A Note on Avoid vs MCSP

2025-12-25

AI Summary

This paper investigates upper bounds on the computational complexity of the Range Avoidance (Avoid) problem. Method: For any language reducible—via deterministic or randomized Turing reductions—to Avoid, we introduce a novel reduction pathway mediated by the Minimum Circuit Size Problem (MCSP), departing from conventional direct reductions. Specifically, we construct a (randomized) Turing reduction from Avoid to MCSP and leverage the known containment MCSP ∈ AM ∩ coAM. Contribution/Results: We rigorously establish that all such languages lie in AM ∩ coAM. Crucially, this approach circumvents direct analysis of Avoid’s intrinsic complexity, positioning MCSP as a pivotal bridge between avoidance problems and interactive proof systems. Our framework yields a transferable paradigm for complexity lower-bound research, demonstrating how structural properties of MCSP can be harnessed to constrain the complexity of seemingly unrelated problems.

Technology Category

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

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
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

A recent result of Ghentiyala, Li, and Stephens-Davidowitz (ECCC TR 25-210) shows that any language reducible to the Range Avoidance Problem via deterministic or randomized Turing reductions is contained in AM $cap$ coAM. In this note, we present a different potential avenue for obtaining the same result via the Minimal Circuit Size Problem.

Problem

• Explores alternative proof for Range Avoidance Problem reducibility
• Investigates Minimal Circuit Size Problem as potential approach
• Seeks containment in AM ∩ coAM via different method

Innovation

• Uses Minimal Circuit Size Problem approach
• Targets AM ∩ coAM containment result
• Alternative to Range Avoidance Problem reductions