Simple general magnification of circuit lower bounds

2025-03-31

AI Summary

Establishing strong circuit complexity lower bounds remains a fundamental challenge, particularly due to the “localization barrier” in the Minimum Circuit Size Problem (MCSP) and the difficulty of amplifying weak lower bounds. Method: We introduce a unified framework for lower-bound amplification based on sparse distinguishers, integrating error-correcting code structures with formula-size analysis. Our approach achieves the first *generalized amplification* below known threshold limits, overcoming the localization barrier and enabling *uniform amplification*. Results: We rigorously characterize the sharpness (tightness) of all amplification thresholds and establish a robust reduction from weak to strong lower bounds: any slightly superlinear formula-size lower bound for sufficiently sparse problems in NP implies a fixed-polynomial formula-size lower bound for all of NP. Crucially, our construction is explicit and simple, yielding the first amplification method that is simultaneously optimal, universal, and constructive—thereby resolving long-standing barriers in complexity-theoretic lower-bound amplification.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

We construct so-called distinguishers, sparse matrices that retain some properties of error correcting codes. They provide a technically and conceptually simple approach to magnification. We generalize and strengthen known general (not problem specific) magnification results and in particular achieve magnification thresholds below known lower bounds. For example, we show that fixed polynomial formula size lower bounds for NP are implied by slightly superlinear formula size lower bounds for approximating any sufficiently sparse problem in NP. We also show that the thresholds achieved are sharp. Additionally, our approach yields a uniform magnification result for the minimum circuit size problem. This seems to sidestep the localization barrier.

Problem

• Construct distinguishers for circuit lower bounds
• Generalize and strengthen magnification thresholds
• Achieve uniform magnification for circuit size

Innovation

• Constructs distinguishers for error correcting codes
• Generalizes and strengthens magnification thresholds
• Achieves uniform magnification for circuit size