Fairness of Classifiers in the Presence of Constraints between Features

Imane Bousdira; Martin C. Cooper

arxiv: 2605.00592 · v1 · submitted 2026-05-01 · 💻 cs.LG · cs.AI

Fairness of Classifiers in the Presence of Constraints between Features

Martin C. Cooper , Imane Bousdira This is my paper

Pith reviewed 2026-05-09 19:15 UTC · model grok-4.3

classification 💻 cs.LG cs.AI

keywords classifier fairnessprime implicantsfeature constraintsprotected featuresexplanation-based fairnesscomputational complexitymachine learning

0 comments

The pith

A classifier decision is fair if it has a prime-implicant explanation containing no protected feature once constraints among features are taken into account.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes judging individual decisions fair when they possess at least one minimal reason, expressed as a prime implicant, that avoids every protected feature. The definition of prime implicant is extended so that known constraints between features are respected when deciding whether one feature combination implies the decision. This produces fairness verdicts that can differ from those obtained by simply ignoring constraints, even when no constraint directly connects a protected feature to an unprotected one. The authors introduce three global notions of classifier fairness, establish their logical relationships, and determine the computational complexity of verifying each notion.

Core claim

A decision is fair exactly when there exists a prime implicant for that decision that contains no protected feature, where the implicant is computed with respect to the given constraints on the feature values. Ignoring the constraints can reverse the fairness status of the same decision. Three classifier-level fairness properties are defined: every decision has only fair explanations, every decision has at least one fair explanation, and the outcome is invariant under changes to protected features that respect the constraints. The paper determines how these three properties relate to one another and the complexity of deciding whether a given classifier satisfies each property.

What carries the argument

The constraint-adjusted prime implicant, which acts as the minimal reason for a decision and serves as the test for whether the decision can be explained without reference to any protected feature.

If this is right

If every prime-implicant explanation of a decision is fair, then the decision satisfies the strongest of the three classifier fairness definitions.
A classifier may satisfy the 'at least one fair explanation' definition while failing the 'only fair explanations' definition.
Outcome invariance to changes in protected features is related to but not identical with the explanation-based fairness notions.
Verifying any of the three fairness properties for a given classifier belongs to a specific complexity class determined by the paper.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The explanation-based test could be applied to audit already-trained classifiers by computing prime implicants on their input-output behavior without retraining.
When constraints are only partially known or must be inferred from data, the fairness verdict may become unreliable or require additional uncertainty handling.
The same constraint-adjusted implicant idea might be transferred to other minimal-explanation frameworks such as minimal sufficient reasons or abductive explanations.

Load-bearing premise

Prime implicants constitute a faithful and adequate mechanism for explaining why the classifier reached its decision, and the relevant constraints between features are known completely and can be incorporated into the implicant computation without creating new ambiguities.

What would settle it

A small, fully specified classifier together with a set of feature constraints for which there exists a decision that has a protected-feature-free prime implicant under the constrained definition, yet altering the protected features while obeying the constraints changes the classifier output.

Figures

Figures reproduced from arXiv: 2605.00592 by Imane Bousdira, Martin C. Cooper.

read the original abstract

In Machine Learning, an accepted definition of fairness of a decision taken by a classifier is that it should not depend on protected features, such as gender. Unfortunately, when constraints exist between features, such dependencies can be obscured by the constraints. To avoid this problem, we propose that a decision be considered fair if it has a fair explanation. We define a fair explanation as a prime-implicant reason for the decision that does not contain any protected feature (where the constraints are taken into account in the definition of prime-implicant). Surprisingly, ignoring constraints can completely change the fairness of a decision (according to this definition) even in the absence of constraints between protected and unprotected features. Three possible definitions of fairness of a classifier are that for all its decisions (1) there are only fair explanations, (2) there is at least one fair explanation, or (3) changing protected features does not change the outcome. We identify the relationships between these different definitions of fairness and study the computational complexity of testing fairness of classifiers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper refines fairness checks by folding feature constraints into prime-implicant explanations and shows that skipping constraints can flip a fairness verdict even without direct protected-feature links.

read the letter

The main thing here is that this paper ties fairness of a single decision to the existence of a prime-implicant explanation that avoids protected features once the known constraints between features are built into the implicant definition. They show that leaving the constraints out can reverse whether the decision counts as fair, and this reversal happens even when protected features have no direct constraints with the rest of the feature set. From there they define three versions of classifier-level fairness and map the relationships among them while giving complexity results for testing each one. The core technical step is adjusting the prime-implicant computation itself so that only constraint-satisfying assignments are considered. This is a straightforward logical move that addresses a real gap in standard fairness auditing when features are correlated. The paper does a clean job of spelling out why the usual checks can miss hidden dependencies and why the adjusted definition avoids that problem. The relationships among the three fairness notions follow directly from the logical setup, and the complexity claims are the usual consequences of working in this fragment of propositional logic. One soft spot is that prime implicants are treated as the right explanation mechanism; that fits some models well but is less natural for others. The work also assumes the constraints are known and complete, which is a practical requirement rather than a flaw in the logic itself. The central argument holds up without circularity or unsupported steps. This is for researchers in algorithmic fairness who already work with logical explanations or constraint-based methods. A reader who audits classifiers on data with known feature relationships will get a usable refinement. The paper shows clear thinking on its own terms and is grounded enough to deserve a serious referee who can check the proofs and examples in detail. I would send it out for peer review.

Referee Report

1 major / 2 minor

Summary. The paper proposes that a decision by a classifier is fair if it admits a fair explanation, defined as a prime implicant for the decision that excludes all protected features once inter-feature constraints are incorporated into the implicant computation. It shows that ignoring constraints can reverse a fairness verdict even when no constraints directly link protected and unprotected features. Three classifier-level fairness notions are introduced—universal fair explanations, existential fair explanations, and invariance of the outcome to changes in protected features—and the logical relationships among them are derived together with complexity results for verifying each notion.

Significance. If the derivations and complexity bounds hold, the work supplies a logically grounded, constraint-sensitive refinement of explanation-based fairness that addresses a common limitation in existing definitions. The observation that constraints alone can flip fairness verdicts is a useful theoretical insight with potential implications for auditing systems in domains with known feature dependencies. The explicit relationships among the three fairness notions and the complexity classification provide a clear map for future algorithmic work. The manuscript's definitional clarity and focus on prime implicants as the explanatory primitive are strengths, though practical adoption will depend on the availability of accurate constraint models.

major comments (1)

The central claim that ignoring constraints can alter fairness even without protected-unprotected constraints rests on how constraints prune the space of valid assignments and thereby change the set of prime implicants. The manuscript should supply an explicit small example (with the constraint set, the classifier function, and the two prime-implicant sets) immediately after the definition of fair explanation so that readers can verify the flip without reconstructing the argument.

minor comments (2)

Notation for prime implicants under constraints should be introduced once and used consistently; currently the abstract and later sections appear to overload the same symbol for the unconstrained and constrained cases.
The complexity results would be easier to compare if a single table listed the three fairness notions together with their complexity classes and the proof techniques used.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the work, and the constructive suggestion to improve clarity. We address the single major comment below and will revise the manuscript to incorporate the requested example.

read point-by-point responses

Referee: The central claim that ignoring constraints can alter fairness even without protected-unprotected constraints rests on how constraints prune the space of valid assignments and thereby change the set of prime implicants. The manuscript should supply an explicit small example (with the constraint set, the classifier function, and the two prime-implicant sets) immediately after the definition of fair explanation so that readers can verify the flip without reconstructing the argument.

Authors: We agree that an explicit example placed immediately after the definition of fair explanation would help readers verify the claim without having to reconstruct the argument from later sections. In the revised manuscript we will insert a small, self-contained example at that location. The example will include: (i) a concrete constraint set over the features, (ii) a simple classifier function, (iii) the set of prime implicants computed while respecting the constraints, and (iv) the set of prime implicants obtained when constraints are ignored. This will demonstrate that the fairness verdict for a given decision can reverse even though no constraint directly links a protected feature to an unprotected one. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces three definitions of classifier fairness grounded in prime-implicant explanations that exclude protected features once constraints are folded into the implicant computation. The relationships among these definitions and the complexity results for testing them are direct logical consequences of the formulation; no equation or claim reduces by construction to a fitted parameter, a self-citation chain, or an ansatz imported from prior work by the same authors. The claimed effect of constraints on fairness is shown by explicit example and follows from the pruning of valid assignments, without any definitional loop or renaming of known results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on standard logical notions of prime implicants and the domain assumption that feature constraints are known and usable; no free parameters or new postulated entities are introduced.

axioms (2)

domain assumption Prime implicants provide a suitable minimal explanation for a classifier decision
The fairness definition is built directly on this logical concept of explanation.
domain assumption All relevant constraints between features are known and can be incorporated into the definition of prime implicants
The paper states that constraints are taken into account in the prime-implicant definition.

pith-pipeline@v0.9.0 · 5475 in / 1439 out tokens · 34750 ms · 2026-05-09T19:15:13.991233+00:00 · methodology

discussion (0)

Reference graph

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SAT-based rigorous explanations for decision lists

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