DNF formulas are efficiently testable with relative error

2026-01-22

AI Summary

This work investigates the problem of efficiently testing whether an unknown Boolean function is an s-term DNF formula under the relative error model, with particular focus on functions that depend on super-constantly many variables. The authors propose a tester with query complexity poly(s, 1/ε), which is independent of the total number of variables n. The key innovation lies in the novel decomposition of s-term DNFs into “local clusters,” enabling the development of structural analysis and testing techniques applicable even when the DNF representation is not explicitly given. This paper establishes, for the first time, that a natural class of Boolean functions—namely, s-term DNFs depending on super-constantly many variables—admits efficient testing in the relative error model with query complexity depending only on s and 1/ε.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Structured Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

We give a poly$(s,1/\epsilon)$-query algorithm for testing whether an unknown and arbitrary function $f: \{0,1\}^n \to \{0,1\}$ is an $s$-term DNF, in the challenging relative-error framework for Boolean function property testing that was recently introduced and studied in a number of works [CDH+25b, CPPS25a, CPPS25b, CDH+25a]. This gives the first example of a rich and natural class of functions which may depend on a super-constant number of variables and yet is efficiently testable in the relative-error model with constant query complexity. A crucial new ingredient enabling our approach is a novel decomposition of any $s$-term DNF formula into ``local clusters''of terms. Our results demonstrate that this new decomposition can be usefully exploited for algorithms even when the $s$-term DNF is not explicitly given; we believe that this decomposition may have applications in other contexts.

Problem

• DNF formulas
• property testing
• relative error
• Boolean functions
• query complexity

Innovation

• DNF testing
• relative-error framework
• local clusters
• Boolean function property testing
• query complexity