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Optimal Depth-Three Circuits for Inner Product

Mohit Gurumukhani, Daniel Kleber, Ramamohan Paturi, Christopher Rosin, Navid Talebanfard

TL;DR

The paper determines the 2-CNF-based depth-3 circuit complexity of the Inner Product function $IP_n$ up to poly$(n)$ factors, showing $\,\mathsf{Size}^2_3(IP_n) \le \mathrm{poly}(n)\cdot (9/5)^n$, matching the recent lower bound. It introduces a general, modular framework that reduces depth-3 circuit construction to designing consistent $k$-CNFs for orbits under the function’s automorphism group and then combining them via disjoint conjunctions. The construction for $IP_n$ is realized through a six-region partition of orbits, region-specific building blocks (notably Matching, 2Imp, and NAND) and computer-assisted optimization to maximize per-region coverage while preserving consistency with IP. The approach blends automorphism-based structural reduction with analytic combinatorics to bound CNF-coefficient growth, and it holds promise for applying to other highly symmetric functions as well as guiding future depth-3 complexity lower bounds.

Abstract

We show that Inner Product in $2n$ variables, $\mathbf{IP}_n(x, y) = x_1y_1 \oplus \ldots \oplus x_ny_n$, can be computed by depth-3 bottom fan-in 2 circuits of size $\mathsf{poly}(n)\cdot (9/5)^n$, matching the lower bound of Göös, Guan, and Mosnoi (Inform. Comput.'24). Our construction is obtained via the following steps. - We provide a general template for constructing optimal depth-3 circuits with bottom fan-in $k$ for an arbitrary function $f$. We do this in two steps. First, we partition $f^{-1}(1)$ into orbits of its automorphism group. Second, for each orbit, we construct one $k$-CNF that (a) accepts the largest number of inputs from that orbit and (b) rejects all inputs rejected by $f$. - We instantiate the template for $\mathbf{IP}_n$ and $k = 2$. Guided by the intuition (which we call modularity principle) that optimal 2-CNFs can be constructed by taking the conjunction of variable-disjoint copies of smaller $2$-CNFs, we use computer search to identify a small set of building block 2-CNFs over at most 4 variables. - We again use computer search to discover appropriate combinations (disjoint conjunctions) of building blocks to arrive at optimal 2-CNFs and analyze them using techniques from analytic combinatorics. We believe that the approach outlined in this paper can be applied to a wide range of functions to determine their depth-3 complexity.

Optimal Depth-Three Circuits for Inner Product

TL;DR

The paper determines the 2-CNF-based depth-3 circuit complexity of the Inner Product function up to poly factors, showing , matching the recent lower bound. It introduces a general, modular framework that reduces depth-3 circuit construction to designing consistent -CNFs for orbits under the function’s automorphism group and then combining them via disjoint conjunctions. The construction for is realized through a six-region partition of orbits, region-specific building blocks (notably Matching, 2Imp, and NAND) and computer-assisted optimization to maximize per-region coverage while preserving consistency with IP. The approach blends automorphism-based structural reduction with analytic combinatorics to bound CNF-coefficient growth, and it holds promise for applying to other highly symmetric functions as well as guiding future depth-3 complexity lower bounds.

Abstract

We show that Inner Product in variables, , can be computed by depth-3 bottom fan-in 2 circuits of size , matching the lower bound of Göös, Guan, and Mosnoi (Inform. Comput.'24). Our construction is obtained via the following steps. - We provide a general template for constructing optimal depth-3 circuits with bottom fan-in for an arbitrary function . We do this in two steps. First, we partition into orbits of its automorphism group. Second, for each orbit, we construct one -CNF that (a) accepts the largest number of inputs from that orbit and (b) rejects all inputs rejected by . - We instantiate the template for and . Guided by the intuition (which we call modularity principle) that optimal 2-CNFs can be constructed by taking the conjunction of variable-disjoint copies of smaller -CNFs, we use computer search to identify a small set of building block 2-CNFs over at most 4 variables. - We again use computer search to discover appropriate combinations (disjoint conjunctions) of building blocks to arrive at optimal 2-CNFs and analyze them using techniques from analytic combinatorics. We believe that the approach outlined in this paper can be applied to a wide range of functions to determine their depth-3 complexity.
Paper Structure (31 sections, 17 theorems, 61 equations, 1 table)

This paper contains 31 sections, 17 theorems, 61 equations, 1 table.

Table of Contents

  1. Introduction
  2. A recipe for constructing depth-3 circuits
  3. Depth-3 upper bound - Proof of Lemma 2.3
  4. Construction for IP
  5. Orbits of IP and spectra of 2-CNFs
  6. Building blocks for our constructions
  7. Reducing to infinitely often case
  8. Splitting orbits into regions
  9. An overview of constructions
  10. Sketch of the construction and analysis for $\mathcal{R}_1$.
  11. Conclusions
  12. How did we discover these circuits?
  13. Modular constructions
  14. Searching for better circuits for small $n$.
  15. Search for small Pareto-optimal building blocks
  16. ...and 16 more sections

Key Result

Theorem 1

$\mathop{\mathrm{\mathsf{Size}}}\nolimits^2_3(\IP_n) \le \poly(n)\cdot (9/5)^n$

Theorems & Definitions (49)

  • Theorem 1: Main result
  • Definition 2.1: Automorphism group and orbits
  • Definition 2.2
  • Lemma 2.3
  • Claim 2.4
  • proof : Proof of \ref{['lem:circuitOrbit']}
  • proof : Proof of \ref{['lem:orbitCover']}
  • Lemma 3.1
  • Definition 3.2: Orbits of $\IP^1_n$ and $\IP^0_n$
  • Definition 3.4: Spectrum of 2-CNF
  • ...and 39 more