AC^0[p]-Frege Cannot Efficiently Prove that Constant-Depth Algebraic Circuit Lower Bounds are Hard

2025-09-20

AI Summary

This paper investigates the provability of constant-depth algebraic circuit lower bounds for the Permanent over finite fields within the AC⁰[p]-Frege proof system. It introduces a family of DNF formulas that characterize the difficulty of proving such lower bounds, and establishes—unconditionally—that this family admits no polynomial-size AC⁰[p]-Frege proofs for infinitely many input lengths. Methodologically, it refines the Santhanam–Tzameret diagonalization framework by incorporating unprovability assumptions about weak algebraic properties (e.g., tensor rank) and connecting them to the Razborov–Krajíček conjecture. A key conceptual contribution is the identification of “existential depth amplification”: if no short algebraic proof exists at a fixed depth, then none exists at any constant depth—a necessary step for proving lower bounds in algebraic proof systems. The results confirm that the DNF family is not efficiently provable in AC⁰[p]-Frege, supporting its status as either a hard tautology or an unprovable statement; several weaker variants are rigorously shown to be true, and the truth of the original formulas is verified under plausible assumptions.

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

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Data transparency and provenance
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We study whether lower bounds against constant-depth algebraic circuits computing the Permanent over finite fields (Limaye-Srinivasan-Tavenas, J. ACM 2025; Forbes, CCC 2024) are hard to prove in certain proof systems. We focus on a DNF formula that expresses that such lower bounds are hard for constant-depth algebraic proofs. Using an adaptation of the diagonalization framework of Santhanam and Tzameret (SIAM J. Comput. 2025), we show unconditionally that this family of DNF formulas does not admit polynomial-size propositional AC0[p]-Frege proofs infinitely often. This rules out the possibility that the DNF family is easy, and establishes that its status is either that of a hard tautology for AC0[p]-Frege or else unprovable (not a tautology). While it remains open whether the DNFs in question are tautologies, we provide evidence in this direction. In particular, under the plausible assumption that certain weak properties of multilinear algebra, specifically those involving tensor rank, do not admit short constant-depth algebraic proofs, the DNFs are tautologies. We also observe that several weaker variants of the DNF formula are provably tautologies, and we show that the question of whether the DNFs are tautologies connects to conjectures of Razborov (ICALP 1996) and Krajicek (J. Symb. Log. 2004). Our result has two additional features. (i) Existential depth amplification: the DNF formula is parameterised by a constant depth d bounding the depth of the algebraic proofs. We show that there exists some fixed depth d such that if there are no small depth-d algebraic proofs of certain circuit lower bounds for the Permanent, then there are no such small algebraic proofs in any constant depth. (ii) Necessity: we show that our result is a necessary step towards establishing lower bounds against constant-depth algebraic proofs, and more generally against any sufficiently strong proof system.

Problem

• Studying whether constant-depth algebraic circuit lower bounds are hard to prove
• Showing DNF formulas lack polynomial-size AC0[p]-Frege proofs unconditionally
• Establishing connections between tautology status and algebraic proof complexity

Innovation

• Adapted diagonalization framework for proof complexity
• Used DNF formulas to express lower bound hardness
• Applied constant-depth algebraic circuit assumptions