depth hierarchy theorem for Boolean circuits over the standard basis of $\mathsf{AND}$, $\mathsf{OR}$, and $\mathsf{NOT}$ gates. Our hierarchy theorem says that for every $d \geq 2$, 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 $(1/2 + o_n(1))$ fraction of all inputs must have size $\exp({n^{\Omega(1/d)}}).$ This answers an open question posed by H{\aa}stad in his Ph.D. thesis. 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 H{\aa}stad, Cai, and Babai. We also use our result to show that there is no "approximate converse" to the results of Linial, Mansour, Nisan and Boppana on the total influence of small-depth circuits, thus answering a question posed by O'Donnell, Kalai, and Hatami. A key ingredient in our proof is a notion of \emph{random projections} which generalize random restrictions.

• The paper establishes an average-case depth hierarchy theorem for Boolean circuits, showing that depth d circuits can compute functions much harder for depth d-1 circuits to approximate on average.
• A novel method using random projections extends traditional circuit simplification techniques to analyze average-case complexity scenarios more effectively.
• This work implies that the polynomial hierarchy is infinite relative to a random oracle with probability one, confirming theoretical conjectures and impacting areas like pseudorandomness and proof complexity.

An Average-case Depth Hierarchy Theorem for Boolean Circuits

The paper by Rossman, Servedio, and Tan addresses a pivotal aspect of computational complexity theory, focusing on Boolean circuits. Its primary contribution is the establishment of an average-case depth hierarchy theorem for Boolean circuits using gates from the standard basis of $$. This work provides valuable insights into the computational power landscape of small-depth circuits.

Summary of Main Results

The central result of the paper is that for every depth $d \geq 2$, there exists a function $f$ computed by a linear-size depth-$d$ formula which cannot be effectively approximated by any depth-$(d-1)$ circuit unless the circuit grows exponentially in size with respect to the number of inputs. Specifically, any depth-$(d-1)$ circuit that attempts to agree with $f$ on more than half of the inputs plus a diminishing fraction must have a size of at least $\exp(n^{\Omega(1/d)})$. This result builds directly upon a conjecture from Håstad's Ph.D. thesis, providing a definitive answer in the field of average-case complexity.

Moreover, an implication of this average-case depth hierarchy theorem is that the polynomial hierarchy becomes infinite relative to a random oracle with probability one. This remarkable finding solidifies the theorized conjectures by Håstad, Cai, and Babai, enhancing our understanding of relativized computational complexity.

Technical Insights

The authors introduce a novel method involving random projections—a generalization of random restrictions—critical to their approach. Traditional random restriction methods simplify the complexities of small-depth circuits but often fail to address average-case scenarios effectively. Random projections overcome this limitation by maintaining the structural integrity needed for worst-case circuit simplifications while allowing the formulation to extend to average-case complexity. This method also adapts the distributional framework to ensure that circuit simplifications complete to the uniform distribution.

Theoretical and Practical Implications

The theoretical implications are profound, confirming fundamental aspects of computational complexity. By establishing the infinitude of the polynomial hierarchy relative to random oracles, the authors provide key insights into the limits and capabilities of computational models, impacting areas such as pseudorandomness and proof complexity. Practically, this research may influence circuit design and complexity theory's application in cryptography, learning theory, and algorithmic construction.

Future Directions

Future explorations may explore analyzing specific types of circuits that could resist the depth hierarchy findings, possibly demanding modifications or extensions of the random projection technique. Additionally, investigating the versatility of these techniques in broader computational models or other bases of logic gates could yield further foundational insights. There remains significant opportunity to leverage this work in understanding complexity barriers and expanding the structural capabilities of computational models.

In conclusion, this paper makes a substantial contribution to complexity theory, revealing deep insights into the nuanced behaviors of Boolean circuits in average-case scenarios and offering compelling avenues for future research.