Separation Results for Constant-Depth and Multilinear Ideal Proof Systems

2026-01-09

Electron. Colloquium Comput. Complex.

AI Summary

This work investigates the separation between constant-depth and multilinear subsystems of the Ideal Proof System (IPS), revealing fundamental differences in their proof complexity. By leveraging functional and multiplicity methods, together with properties of multilinear polynomials over the Boolean cube and tools from algebraic circuit complexity, the authors establish the first depth hierarchy theorem for constant-depth IPS and achieve an unconditional separation within multilinear IPS analogous to Raz’s result for algebraic circuits. Specifically, they construct unsatisfiable instances efficiently refutable by depth-Δ IPS but requiring superpolynomial size for depth-Δ/10 IPS. Furthermore, they show that multilinear-NC² can efficiently refute these instances via functional proofs, whereas multilinear-NC¹ IPS requires superpolynomial size, thereby significantly advancing the theoretical boundaries of algebraic proof complexity.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Multiagent Systems: Other Foundations of Multi Agent Systems
• Machine Learning: Multi-instance/Multi-view Learning

Application Category

• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

In this work, we establish separation theorems for several subsystems of the Ideal Proof System (IPS), an algebraic proof system introduced by Grochow and Pitassi (J. ACM, 2018). Separation theorems are well-studied in the context of classical complexity theory, Boolean circuit complexity, and algebraic complexity. In an important work of Forbes, Shpilka, Tzameret, and Wigderson (ToC, 2021), two proof techniques were introduced to prove lower bounds for subsystems of the IPS, namely the functional method and the multiples method. We use these techniques and obtain the following results. Hierarchy theorem for constant-depth IPS: Recently, Limaye, Srinivasan, and Tavenas (J. ACM 2025) proved a hierarchy theorem for constant-depth algebraic circuits. We adapt the result and prove a hierarchy theorem for constant-depth $\mathsf{IPS}$. We show that there is an unsatisfiable multilinear instance refutable by a depth-$\Delta$ $\mathsf{IPS}$ such that any depth-($\Delta/10)$ $\mathsf{IPS}$ refutation for it must have superpolynomial size. This result is proved by building on the multiples method. Separation theorems for multilinear IPS: In an influential work, Raz (ToC, 2006) unconditionally separated two algebraic complexity classes, namely multilinear $\mathsf{NC}^{1}$ from multilinear $\mathsf{NC}^{2}$. In this work, we prove a similar result for a well-studied fragment of multilinear-$\mathsf{IPS}$. Specifically, we present an unsatisfiable instance such that its functional refutation, i.e., the unique multilinear polynomial agreeing with the inverse of the polynomial over the Boolean cube, has a small multilinear-$\mathsf{NC}^{2}$ circuit. However, any multilinear-$\mathsf{NC}^{1}$ $\mathsf{IPS}$ refutation ($\mathsf{IPS}_{\mathsf{LIN}}$) for it must have superpolynomial size. This result is proved by building on the functional method.

Problem

• Ideal Proof System
• separation theorems
• constant-depth
• multilinear
• algebraic proof complexity

Innovation

• Ideal Proof System
• constant-depth hierarchy
• multilinear IPS
• functional method
• multiples method