DNF formulas are efficiently testable with relative error

TL;DR

The paper develops a poly$(s,1/ε)$-query tester for relative-error testing of s-term DNFs, independent of the ambient dimension $n$, by introducing a novel K-clustering decomposition that partitions the DNF terms into clusters each corresponding to a factored-DNF. It couples this structural decomposition with a two-part approach: Find-Factored-DNFs to produce approximate simulators for the cluster-based components, and Test-Factored-DNFs to test these components under relative error, while handling imperfect simulators via robust learning and consistency checking. The authors integrate a DNFLearner and ConsCheck to build and validate oracle simulators for the factored components, ensuring the overall tester accepts yes-instances with high probability and rejects far/no-instances with high probability. This framework yields the first efficient relative-error tester for a natural, expressive class that can depend on many variables, suggesting broader applicability of the clustering decomposition to other testing and learning tasks. The work thus bridges sparse-function testing with multiplicative accuracy guarantees and opens avenues for applying the local-clustering structure to other Boolean-function classes.

Abstract

We give a poly$(s,1/ε)$-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.

Figures (20)

• Figure 1: \ref{['Sim-SAMP']}
• Figure 2: Procedure to construct the $K$-clustering of a DNF $f$
• Figure 3: \ref{['Test-DNF']}
• Figure 4: \ref{['Test-Equivalence']}
• Figure 5: \ref{['In-Pool']}

Theorems & Definitions (148)

• Theorem 1: Relative-error testing of $s$-term DNF
• Remark 2
• Lemma 3: Approximate symmetry of relative distance
• Lemma 4: Approximate triangle inequality for relative distance
• Definition 5
• Definition 7: Cube
• Definition 8: Distance from a point to a term $T$
• Lemma 9
• proof
• Remark 10