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KRW Composition Theorems via Lifting

Susanna F. de Rezende, Or Meir, Jakob Nordström, Toniann Pitassi, Robert Robere

TL;DR

The KRW conjecture links depth lower bounds to the composition of Boolean functions, and this work broadens the scope by proving a monotone KRW composition theorem for a wide class of inner monotone functions via lifting from query complexity, as well as a semi-monotone version leveraging Razborov's rank method. The authors introduce a generalized lifting theorem that preserves hardness on restricted input sets defined by average-degree conditions, enabling the handling of inner functions such as $\textit{s{-}t}$-connectivity, clique, and generation. They also implement a semi-monotone framework that uses a rank-based construction with a symmetric matrix $A$ satisfying $A^2=I$, deriving lower bounds through monochromatic-rectangle rank analyses. Finally, they apply these results to classical functions and CNF-based reductions (e.g., $stConn$, bitPHP, Gen) to obtain concrete lower bounds, illustrating the potential to advance depth lower bounds toward $\mathbf{P}\not\subseteq \mathbf{NC}^1$ through broader inner-function families.

Abstract

One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., $\mathbf{P}\not\subseteq\mathbf{NC}^1$). Karchmer, Raz, and Wigderson (Computational Complexity 5(3/4), 1995) suggested to approach this problem by proving that depth complexity behaves "as expected" with respect to the composition of functions $f\diamond g$. They showed that the validity of this conjecture would imply that $\mathbf{P}\not\subseteq\mathbf{NC}^1$. Several works have made progress toward resolving this conjecture by proving special cases. In particular, these works proved the KRW conjecture for every outer function $f$, but only for few inner functions $g$. Thus, it is an important challenge to prove the KRW conjecture for a wider range of inner functions. In this work, we extend significantly the range of inner functions that can be handled. First, we consider the $\textit{monotone}$ version of the KRW conjecture. We prove it for every monotone inner function $g$ whose depth complexity can be lower bounded via a query-to-communication lifting theorem. This allows us to handle several new and well-studied functions such as the $s\textbf{-}t$-connectivity, clique, and generation functions. In order to carry this progress back to the $\textit{non-monotone}$ setting, we introduce a new notion of $\textit{semi-monotone}$ composition, which combines the non-monotone complexity of the outer function $f$ with the monotone complexity of the inner function $g$. In this setting, we prove the KRW conjecture for a similar selection of inner functions $g$, but only for a specific choice of the outer function $f$.

KRW Composition Theorems via Lifting

TL;DR

The KRW conjecture links depth lower bounds to the composition of Boolean functions, and this work broadens the scope by proving a monotone KRW composition theorem for a wide class of inner monotone functions via lifting from query complexity, as well as a semi-monotone version leveraging Razborov's rank method. The authors introduce a generalized lifting theorem that preserves hardness on restricted input sets defined by average-degree conditions, enabling the handling of inner functions such as -connectivity, clique, and generation. They also implement a semi-monotone framework that uses a rank-based construction with a symmetric matrix satisfying , deriving lower bounds through monochromatic-rectangle rank analyses. Finally, they apply these results to classical functions and CNF-based reductions (e.g., , bitPHP, Gen) to obtain concrete lower bounds, illustrating the potential to advance depth lower bounds toward through broader inner-function families.

Abstract

One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., ). Karchmer, Raz, and Wigderson (Computational Complexity 5(3/4), 1995) suggested to approach this problem by proving that depth complexity behaves "as expected" with respect to the composition of functions . They showed that the validity of this conjecture would imply that . Several works have made progress toward resolving this conjecture by proving special cases. In particular, these works proved the KRW conjecture for every outer function , but only for few inner functions . Thus, it is an important challenge to prove the KRW conjecture for a wider range of inner functions. In this work, we extend significantly the range of inner functions that can be handled. First, we consider the version of the KRW conjecture. We prove it for every monotone inner function whose depth complexity can be lower bounded via a query-to-communication lifting theorem. This allows us to handle several new and well-studied functions such as the -connectivity, clique, and generation functions. In order to carry this progress back to the setting, we introduce a new notion of composition, which combines the non-monotone complexity of the outer function with the monotone complexity of the inner function . In this setting, we prove the KRW conjecture for a similar selection of inner functions , but only for a specific choice of the outer function .

Paper Structure

This paper contains 48 sections, 33 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Our results
  3. The monotone composition theorem
  4. The semi-monotone composition theorem
  5. Our techniques
  6. The monotone composition theorem
  7. The semi-monotone composition theorem
  8. Preliminaries
  9. Communication complexity
  10. Subadditive measures on trees
  11. Monotone formulas and Karchmer-Wigderson relations
  12. Decision trees
  13. The Razborov rank measure
  14. The Nullstellensatz proof system
  15. Lifting theorems
  16. ...and 33 more sections

Key Result

Theorem 1.1

Let $f:\left\{ 0,1 \right \}^{m}\to\left\{ 0,1 \right \}$ and $g:\left\{ 0,1 \right \}^{n}\to\left\{ 0,1 \right \}$ be non-constant monotone functions. If there is a lifted search problem $S\diamond\mathrm{gd}$ that reduces to $\textit{mKW}_{g}$ and satisfies the conditions of the theorem of CFKMP19 In particular, if $\mathsf{CC}(\textit{mKW}_{g})=\tilde{O}\left(\mathsf{Q}(S)\cdot t\right)$, then

Theorems & Definitions (77)

  • Conjecture 1: The KRW conjecture
  • Theorem 1.1: monotone composition theorem, informal
  • Definition 1: Semi-monotone composition
  • Conjecture 2: Semi-monotone KRW conjecture
  • Theorem 1.2: semi-monotone composition theorem, informal
  • Theorem 1.3: informal
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • ...and 67 more