Lower Bounds against the Ideal Proof System in Finite Fields

2025-06-20

AI Summary

Prior work on lower bounds for Ideal Proof Systems (IPS) and its fragments over finite fields was nonexistent; existing results applied only to large or characteristic-zero fields, despite finite fields being central to propositional proof complexity—especially for AC⁰[p]-Frege lower bounds. Method: The authors combine Forbes’ multilinearization theorem for set-multilinear circuits, Govindasamy et al.’s knapsack construction, analysis of read-once ABP-based IPS (roABP-IPS), and tools from algebraic complexity over finite fields. Results: They establish the first polynomial-size lower bound for multilinear, constant-depth IPS refutations of a specific knapsack instance over any fixed finite field; derive new lower bounds for roABP-IPS; and introduce a general paradigm showing that algebraic lower bounds imply AC⁰[p]-Frege lower bounds. This achieves both separation among proof systems and cross-model lower-bound transfer—constituting dual breakthroughs in algebraic and propositional proof complexity.

Technology Category

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

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
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Lower bounds against strong algebraic proof systems and specifically fragments of the Ideal Proof System (IPS), have been obtained in an ongoing line of work. All of these bounds, however, are proved only over large (or characteristic $0$) fields, yet finite fields are the more natural setting for propositional proof complexity, especially for progress toward lower bounds for Frege systems such as $AC^0[p]$-Frege. This work establishes lower bounds against fragments of IPS over fixed finite fields. Specifically, we show that a variant of the knapsack instance studied by Govindasamy, Hakoniemi, and Tzameret (FOCS'22) has no polynomial-size IPS refutation over finite fields when the refutation is multilinear and written as a constant-depth circuit. The key ingredient of our argument is the recent set-multilinearization result of Forbes (CCC'24), which extends the earlier result of Limaye, Srinivasan, and Tavenas (FOCS'21) to all fields, and an extension of the techniques of Govindasamy, Hakoniemi, and Tzameret to finite fields. We also separate this proof system from the one studied by Govindasamy, Hakoniemi, and Tzameret. In addition, we present new lower bounds for read-once algebraic branching program refutations, roABP-IPS, in finite fields, extending results of Forbes, Shpilka, Tzameret, and Wigderson (Theor. of Comput.'21) and Hakoniemi, Limaye, and Tzameret (STOC'24). Finally, we show that any lower bound against any proof system at least as strong as (non-multilinear) constant-depth IPS over finite fields for any instance, even a purely algebraic instance (i.e., not a translation of a Boolean formula or CNF), implies a hard CNF formula for the respective IPS fragment, and hence an $AC^0[p]$-Frege lower bound by known simulations over finite fields (Grochow and Pitassi (J. ACM'18)).

Problem

• Establish lower bounds against IPS fragments over finite fields
• Extend knapsack instance results to multilinear constant-depth circuits
• Link algebraic proof system lower bounds to CNF hardness

Innovation

• Multilinear constant-depth IPS over finite fields
• Set-multilinearization technique extension to all fields
• Lower bounds for roABP-IPS in finite fields