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Feature Relevancy, Necessity and Usefulness: Complexity and Algorithms

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A feature is necessary for a prediction exactly when some counterfactual edit flips the model's output.

desk verdict Theorem 18 is a genuinely neat characterization, but the paper undercuts itself with a wrong counting formula and an inverted FBDD criterion. read the letter →

arxiv 2505.09640 v1 pith:QFCLFARS submitted 2025-05-06 cs.AI

Tomás Capdevielle , Santiago Cifuentes This is my paper · ORCID
classification cs.AI
keywords featurerelevancynecessityusefulnesssufficientreasonshittingsetsexplainableAIdecisiontreesFBDD
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies logic-based feature importance: whether a feature is relevant to a given prediction (it appears in at least one minimal set of features that determines the prediction) or necessary (it appears in every such set). Its central result is a characterization: a feature is necessary exactly when changing that feature alone to some value changes the model's output, written $M(e)\neq M(e^{x=b})$. That condition is a small set of model evaluations, so necessity becomes easy to certify for any efficiently evaluable model with bounded feature domains; the paper also gives linear-time procedures for decision trees and read-once binary decision diagrams (FBDDs). It then introduces a global notion, usefulness, and proves a feature is useful exactly when it is necessary, equivalently relevant, for some entity. Experiments on three public datasets indicate that the resulting feature-ranking score agrees with expected importance.

What carries the argument

The load-bearing mechanism is the sufficient-reason/hitting-set correspondence together with the new flip-test characterization. A sufficient reason is a minimal set of features that already determines the prediction; by the paper's Lemma 10, sufficient reasons are exactly the minimal hitting sets of the hypergraph formed by the clauses of the model restricted to the entity. That correspondence carries the relevancy algorithms and the hardness results. The new step is Theorem 18: $x$ is necessary exactly when $X\setminus\{x\}$ is not a reason, which turns a set-membership question into one-feature counterfactual checks. For the linear-time bounds, the algorithm decomposes the tree path into intervals of values per feature and evaluates a tautology-checking recursion on each interval, while for FBDDs it uses the read-once property to inspect a single node per feature.

What would settle it

Take a real-valued classifier that is not comparison-based, such as a small network with a periodic activation, and pick an entity near a decision boundary so that a tiny change in one feature flips the class but no threshold captured by the node set does; if the discretization route declares the feature non-necessary while the direct flip test $M(e)\neq M(e^{x=b})$ succeeds for some real $b$, the numerical extension is not universal.

Watch

Extended reading notes

Core claim

On its own terms, the central discovery is that necessity is a flip test. For any model $M$, entity $e$, and feature $x$, the feature is necessary for the prediction $M(e)$ if and only if there is some value $b$ with $M(e)\neq M(e^{x=b})$. The proof is minimality: if $X\setminus\{x\}$ were a reason for the prediction, then some sufficient reason would avoid $x$; conversely, if flipping $x$ can change the output, every sufficient reason must contain $x$. This reduces necessity to direct model evaluations, $O(\mathrm{eval}(M)|D_x|)$ for bounded domains, and it underlies the paper's linear-time all-necessary-features algorithms for decision trees and FBDDs. The same characterization drives the new global notion of usefulness: $x$ is useful exactly when it is necessary for some entity, equivalently when it is relevant for some entity, and deciding usefulness is equivalent to deciding model equivalence, giving tractability for decision trees and ordered binary decision diagrams (OBDDs) and coNP-completeness for disjunctive normal form formulas (DNFs).

Load-bearing premise

The numerical-feature results assume that any comparison-based model can be discretized into finitely many intervals without changing its predictions (the paper's Section 4.2, citing [14] without proof or bounds); if that step fails, the results for continuous features and the abstract's neural-network claim lose their foundation.

Editorial extensions

If this is right

  • For any model that can be evaluated in polynomial time and whose features have bounded domains, checking whether a feature is necessary costs $O(\mathrm{eval}(M)|D_x|)$, so necessity queries stop being a search problem.
  • All necessary features of a decision tree with categorical and numerical features can be listed in $O(|X|+|T|)$ time, and the same linear bound holds for FBDDs and OBDDs.
  • A feature that is useful in the new global sense is one whose value matters for some prediction, and deciding usefulness is polynomial-time equivalent to model equivalence, yielding tractability for decision trees and OBDDs and coNP-completeness for DNFs.
  • The usefulness score, computed by counting entities for which a feature is necessary, is computable in quadratic time for boolean decision trees with two model-counting calls, and its ranking agrees with established importance on three public datasets.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Read narrowly: the abstract's claim that necessity is detectable efficiently in neural networks is supported only through the bounded-domain or comparison-based route; real-valued networks with arbitrary activations are not covered by the discretization argument.
  • If the flip-test characterization holds broadly, an exact model-agnostic necessity oracle is available by querying the original black box directly, without compiling it to a circuit or building a surrogate model, which could make local explanations cheaper than Shapley-value approximations in practice.
  • Counting entities where a single-feature flip changes the prediction yields a global importance ranking defined entirely from the classifier's behavior; this could serve as a sanity check for other importance scores, since it does not rely on cooperative-game axioms.
  • A testable next step is to compare usefulness rankings against leave-one-feature-out accuracy on datasets not used in the paper, which would separate ranking quality from agreement with reported importance.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper studies logic-based explanations for classification models, centered on the notions of relevant, necessary, and useful features. It proves a characterization of necessary features (Theorem 18) and of the new global notion of usefulness (Theorem 22), gives algorithms for decision trees and FBDDs, proves several NP-completeness and #P-completeness results, and reports experiments comparing a usefulness-based ranking with SHAP and with informal ground truth from Kaggle. The paper also claims, in the abstract, that necessity can be detected efficiently in complex models including neural networks.

Significance. The conceptual core of the paper is valuable: Theorem 18 gives a clean, correct characterization of necessity by a small number of model evaluations, and Theorem 22 correctly ties the global notion of usefulness to the local notions of relevancy and necessity. If the algorithmic and counting claims were correct, the paper would offer a practical scoring method for decision trees. However, several load-bearing results are wrong as stated: Proposition 24's counting formula is false, Proposition 21's FBDD criterion is inverted, and Algorithm 1 contains a recursion bug. The numerical-feature and neural-network claims are also unsupported. These issues must be fixed before the results can be relied upon.

major comments (4)
  1. [§4.3, Proposition 24] The formula for the number of entities for which x is not necessary is incorrect. The paper states this number as CT(∧_{b∈Dx} M_{x=b}) + CT(¬∧_{b∈Dx} M_{x=b}). The second term is the complement of the first, so the sum is always |ent(X)|, independent of M and x. The correct second term is CT(∧_{b∈Dx} ¬M_{x=b}), i.e. the entities all of whose x-variants are classified 0. This is consistent with the k-class generalization given later in the same section, which uses the correct form. As a counterexample, for the one-feature identity model M(e)=e(x), x is necessary for every entity, so the not-necessary count should be 0, while the printed formula gives CT(x)+CT(¬x)=|ent(X)|. Corollary 25 and the usefulness score in Section 5 inherit this error, so the printed score definition is invalid.
  2. [§4.2, Proposition 21] The criterion for necessity in FBDDs is stated backwards. For the unique node v on the path P with feat(v)=x, D(e^{x=0}) = D_left(v)(e) and D(e^{x=1}) = D_right(v)(e). By Theorem 18, x is necessary precisely when D(e^{x=0}) ≠ D(e^{x=1}), i.e. when D_left(v)(e) ≠ D_right(v)(e). The printed condition D_left(v)=D_right(v)=D(e) identifies the opposite situation, where the feature has no effect on the prediction, and is therefore not a necessary feature unless the model is degenerate. The claimed linear-time FBDD algorithm is not correct as stated.
  3. [§4.2, Proposition 20 and Algorithm 1] Algorithm 1 in Figure 3 contains a recursion bug: in line 13, both recursive calls use w1, the left child, so the right child is never explored for values above value(v). The second call should be on w2. The same function, in the categorical case (Algorithm 2 in Figure 4), tests 'value(v) ∈ (a, b)' on line 12 even though v is a categorical node; the condition should involve values(v), and the partition of Z should use values(v) and Dx \ values(v). As printed, these algorithms do not justify the O(|X|+|T|) claim.
  4. [Abstract and §4.2 (numerical features)] The abstract's claim that necessity can be detected efficiently in 'complex models such as neural networks' is not supported by the paper. Corollary 19 applies only when each feature domain is small. The subsequent discussion covers 'comparison-based models' and relies on discretization 'using well-known ideas [14]' without a theorem, proof, or complexity bound, and neural networks are not comparison-based in the sense used there. The authors should either state a precise model class for which the discretization argument is proved or remove the neural-network claim from the abstract.
minor comments (5)
  1. [§4.1, Proposition 17] The proof of hardness cites 'Lemma 13' for the NP-hardness of deciding whether a minimal hitting set contains a given set of nodes, but Lemma 13 is an algorithmic result; the hardness result is Proposition 14. Please correct the reference.
  2. [§4.2, text after Corollary 19] The sentence 'the number of classes that have to be considered grows linearly with the number of nodes, and thus we conclude tractability' is stated informally. A formal statement of the discretization, including how to handle repeated thresholds and boundary cases, would strengthen the paper.
  3. [§3, Lemma 13] The notation 'B\{x}' is used in the proof of Lemma 13, but the node under consideration is v, not x. This is a typo and should be fixed for readability.
  4. [§5, Ground truth] The ground-truth rankings are taken from Kaggle posts and selected papers and are treated as authoritative. Since these are informal and may disagree with each other, the experiments would be more convincing if the comparison were formulated as a hypothesis test or at least accompanied by a sensitivity analysis over the choice of ground truth.
  5. [§2, Definition 5] It is stated without proof that 'there are no numerical nodes with labels of the form (x, Mx)'. This is a WLOG assumption that should be justified, because Mx can be finite and the comparison x ≤ Mx would then be tautological, which may affect the size of the tree after removing such nodes.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central characterization is derived from definitions, and the experimental comparison is externally anchored.

full rationale

The paper's main derivation chain is self-contained. Theorem 18 is proved directly from Definition 1 and the meaning of X\{x} being a reason; Corollary 19 is an immediate evaluation bound and does not presuppose the target. Proposition 20 and the FBDD algorithm are algorithmic consequences of the same characterization, aside from apparent correctness defects noted below. Theorem 22 derives usefulness from the same characterization and from the definition of relevant features; nothing is defined in terms of the result it is used to prove. Complexity claims rest on standard reductions and external hardness results, and the experiments compare the proposed score with externally sourced ground-truth rankings and SHAP, so there is no fitted-input-called-prediction loop. The numerical discretization step is an unsupported external citation to [14], but a citation is not circular when it is not by the present authors and the assumption is substantial; this is a correctness risk. Proposition 24's counting formula appears invalid because CT(A)+CT(¬A) is the total entity count, and Proposition 21 contains a sign error in its FBDD necessity criterion, but these are mathematical errors that invalidate the reported score and algorithm, not circular derivations of the score from the score. Accordingly, no circular step is exhibited and the circularity score is 0.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The theoretical complexity results depend only on standard complexity assumptions and closure properties; the experimental conclusions ('score consistent with ground truth') depend on hand-chosen discretization and tree-size hyperparameters, and on a ground-truth ranking assembled from blogs and papers rather than a fixed benchmark.

free parameters (3)
  • bin count = 3, 4, 5, 6
    Experimental evaluation discretizes numerical features into 3-6 bins; results vary with bin count (e.g., AveOccup rank changes at 3 bins). The choice is arbitrary and affects the claimed consistency with ground truth.
  • max leaves = 100 times the bin count for California and Bike, 150 times for Adult
    Tree size regularization hyperparameter chosen by hand; different sizes would change which features appear and thus the usefulness scores.
  • number of trained trees = 20 per bin size
    Used to average scores; not fitted but chosen for statistical stability.
assumptions (4)
  • standard math P is not NP and #P-hardness assumptions underlying the NP-completeness and #P-completeness results
    Used implicitly in Propositions 14, 16, 17 and the coNP-completeness of USEFUL_DNF.
  • domain assumption Model classes are closed under conditioning, disjoint disjunction, negation, and conjunction as stated in Section 2
    Used in Propositions 23 and 24; not all real model classes satisfy these, so results are conditional on the closure properties.
  • domain assumption Comparison-based models can be discretized into finitely many intervals without changing behavior
    Invoked in Section 4.2 to extend necessity detection to numerical features; no general proof or bound is given.
  • ad hoc to paper The 'ground truth' importance rankings from Kaggle posts and selected papers are accurate
    Used in Section 5 to judge the usefulness ranking; the sources are informal and chosen by the authors.

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Cite this review

Pith. "Pith review of Feature Relevancy, Necessity and Usefulness: Complexity and Algorithms." pith.science (2026). https://pith.science/paper/QFCLFARS

@misc{pith2026250509640,
  author       = {Pith},
  title        = {Pith review of: Feature Relevancy, Necessity and Usefulness: Complexity and Algorithms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QFCLFARS}},
  note         = {Machine review of arXiv:2505.09640}
}
read the original abstract

Given a classification model and a prediction for some input, there are heuristic strategies for ranking features according to their importance in regard to the prediction. One common approach to this task is rooted in propositional logic and the notion of \textit{sufficient reason}. Through this concept, the categories of relevant and necessary features were proposed in order to identify the crucial aspects of the input. This paper improves the existing techniques and algorithms for deciding which are the relevant and/or necessary features, showing in particular that necessity can be detected efficiently in complex models such as neural networks. We also generalize the notion of relevancy and study associated problems. Moreover, we present a new global notion (i.e. that intends to explain whether a feature is important for the behavior of the model in general, not depending on a particular input) of \textit{usefulness} and prove that it is related to relevancy and necessity. Furthermore, we develop efficient algorithms for detecting it in decision trees and other more complex models, and experiment on three datasets to analyze its practical utility.

Figures

Figures reproduced from arXiv: 2505.09640 by the authors.

Figure 1
Figure 1. Example of a decision tree representing a recommendation system for a film database. The set of features [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. An FBDD representing the CNF formula from Example 1. Note that each directed path does not contain two nodes with the same feature. Let P 0 T ⊂ PT be the subset of paths of T that end at a node with label 0. Then, it holds that T ≡ ¬ W p∈P − T Fp = V p∈P − T Cp. We denote the boolean formula V p∈P − T Cp as cnf(T). It holds that |cnf(T)| = O(|X||T|). Even though any decision tree is represented by a succinct (i.e. w… view at source ↗
Figure 3
Figure 3. Algorithm to decide whether Tv(xx=c) = 1 for all c ∈ (a, b] Algorithm 2 tautT (v, e, x, Z) Require: v is a node from T, e an entity for T and x a categorical feature 1: if Z = ∅ then 2: return 1 3: else if v is a leaf with label b then 4: return b 5: end if 6: w1 ← left child of v 7: w2 ← right child of v 8: if feat(v) ̸= x ∧ l(v)(e) = 1 then 9: return tautT (w1, e, x, Z 10: else if feat(v) ̸= x ∧ l(v)(e) = 0 then 1… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Algorithm to decide whether Tv(ex=b) = 1 for all b ∈ Z. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: Results for the California Housing Dataset. For each number of bins we train 20 models and show the average [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]
Figure 6
Figure 6. Figure 6: Results for the Bike Sharing Dataset. The displayed plots are analogous to the ones from Figure [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: Results for the Adult Income Dataset. The displayed plots are analogous to the ones from Figure [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.