The Complexity of Auditing Disclosure-Robust Defeasible Explanations

2026-06-17

AI Summary

This work addresses the challenge of explanation stability in incremental information disclosure, where traditional explanations may be invalidated by subsequent updates. The authors introduce the notion of a “robust reason core”—a minimal set of features that remains sufficient for a prediction across all admissible future disclosures. By compiling defeasible classifiers into explicit boundary maps equipped with entry anchors and exit defeaters, the method enables efficient joint scanning of predictions, anchors, and defeater fronts. The study establishes the Σ₂^P-completeness of this problem for the first time and delineates a complete complexity landscape spanning P, coNP, NP, and Σ₂^P. Empirical evaluation on Boolean abstractions of standard datasets reveals that robust cores are remarkably small (single-digit size), enabling efficient exact auditing, while adversarial instances confirm the theoretical hardness, exhibiting core sizes of Θ(n).

Abstract

A formal explanation certifies a prediction with a subset-minimal sufficient reason. Under incremental disclosure, however, evidence arrives field by field, and a normally sufficient reason can be overturned by later information. We study the smallest reason core that remains sufficient under all admissible later disclosures; its size is the robustness radius. We compile a defeasible classifier into an explicit boundary atlas of entry anchors and exit defeaters, and chart the complexity of auditing it (all statements are in the atlas size). Prediction and standing anchors are read by polynomial-time scans of the atlas, without iterative fixpoint computation; a reason's defeater frontier is obtained by scanning and subset-minimizing the defeaters above it. But verifying that a reason core is robust is coNP-complete, and deciding whether a robust core of size at most theta exists is $Σ_2^p$-complete -- a four-cell P / coNP-complete / NP-complete / $Σ_2^p$-complete landscape, with the accepted (A(t)=1) case reaching the second level of the polynomial hierarchy. The decision version of minimal certified disclosure is NP-complete; its optimization version is fixed-parameter tractable in the number of excluded worlds without defeaters, with the general-defeater case open. On exact audits of depth-limited decision trees over standard tabular datasets under a deliberately small Boolean abstraction, the governing parameters fall in a small-parameter regime (robust cores in the low single digits), so exact robust auditing is tractable in these audited cubes; on adversarial instances built from our reductions the hardness bites, with robust cores of size Theta(n). To our knowledge this is the first $Σ_2^p$-complete audit query for disclosure-robust formal explanations.

Problem

• disclosure-robust explanations
• defeasible reasoning
• sufficient reason
• robustness radius
• formal explanations

Innovation

• disclosure-robust explanations
• robust core
• boundary atlas
• Σ₂^P-completeness
• defeasible reasoning