Functional Lower Bounds in Algebraic Proofs: Symmetry, Lifting, and Barriers

2024-06-10

Electron. Colloquium Comput. Complex.

AI Summary

This paper investigates lower bounds on proof complexity for the Ideal Proof System (IPS) under symmetry and invariance constraints. For Boolean instances and symmetric polynomials, it establishes the first exponential-size lower bounds for IPS fragments over finite fields; it further exposes the fundamental failure of functional lower-bound techniques in the Boolean setting and delineates their precise applicability limits. Methodologically, the work integrates symmetry analysis, Nullstellensatz degree lower bounds, lifting lemmas, oblivious read-once algebraic branching programs (roABPs), multilinear formulas, and the placeholder IPS framework. Key contributions include: (1) a degree lower bound exceeding $n$ for $n$-variable symmetric polynomials; (2) exponential-size lower bounds for roABP-IPS proofs; and (3) the first IPS lower bounds for invariant polynomials over positive-characteristic fields, along with strengthened constant-depth IPS lower bounds. Collectively, these results systematically characterize symmetry-induced complexity barriers in algebraic proof systems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• 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: 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

Strong algebraic proof systems such as IPS (Ideal Proof System; Grochow-Pitassi [J. ACM, 65(6):37:1–55, 2018]) offer a general model for deriving polynomials in an ideal and refuting unsatisfiable propositional formulas, subsuming most standard propositional proof systems. A major approach for lower bounding the size of IPS refutations is the Functional Lower Bound Method (Forbes, Shpilka, Tzameret and Wigderson [Theory Comput., 17: 1-88, 2021]), which reduces the hardness of refuting a polynomial equation f(x)=0 with no Boolean solutions to the hardness of computing the function 1/f(x) over the Boolean cube with an algebraic circuit. Using symmetry we provide a general way to obtain many new hard instances against fragments of IPS via the functional lower bound method. This includes hardness over finite fields and hard instances different from Subset Sum variants both of which were unknown before, and stronger constant-depth lower bounds. Conversely, we expose the limitation of this method by showing it cannot lead to proof complexity lower bounds for any hard Boolean instance (e.g., CNFs) for any sufficiently strong proof systems. Specifically, we show the following: Nullstellensatz degree lower bounds using symmetry: Extending [Forbes et al. Theory Comput., 17: 1-88, 2021] we show that every unsatisfiable symmetric polynomial with n variables requires degree >n refutations (over sufficiently large characteristic). Using symmetry again, by characterising the n/2-homogeneous slice appearing in refutations, we show that unsatisfiable invariant polynomials of degree n/2 require degree ≥ n refutations. Lifting to size lower bounds: Lifting our Nullstellensatz degree bounds to IPS-size lower bounds, we obtain exponential lower bounds for any poly-logarithmic degree symmetric instance against IPS refutations written as oblivious read-once algebraic programs (roABP-IPS). For invariant polynomials, we show lower bounds against roABP-IPS and refutations written as multilinear formulas in the placeholder IPS regime (studied by Andrews and Forbes [54th Ann. Symp. Theory Comput., STOC 2022]), where the hard instances do not necessarily have small roABPs themselves, including over positive characteristic fields. This provides the first IPS-fragment lower bounds over finite fields. By an adaptation of the work of Amireddy, Garg, Kayal, Saha and Thankey [50th Intl. Colloq. Aut. Lang. Prog., ICALP 2023], we strengthen the constant-depth IPS lower bounds obtained recently in Govindasamy, Hakoniemi and Tzameret [63rd IEEE Ann. Symp. Found. Comput. Sci., FOCS 2022]. Barriers for Boolean instances: While lower bounds against strong propositional proof systems were the original motivation for studying algebraic proof systems in the 1990s [Beame et al. Proc. London Math. Soc. (3) 73, 1 (1996), 1–26; Buss et al. Computational Complexity 6, 3 (1996), 256–298] we show that the functional lower bound method alone cannot establish any size lower bound for Boolean instances for any sufficiently strong proof systems, and in particular, cannot lead to lower bounds against AC0[p]-Frege and TC0-Frege.

Problem

• Algebraic Proof Systems
• Symmetry-related Mathematical Problems
• Boolean Problems

Innovation

• Symmetry Principle
• Algebraic Proof Systems
• Hard Problem Discovery