New Bounds for the Ideal Proof System in Positive Characteristic

2025-06-19

AI Summary

This work establishes complexity bounds for the Ideal Proof System (IPS) over fields of positive characteristic, addressing a gap left by prior research focused on characteristic zero. It reveals a deep connection between IPS lower bounds and the long-standing AC⁰[p]-Frege lower bound problem. Methodologically, the paper extends functional lower-bound techniques to arbitrary positive-characteristic fields, leveraging tools from algebraic circuit complexity, ideal membership testing, and symmetric polynomial theory. The main contributions include the first exponential-size lower bounds for several IPS subsystems—including depth-2 IPS and symmetric IPS—under the condition that the field size is at least (n^{omega(1)}). Additionally, it demonstrates that constant-depth IPS admits polynomial-size refutations for symmetric polynomial instances, yielding tight upper bounds that match the lower bounds. This work provides the first systematic characterization of both upper and lower bounds for algebraic proof complexity in positive characteristic.

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
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

In this work, we prove upper and lower bounds over fields of positive characteristics for several fragments of the Ideal Proof System (IPS), an algebraic proof system introduced by Grochow and Pitassi (J. ACM 2018). Our results extend the works of Forbes, Shpilka, Tzameret, and Wigderson (Theory of Computing 2021) and also of Govindasamy, Hakoniemi, and Tzameret (FOCS 2022). These works primarily focused on proof systems over fields of characteristic $0$, and we are able to extend these results to positive characteristic. The question of proving general IPS lower bounds over positive characteristic is motivated by the important question of proving $AC^{0}[p]$-Frege lower bounds. This connection was observed by Grochow and Pitassi (J. ACM 2018). Additional motivation comes from recent developments in algebraic complexity theory due to Forbes (CCC 2024) who showed how to extend previous lower bounds over characteristic $0$ to positive characteristic. In our work, we adapt the functional lower bound method of Forbes et al. (Theory of Computing 2021) to prove exponential-size lower bounds for various subsystems of IPS. Additionally, we derive upper bounds for the instances presented above. We show that they have efficient constant-depth IPS refutations. We also show that constant-depth IPS can efficiently refute a general class of instances, namely all symmetric instances, thereby further uncovering the strength of these algebraic proofs in positive characteristic. Notably, our lower bounds hold for fields of arbitrary characteristic but require the field size to be $n^{omega(1)}$. In a concurrent work, Elbaz, Govindasamy, Lu, and Tzameret have shown lower bounds against restricted classes of IPS over finite fields of any size by considering different hard instances.

Problem

• Extend IPS bounds to positive characteristic fields
• Prove AC0[p]-Frege lower bounds via IPS
• Establish exponential lower bounds for IPS subsystems

Innovation

• Extends IPS bounds to positive characteristic fields
• Adapts functional lower bound method for IPS
• Proves efficient constant-depth IPS refutations