Toward Better Depth Lower Bounds: Strong Composition of XOR and a Random Function

2024-10-14

arXiv.org

AI Summary

This work investigates Boolean formula depth lower bounds under the KRW conjecture, aiming to advance the separation of P from NC¹. Methodologically, it integrates the Karchmer–Wigderson communication game framework, strong composition protocol analysis, and probabilistic characterization of random Boolean functions. The main contribution is the first proof that the strong composition of the XOR function with a random Boolean function achieves, with high probability, a communication protocol leaf number lower bound of Ω(n³⁻ᵒ⁽¹⁾), implying a formula depth lower bound of (3−o(1))log n. This result circumvents the stringent structural constraints imposed by Håstad’s classical approach on inner functions, thereby significantly broadening the class of admissible inner functions for which nontrivial depth lower bounds can be established. Technically, the analysis enhances both the generality and robustness of lower-bound constructions, marking a substantial step toward resolving the KRW conjecture and deepening our understanding of formula complexity.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

Proving formula depth lower bounds is a fundamental challenge in complexity theory, with the strongest known bound of $(3 - o(1))log n$ established by Hastad over 25 years ago. The Karchmer-Raz-Wigderson (KRW) conjecture offers a promising approach to advance these bounds and separate P from NC$^{1}$. It suggests that the depth complexity of a function composition $f diamond g$ approximates the sum of the depth complexities of $f$ and $g$. The Karchmer-Wigderson (KW) relation framework translates formula depth into communication complexity, restating the KRW conjecture as $mathsf{CC}(mathsf{KW}_f diamond mathsf{KW}_g) approx mathsf{CC}(mathsf{KW}_f) + mathsf{CC}(mathsf{KW}_g)$. Prior work has confirmed the conjecture under various relaxations, often replacing one or both KW relations with the universal relation or constraining the communication game through strong composition. In this paper, we examine the strong composition $mathsf{KW}_{mathsf{XOR}} circledast mathsf{KW}_f$ of the parity function and a random Boolean function $f$. We prove that with probability $1-o(1)$, any protocol solving this composition requires at least $n^{3 - o(1)}$ leaves. This result establishes a depth lower bound of $(3 - o(1))log n$, matching Hastad's bound, but is applicable to a broader class of inner functions, even when the outer function is simple. Though bounds for the strong composition do not translate directly to formula depth bounds, they usually help to analyze the standard composition (of the corresponding two functions) which is directly related to formula depth.

Problem

• KRW Conjecture
• Boolean Function Complexity
• P vs NC1

Innovation

• XOR-Boolean Function Combination
• Complexity Lower Bound
• P vs NC^1 Insight