Unifying Formal Explanations: A Complexity-Theoretic Perspective

Published as a conference paper at ICLR 2026 U N I F Y I N G F O R M A L E X P L A N A T I O N S : A C O M P L E X I T Y - T H E O R E T I C P E R S P E C T I V E Shahaf Bassan 1 ∗ , Xuanxiang Huang 2 ∗ , Guy Katz 1 The Hebrew Uni versity of Jerusalem 1 , Nanyang T echnological Uni versity 2 shahaf.bassan@mail.huji.ac.il, xuanxiang.huang@ntu.edu.sg, g.katz@mail.huji.ac.il A B S T R AC T Previous work has explored the computational complexity of deriving two fun- damental types of explanations for ML model predictions: (i) sufﬁcient r easons , which are subsets of input features that, when ﬁxed, determine a prediction, and (ii) contrastive r easons , which are subsets of input features that, when modiﬁed, alter a prediction. Prior studies hav e examined these e xplanations in different con- texts, such as non-probabilistic versus probabilistic frame works and local versus global settings. In this study , we introduce a uniﬁed framework for analyzing these explanations, demonstrating that they can all be characterized through the minimization of a uniﬁed probabilistic v alue function. W e then prove that the complexity of these computations is inﬂuenced by three ke y properties of the value function: (i) monotonicity , (ii) submodularity , and (iii) supermodularity — which are three fundamental properties in combinatorial optimization . Our ﬁndings un- cov er some counterintuitiv e results regarding the nature of these properties within the explanation settings examined. For instance, although the local value functions do not exhibit monotonicity or submodularity/supermodularity whatsoev er , we demonstrate that the global v alue functions do possess these properties. This distinction enables us to prov e a series of nov el polynomial-time results for com- puting v arious explanations with pro vable guarantees in the global explainability setting, across a range of ML models that span the interpretability spectrum, such as neural networks, decision trees, and tree ensembles. In contrast, we sho w that ev en highly simpliﬁed versions of these explanations become NP-hard to compute in the corresponding local explainability setting. 1 I N T RO D U C T I O N Despite substantial progress in methods for explaining ML model decisions, the literature has consistently unfortunately found that many desirable explanation types with different prov able guarantees are computationally hard to obtain (Barceló et al., 2020; V an den Broeck et al., 2022), with the dif ﬁculty typically w orsening in comple x or highly non-linear models (Barceló et al., 2020; Adolﬁ et al., 2025; 2024). As a result, the computational complexity of obtaining explanations has become a central theoretical focus, with man y recent works aiming to chart which types of e xplanations can be ef ﬁciently obtained for different kinds of models, and which remain out of reach (Barceló et al., 2020; Wäldchen et al., 2021; Arenas et al., 2022; 2023; Marzouk & De La Higuera, 2024; Ordyniak et al., 2023; Laber, 2024; Bhattacharjee & Luxbur g, 2024; Blanc et al., 2021; 2022). From sufﬁcient to contrastive r easons. Among studies on the computational complexity of generat- ing explanations, two fundamental types of e xplanations for ML models were extensiv ely examined: (i) sufﬁcient r easons and (ii) contrastive r easons (Barceló et al., 2020; Arenas et al., 2022; Audemard et al., 2022a; Arenas et al., 2021; Barceló et al., 2025; Marques-Silva & Ignatiev, 2022; Ignatiev et al., 2020b; Darwiche & Hirth, 2020). A sufﬁcient r eason is a subset of input features S such that when these features are ﬁxed to speciﬁc v alues, the model’ s prediction remains unchanged, reg ardless of the values assigned to the complementary set S . A contrastive r eason is a subset of input features S such that modifying these features leads to a change in the model’ s prediction. ∗ Equal contribution. 1 Published as a conference paper at ICLR 2026 Unlike additi ve attrib ution methods, which allocate importance scores across features b ut are often hard for humans to interpret (Kumar et al., 2020) or lack actionability (Bilodeau et al., 2024), suf ﬁcient and contrasti ve reasons pro vide discr ete, condition-like e xplanations that directly answer “what is enough to justify this prediction?” or “what must change to ﬂip it?”. Their intuitive nature has giv en them a central role in many classic XAI methods (Ribeiro et al., 2018; Carter et al., 2019; Ignatiev et al., 2019; Dhurandhar et al., 2018), shown greater effecti veness in improving human prediction ov er additiv e models (Ribeiro et al., 2018; Y in & Neubig, 2022; Dhurandhar et al., 2018), and prov ed useful for do wnstream tasks such as bias detection (Balkir et al., 2022; Carter et al., 2021; La Malfa et al., 2021; Muthukumar et al., 2018), model debugging (Jaco vi et al., 2021), and anomaly detection (Da vidson et al., 2025). A well-established principle further holds that smaller suf ﬁcient and contrasti ve e xplanations enhance interpretability , making minimality a central guarantee of interest (Ribeiro et al., 2018; Lopardo et al., 2023; Carter et al., 2019; Barceló et al., 2020; Arenas et al., 2022; Blanc et al., 2021; Wäldchen et al., 2021). Barceló et al. (2020) conducted one of the earliest studies on the complexity of deri ving sufﬁcient and contrastiv e reasons. Their work established that ﬁnding the minimal-sized sufﬁcient reason for a decision tree is NP-hard, with the complexity further increasing to Σ P 2 -hard for neural networks. In the case of contrasti ve reasons, they demonstrated that computing the smallest possible e xplanation is solvable in polynomial time for decision trees but becomes NP-Hard for neural networks. Similar hardness results were later sho wn to hold for tree ensembles as well (Izza & Marques-Silv a, 2021; Ordyniak et al., 2024; Audemard et al., 2022b). From non-pr obabilistic to probabilistic explanations. A common criticism of the classic deﬁnition of sufﬁcient and contrasti ve reasons is their rigidity and lack of ﬂexibility , as they hold in an absolute sense ov er entire domains and can thus lead to e xcessively large or uninformati ve explanations (Ig- natiev et al., 2019; 2020a; Arenas et al., 2022; Wäldchen et al., 2021). T o address these limitations, the literature has shifted to wards a more general deﬁnition that incorporates a pr obabilistic perspecti ve on suf ﬁcient and contrastiv e reasons (Wäldchen et al., 2021; Arenas et al., 2022; Blanc et al., 2021; Izza et al., 2023; Xue et al., 2023). Under this framew ork, the goal is to identify subsets of input features that inﬂuence a prediction with a probability exceeding a gi ven threshold δ . Wäldchen et al. (2021) were the ﬁrst to study the complexity of probabilistic sufﬁcient reasons, showing that for CNF classiﬁers the problem is NP PP -hard. This hardness extends to tree ensembles and neural networks (Barceló et al., 2020; Ordyniak et al., 2023). A central theoretical insight in the probabilistic setting is the lack of monotonicity in the probability function (Arenas et al., 2022; Izza et al., 2023), which makes e ven subset minimal explanations computationally hard. Strikingly , (Arenas et al., 2022) sho w that ﬁnding a subset minimal probabilistic suf ﬁcient reason is NP-hard e ven for decision trees — unlike in the non-probabilistic case, where such explanations are computable in polynomial time (Huang et al., 2021; Barceló et al., 2020). From local to global explanations. In a more recent study , Bassan et al. (2024) extend the complexity analysis from the local (non-probabilistic) setting — where explanations are tied to individual predictions — to the global (non-probabilistic) setting, which seeks sufﬁcient or contrasti ve reasons ov er entire domains. Howe ver , as with other non-probabilistic methods (Ignatie v et al., 2019; Barceló et al., 2020; Arenas et al., 2021; Darwiche & Hirth, 2020), the criteria are e xtremely strict — ar guably ev en more so in the global setting than in the local one. For instance, an y feature excluded from a global sufﬁcient reason is deemed strictly redundant (Bassan et al., 2024), often making the explanation span nearly all input features and thus less informati ve. O U R C O N T R I B U T I O N S 1. W e unify pre vious explanation computation problems — including suf ﬁcient, contrasti ve, probabilistic, non-probabilistic, as well as local and global — into one framew ork, described as a minimization task over a uniﬁed value function. W e then identify three fundamen- tal properties of the value function that signiﬁcantly impact the complexity of this task: (a) monotonicity , (b) supermodularity . and (c) submodularity . 2. Interestingly , we sho w that these properties behav e in strikingly dif fer ent manners depending on the structure of the v alue function. In particular , we demonstrate the surprising result that while the local value functions for both suf ﬁcient and contrasti ve reasons are non-monotonic , their global counterparts are monotonic non-decr easing . Moreover , we identify additional 2 Published as a conference paper at ICLR 2026 intriguing properties unique to the global setting: the global sufﬁcient value function is supermodular , whereas the global contrastiv e value function is submodular — in contrast to the local setting, where neither property holds. 3. W e le verage these properties to deri ve ne w complexity results for explanation computation, rev ealing the intriguing ﬁnding that global e xplanations with guarantees can be computed efﬁciently , e ven though computing their local counterparts remains computationally hard. W e demonstrate these ﬁndings across three widely used model types that span the inter- pretability spectrum: (i) neural networks, (ii) decision trees, and (iii) tree ensembles. First, we prov e that while computing a subset-minimal local suf ﬁcient/contrastiv e probabilistic explanation is NP-hard ev en for decision trees (Arenas et al., 2022), its global counter- part can be computed in polynomial time. W e further extend this result to any black-box model (including complex models such as neural networks and tree ensembles) when using empirical distributions. Speciﬁcally , we show that obtaining a subset-minimal global suf ﬁ- cient/contrastiv e explanation is achiev able in polynomial time, whereas the local version remains NP-hard for these models. 4. Finally , we present an ev en stronger complexity result by exploiting the submodular and supermodular properties of the value functions in the global setting — properties that do not hold in the local case. Speciﬁcally , we show that it is possible to achiev e provable constant- factor approximation guarantees for computing cardinally minimal global explanations — ev en for complex models like neural networks or tree ensembles — when the empirical distribution is ﬁx ed. In sharp contrast, we establish strong inapproximability results for the local setting, demonstrating that no bounded approximation is possible, even under very simple assumptions. Owing to space constraints, we pro vide an ov erview of our main theorems and corollaries in the main text, with full proofs deferred to the appendix. 2 P R E L I M I NA R I E S Setting. W e consider an input space of dimension n ∈ N , with input v ectors x := ( x 1 , x 2 , . . . , x n ) . Each coordinate x i may take v alues from its corresponding domain X i , which can be either discrete or continuous. The full input feature space is therefore F := X 1 × X 2 × . . . × X n . W e consider classiﬁcation models f : F → [ c ] where c ∈ N is the number of classes. Moreover , we consider a generic distribution D : F → [0 , 1] ov er the input space. In many settings, ho wev er , accurately approximating D is computationally infeasible. A natural alternative is to instead work with a ﬁxed empirical dataset D := { z 1 , z 2 , . . . , z | D | } , which serves as a practical proxy for the “true” distribution, for instance, by taking D as a sampled subset from the av ailable training data. The explanations that we study are either local or global. In the local case, the explanations target a speciﬁc prediction x ∈ F , providing a form of reasoning for why the model predicted f ( x ) for that instance. In the global case, explanations aim to reﬂect the general reasoning behind the behavior of f across a wider re gion of the input space, independent of an y speciﬁc x , and to characterize its ov erall decision-making logic. Models. While many results presented in this work are general (e.g., inherent properties of value functions), we also provide some model-speciﬁc complexity results for widely-used ML models. T o broadly address the interpretability spectrum, we chose to analyze models ranging from those typically considered “black-box” to those commonly regarded as “interpretable”. Speciﬁcally , we focus on: (i) decision trees, (ii) neural networks (all architectures at least as expressi ve as feed-forw ard ReLU networks), and (iii) tree ensembles, including majority-voting ensembles (e.g., random forests) and weighted-voting ensembles (e.g., XGBoost). Formal deﬁnitions are provided in the Appendix. Distributions. W e emphasize that in probabilistic explanation settings, the complexity can v ary sig- niﬁcantly with the input distribution D . Particularly , we focus on three distribution types: (i) general distributions , which mak e no speciﬁc assumptions over D and thus encompass all possible distribu- tions. W e use these distributions mainly in proofs of properties that hold uni versally; (ii) empirical distributions , which include all distrib utions derived from the ﬁnite dataset D — an approach com- monly employed in XAI (Lundberg & Lee, 2017; V an den Broeck et al., 2022); and (iii) independent distributions , which assume that features in D are mutually independent — another widely adopted 3 Published as a conference paper at ICLR 2026 assumption in XAI literature (Arenas et al., 2022; 2023; Lundberg & Lee, 2017; Ribeiro et al., 2018). W e note that empirical distrib utions do not necessarily imply feature independence; rather, the y can represent complex dependencies extracted from ﬁnite datasets (V an den Broeck et al., 2022). The complete formal deﬁnitions of these distributions are pro vided in the Appendix. 3 F O R M S O F R E A S O N S In this section, we introduce the explanation types studied in this work, starting with the strict non-probabilistic deﬁnitions of (local/global) suf ﬁcient and contrastiv e reasons, and then extending to their more ﬂexible and generalizable probabilistic counterparts. 3 . 1 Non-Pr obabilistic S U FFI C I E N T A N D C O N T R A S T I V E R E A S O N S Sufﬁcient reasons. In the conte xt of feature selection , users often select the top k features that contribute to a model’ s decision. W e examine the well-established sufﬁciency criterion for this selection, which aligns with commonly used explainability methods (Ribeiro et al., 2018; Carter et al., 2019; Ignatiev et al., 2019; Dasgupta et al., 2022). This feature selection can be carried out either locally — focusing on a single prediction — or globally — across the entire input domain. Following standard con ventions, we deﬁne a local sufﬁcient r eason as a subset of features S ⊆ { 1 , . . . , n } such that when the features in S are ﬁxed to their corresponding v alues in x , the model’ s prediction remains f ( x ) regardless of the values assigned to the remaining features S . Formally , S is a local sufﬁcient reason for ⟨ f , x ⟩ if f the following condition holds: ∀ z ∈ F , f ( x S ; z ¯ S ) = f ( x ) . (1) Here, ( x S ; z ¯ S ) denotes a vector where features in S take their values from x , and those in S from z . Local suf ﬁcient reasons are closely related to semi-factual explanations (K enny & Keane, 2021; Alfano et al., 2025), which search for alternati ve inputs z ′ that keep the prediction unchanged (i.e., f ( z ′ ) = f ( x ) ) while being as “close” as possible to the original point. When we instantiate z ′ as ( x S ; z ¯ S ) and measure proximity via Hamming distance, the two notions coincide. A global sufﬁcient r eason (Bassan et al., 2024) is a subset of input features S ⊆ { 1 , . . . , n } that serves as a local suf ﬁcient reason for every possible input x ∈ F : ∀ x , z ∈ F , f ( x S ; z ¯ S ) = f ( x ) . (2) Contrastive reasons. Another pre v alent approach to providing explanations is by pinpointing subsets of input features that modify a prediction (Dhurandhar et al., 2018; Mothilal et al., 2020; Guidotti, 2024). This type of explanation aims to determine the minimal changes necessary to alter a prediction. Formally , a subset S ⊆ { 1 , . . . , n } is deﬁned as a local contrastive r eason concerning ⟨ f , x ⟩ iff: ∃ z ∈ F , f ( z S ; x ¯ S )  = f ( x ) . (3) Contrastiv e reasons are also closely connected to counterfactual explanations (Guidotti, 2024; Mothilal et al., 2020), which seek a nearby assignment x ′ for which the prediction ﬂips, i.e., f ( x ′ )  = f ( x ) . When the distance metric is taken to be the Hamming weight, the two notions coincide. Similarly to suf ﬁcient reasons, one can determine a subset of input features that, when altered, changes the prediction for all inputs within the domain of interest. This form of explanation is also closely connected to approaches for identifying bias (Arenas et al., 2021; Bassan et al., 2024; Darwiche & Hirth, 2020), as well as to gr oup counterfactual explanation methods that search for counterfactuals ov er multiple data instances (Carrizosa et al., 2024; W arren et al., 2024). Formally , we deﬁne a subset S as a global contrastive r eason with respect to f iff: ∀ x ∈ F , ∃ z ∈ F , f ( z S ; x ¯ S )  = f ( x ) . (4) 4 Published as a conference paper at ICLR 2026 3 . 2 Pr obabilistic S U FFI C I E N T A N D C O N T R A S T I V E R E A S O N S As discussed in the introduction, non-probabilistic sufﬁcient and contrastiv e reasons are often criticized for imposing signiﬁcantly overly strict conditions, motivating a shift to pr obabilistic deﬁnitions (Arenas et al., 2022; Ribeiro et al., 2018; Izza et al., 2023; Wäldchen et al., 2021; Bounia & K oriche, 2023; Subercaseaux et al., 2025; W ang et al., 2021a; Blanc et al., 2021), which generalize these requirements by demanding that the guarantees hold with probability at least δ ∈ [0 , 1] . The special case δ = 1 recov ers the original non-probabilistic deﬁnitions. Probabilistic sufﬁcient reasons. W e deﬁne S ⊆ { 1 , . . . , n } as a local δ -sufﬁcient r eason with respect to ⟨ f , x ⟩ if, when the features in S are ﬁxed to their corresponding values in x and the remaining features are sampled from a distrib ution D , the classiﬁcation remains unchanged with probability at least δ . In other words: Pr z ∼D ( f ( z ) = f ( x ) | z S = x S ) ≥ δ. (5) where z S = x S denotes that the features in S of vector z are ﬁxed to their corresponding values in x . W e adopt the standard notion of global e xplanations — by averaging o ver all inputs in the global domain — and deﬁne a subset S ⊆ { 1 , . . . , n } as a global δ -sufﬁcient r eason with respect to ⟨ f ⟩ if, when taking the expectation of the local sufﬁciency probability ov er samples dra wn from the distribution D , the expectation remains with v alue at least δ . In other words: E x ∼D [ Pr z ∼D ( f ( z ) = f ( x ) | z S = x S )] ≥ δ. (6) Probabilistic contrastive reasons. Similar to the non-probabilistic case, we deﬁne a local δ - contrastive r eason as a subset of input features that changes a prediction with some probability . Here, unlike suf ﬁcient reasons, we set the features of the complementary set S to their respectiv e values in x , and when allo wing the features in S to v ary , we require the prediction to dif fer from the original prediction f ( x ) with a probability exceeding δ . Formally: Pr z ∼D ( f ( z )  = f ( x ) | z ¯ S = x ¯ S ) ≥ δ. (7) For the global setting, we deﬁne a subset S to be a global δ -contrastive r eason , analogous to global sufﬁcient reasons, by computing the e xpectation ov er all local contrastiv e reasons sampled from the distribution D , and requiring that this expected v alue exceeds δ : E x ∼D [ Pr z ∼D ( f ( z )  = f ( x ) | z ¯ S = x ¯ S )] ≥ δ. (8) 4 F O R M S O F M I N I M A L I T Y As discussed in the introduction, across all the explanation forms discussed so far — whether sufﬁcient or contrasti ve, local or global — a common assumption in the literature is that explanations of smaller size are more meaningful, thereby making their minimality a particularly important prov able guarantee. In this study , we explore two central notions of minimality across all our e xplanation types: Deﬁnition 1. Assuming a subset S ⊆ { 1 , . . . n } is an explanation, then: 1. S is a cardinally-minimal explanation (Barceló et al., 2020; Bassan et al., 2024) if f S has the smallest explanation car dinality | S | (i.e., ther e is no explanation S ′ such that | S ′ | < | S | ). 2. S ⊆ { 1 , . . . , n } is a subset-minimal explanation (Ar enas et al., 2022; Ignatiev et al., 2019) iff S is an explanation, and any S ′ ⊆ S is not an e xplanation. W e note that cardinal-minimality is strictly stronger than subset-minimality: e very cardinally minimal S is subset-minimal, but not vice versa (see Appendix B.3 for an example). W e use the terms subset and cardinally minimal, rather than local and global minima, to av oid confusion with local vs. global explanations (input- vs. domain-lev el reasoning). Both notions apply to all explanation forms we analyze. For instance, a cardinally minimal local probabilistic δ sufﬁcient reason is the one with the smallest | S | , while a subset-minimal one is any S where no proper subset S ′ ⊆ S also qualiﬁes. 5 Published as a conference paper at ICLR 2026 5 A U N I FI E D C O M B I N A T O R I A L O P T I M I Z A T I O N T A S K Interestingly , all previously discussed computational problems — local or global, sufﬁcient or contrasti ve, probabilistic or not — can be cast as ﬁnding a minimal-size subset S such that v ( S ) ≥ δ , where in non-probabilistic settings we set δ := 1 . W e no w introduce notation for the value functions: let v ℓ suff denote the local sufﬁciency probability from equation 5, i.e., Pr z ∼D p ( f ( x ) = f ( z ) | x S = z S ) , and deﬁne the global variant as v g suff (equation 6). Similarly , let v ℓ con and v g con denote the local and global contrastiv e probabilities (equations 7 and 73, respectiv ely). Using this notation, we now formally deﬁne the uniﬁed task of ﬁnding a cardinally minimal δ -local/global sufﬁcient/contrasti ve reason: Cardinally Minimal δ -Explanation : Input : Model f , a distribution D , (possibly , an input x ), a general value function v : 2 n → [0 , 1] (deﬁned using f , D , and possibly x ), and some δ ∈ [0 , 1] . Output : A subset S ⊆ [ n ] such that v ( S ) ≥ δ and | S | is minimal. Similarly , for the relaxed condition where the goal is to obtain a subset-minimal rather than a cardinally-minimal local or global sufﬁcient/contrasti ve reason, we deﬁne the following relax ed optimization objectiv e: Subset Minimal δ -Explanation : Input : Model f , a distribution D , (possibly , an input x ), a general value function v : 2 n → [0 , 1] (deﬁned using f , D , and possibly x ), and some δ ∈ [0 , 1] . Output : A subset S ⊆ [ n ] such that v ( S ) ≥ δ and for any S ′ ⊆ S it holds that v ( S ′ ) < δ . Properties of v that affect the complexity . W e no w outline se veral key properties of the value function, which we later show play a crucial role in determining the complexity of generating explanations. The ﬁrst property is non-decr easing monotonicity , which ensures that the marginal contribution v ( S ∪ { i } ) − v ( S ) is consistently non-negati ve. Formally: Deﬁnition 2. W e say that a value function v maintains non-decreasing monotonicity if for any S ⊆ { 1 , . . . , n } and any i ∈ { 1 , . . . , n } it holds that: v ( S ∪ { i } ) ≥ v ( S ) . The other key properties of supermodularity and submodularity pertain to the beha vior of the mar ginal contribution v ( S ∪ { i } ) − v ( S ) . Speciﬁcally , in the supermodular case, this contribution forms a monotone non-decreasing function. In contrast, under the dual deﬁnition of the submodular case, the marginal contrib ution v ( S ∪ { i } ) − v ( S ) is a monotone non-increasing function. F ormally: Deﬁnition 3. Let ther e be some value function v , some S ⊆ S ′ ⊆ { 1 , . . . , n } , and i ∈ S ′ . Then: 1. v maintains supermodularity iff it holds that: v ( S ∪ { i } ) − v ( S ) ≤ v ( S ′ ∪ { i } ) − v ( S ′ ) . 2. v maintains submodularity iff it holds that: v ( S ∪ { i } ) − v ( S ) ≥ v ( S ′ ∪ { i } ) − v ( S ′ ) . 6 U N R A V E L I N G N E W P R O P E RT I E S O F T H E G L O B A L V A L U E F U N C T I O N S Prior work shows that in the local setting of non-probabilistic explanations, subset-minimal sufﬁcient or contrastiv e reasons can be computed thanks to monotonicity , which holds only in the restricted case δ = 1 . This assumption, ho wever , is highly limiting and lacks practical ﬂe xibility . While one might hope to generalize to probabilistic guarantees for arbitrary δ , prior results demonstrate that monotonicity breaks down in this setting, rendering the computation of explanations computationally harder (Arenas et al., 2022; K ozachinskiy, 2023; Izza et al., 2023; Subercaseaux et al., 2025; Izza et al., 2024b). At the ev en more extreme case of the global and non-pr obabilistic setting, Bassan et al. (2024) demon- strated the stringent uniqueness property — i.e., there is exactly one subset-minimal explanation. Howe ver , requiring δ = 1 makes this setting highly restricti ve, especially in the global case, where the explanation conditions must hold for all inputs. In fact, Bassan et al. (2024) pro ves that this unique minimal subset is equiv alent to the subset of all features that are not strictly redundant (which may , in practice, be all of them). W e pro ve that in the general global pr obabilistic setting — for any δ — this uniqueness property actually does not hold, and the number of subsets can ev en be e xponential . 6 Published as a conference paper at ICLR 2026 Proposition 1. While the non-pr obabilistic case ( δ = 1 ) admits a unique subset-minimal global sufﬁcient r eason (Bassan et al., 2024), in the gener al pr obabilistic setting (for arbitrary δ ), ther e e xist functions f that have Θ( 2 n √ n ) subset-minimal global sufﬁcient r easons. Interestingly , although the uniqueness property fails to hold in the general case for arbitrary δ (and is restricted to the special case of δ = 1 ), we show that the crucial monotonicity property holds for the global value function across all v alues of δ . This applies to both the suf ﬁcient ( v g suff ) and contrastiv e ( v g con ) value functions. This ﬁnding is surprising as it stands in sharp contrast to the local setting, where the corresponding value functions ( v ℓ suff and v ℓ con ) do not satisfy this property: Proposition 2. While the local pr obabilistic setting (for any δ ) lacks monotonicity — i.e ., the value functions v ℓ con and v ℓ suff ar e non-monotonic (Arenas et al., 2022; Izza et al., 2023; Subercaseaux et al., 2025; Izza et al., 2024b) — in the global pr obabilistic setting (also for any δ ), both value functions v g con and v g suff ar e monotonic non-decr easing. Beyond the surprising insight that monotonicity holds for the global value functions — but fails to hold in the local one — we identify additional structural properties unique to the global setting, including submodularity or supermodularity . In particular , we show that under the common assumption of feature independence, the global sufﬁcient v alue function v g suff exhibits supermodularity . In contrast, its local counterpart v ℓ suff fails to e xhibit this property e ven under the much more restricti ve assumption of a uniform (and hence independent) input distribution. Speciﬁcally: Proposition 3. While the local pr obabilistic sufﬁcient setting (for any δ ) lacks supermodularity — even when D is uniform, i.e., the value function v ℓ suff is not supermodular — in the global pr oba bilistic setting (also for any δ ), when D exhibits featur e independence, the value function v g suff is supermodular . Interestingly , for the second family of explanation settings — speciﬁcally , that of obtaining a global probabilistic contrastive reason — we show that the corresponding v alue function is not supermodular , but rather submodular . This result is particularly surprising when contrasted with the local setting, where the value function is neither submodular nor supermodular . Proposition 4. While the local pr obabilistic contrastive setting (for any δ ) lacks submodularity — even when D is uniform, i.e., the value function v ℓ con is not submodular — in the global pr obabilistic setting (also for any δ ), when D exhibits featur e independence, the value function v g con is submodular . 7 C O M P U T A T I O N A L C O M P L E X I T Y R E S U L T S 7 . 1 S U B S E T M I N I M A L E X P L A N A T I O N S In this section, we e xamine the complexity of obtaining subset minimal explanations (local/global, sufﬁcient/contrasti ve) across the different model types analyzed. The key property at play here is the monotonicity of the value function. The previous section established that monotonicity holds for both global v alue functions, v g con and v g suff , but does not hold for the local v alue functions, v ℓ con and v ℓ suff . This distinction is crucial in showing that a greedy algorithm con ver ges to a subset minimal explanation in the global setting but fails in the local setting. As a result, we will showcase the surprising ﬁnding that computing v arious local subset-minimal explanation forms is hard, whereas computing subset-minimal global explanation forms is tractable (polynomial-time solvable). W e will begin by introducing the follo wing generalized greedy algorithm: Algorithm 1 Subset Minimal Explanation Search Input V alue function v , and some δ ∈ [0 , 1] 1: S ← { 1 , . . . , n } 2: while min i ∈ S v ( S \ { i } ) ≥ δ do 3: j ← argmax i ∈ S v ( S \ { i } ) 4: S ← S \ { j } 5: end while 6: return S ▷ S is a (subset minimal?) δ -explanation Algorithm 1 aims to obtain a subset-minimal δ -explanation. W e begin the algorithm with the subset S initialized as the entire input space { 1 , . . . , n } . Iterati vely , we check whether the minimal value that 7 Published as a conference paper at ICLR 2026 v ( S \ { i } ) can attain exceeds δ . In each iteration, we remo ve a feature j from S that minimizes the decrease in the value function, selecting the feature j that maximizes v ( S \ { j } ) . Once this iterativ e process concludes, we return S . The key determinant of whether Algorithm 1 yields a subset-minimal explanation is the monotonicity property of the v alue function v . The algorithm concludes with a phase in which removing any individual feature from S results in v ( S \ { i } ) being smaller than δ . Howe ver , monotonicity ensures that this holds for any v ( S \ S ′ ) , providing a signiﬁcantly stronger guarantee. Gi ven the monotonicity property of the value functions established in the previous sections, we derive the following proposition: Proposition 5. Computing Algorithm 1 with the local value functions v ℓ con and v ℓ suff does not always con ver ge to a subset minimal δ -sufﬁcient/contr astive r eason. However , computing it with the global value functions v g con or v g suff necessarily pr oduces subset minimal δ -sufﬁcient/contr astive r easons. Building on this result, we proceed to establish ne w complexity ﬁndings for deri ving various forms of subset-minimal explanations within our frame work, considering the dif ferent analyzed model types. Decision trees. W e begin by e xamining a highly simpliﬁed and ostensibly “interpretable” scenario. Speciﬁcally , we assume that the model f is a decision tree and that the distribution D is independent. W ithin this simpliﬁed setting, we demonstrate a strict separation: Arenas et al. (2022) established the surprising intractability result that, unless PTIME = NP , no polynomial-time algorithm exists for computing a subset-minimal local δ -sufﬁcient reason for decision trees under independent distributions (e ven under the uniform distribution). In contrast, we demonstrate the unexpected result that this exact problem can be solved ef ﬁciently in the global setting, meaning that a subset-minimal global δ -sufﬁcient reason for decision trees can indeed be computed in polynomial time. Formally: Theorem 1. If f is a decision tr ee and the pr obability term v g suff can be computed in polynomial time given the distrib ution D (which holds for independent distributions, among other s), then obtaining a subset-minimal global δ -sufﬁcient r eason can be obtained in polynomial time . However , unless PTIME = NP , no polynomial-time algorithm exists for computing a local δ -sufﬁcient r eason for decision tr ees even under independent distrib utions. Extension to other tractable models. W e additionally note that a similar tractability guarantee for computing global explanations also holds for orthogonal DNFs (Crama & Hammer, 2011), which we brieﬂy recall are Disjuncti ve Normal F orm formulas whose terms are pairwise mutually e xclusiv e. Establishing the result for this class is useful because orthogonal DNFs gener alize decision trees while preserving their clean structural properties, allowing our guarantees to extend be yond tree-structured models. For completeness, we pro vide the full argument in Appendix N. Neural networks, tree ensembles, and other complex models. W e now extend our pre vious results to more complex models beyond decision trees. Speciﬁcally , we will demonstrate that when the distribution D is deriv ed from empirical distributions, and under the fundamental assumption that the model f allows polynomial-time inference, it follo ws that computing a subset-minimal global δ -sufﬁcient and contrasti ve reason can be done in polynomial time. Proposition 6. F or any model f , and empirical distribution D — computing a subset-minimal global δ -sufﬁcient or δ -contrastive r eason for f can be done in polynomial time. This strong complexity outcome, which holds for any model under an empirical data distribution assumption, allows us to further dif ferentiate the complexity of local and global explanation settings. Speciﬁcally , for certain complex models, computing subset-minimal local explanations remains intractable even when restricted to empirical distributions. W e establish this fact for both neural networks and tree ensembles, leading to the follo wing theorem on a strict complexity separation: Theorem 2. Assuming f is a neural network or a tr ee ensemble, and D is an empirical distribution — ther e exist polynomial-time algorithms for obtaining subset minimal global δ -sufﬁcient and con- trastive r easons. However , unless PTIME = NP , ther e is no polynomial time algorithm for computing a subset minimal local δ -sufﬁcient r eason or a subset minimal local δ -contrastive r eason. 7 . 2 A P P R OX I M AT E C A R D I NA L L Y M I N I M A L E X P L A N A T I O N S In this subsection, we shift our focus from subset-minimal sufﬁcient/contrasti ve reasons to the ev en more challenging task of ﬁnding a cardinally minimal δ -sufﬁcient/contrasti ve reason. W e 8 Published as a conference paper at ICLR 2026 will demonstrate that in the global setting, the interplay between supermodularity/submodularity and monotonicity of the value function enables us to establish nov el pr ovable appr oximations for computing explanations. In contrast, we will show that computing these explanations in the local setting remains intractable. This result further strengthens the surprising distinction between the tractability of computing global explanations v ersus local ones. A uniﬁed greedy approximation algorithm. When working with a non-decreasing monotonic submodular function, the problem of identifying a cardinally minimal explanation closely aligns with the submodular set cover problem (W olsey, 1982). In contrast, employing a supermodular function leads to a non-submodular variation of this problem (Shi et al., 2021). These problems have garnered signiﬁcant interest due to their strong approximation guarantees (W olsey, 1982; Iyer & Bilmes, 2013; Chen & Crawford, 2023). In the context of submodular set cover optimization, a standard approach in volv es using a classic greedy algorithm, which we will ﬁrst outline (Algorithm 2). This algorithm serves as the foundation for approximating a cardinally minimal suf ﬁcient or contrastiv e δ -reason, and we will later examine its speciﬁc guarantees in both cases. Algorithm 2 Cardinally Minimal Explanation Approximation Search Input V alue function v , and some δ ∈ [0 , 1] 1: S ← ∅ 2: while max i ∈ S v ( S ∪ { i } ) < δ do 3: j ← argmax i ∈ S v ( S ∪ { i } ) 4: S ← S ∪ { j } 5: end while 6: return S ▷ | S | is a (prov able approximation?) of a cardinally minimal δ -explanation Algorithm 2 closely resembles Algorithm 1, but works bottom-up rather than top-down. It starts with an empty subset and incrementally adds features, each time selecting the one that minimizes the increase in v ( S ∪ i ) , stopping when adding any feature would push the v alue ov er δ . Cardinally minimal contrastive r easons. W e begin with global contrasti ve reasons, where mono- tonicity and submodularity hold, reducing the task to the classic submodular set cover problem and allowing us to apply W olsey (1982)’ s classic guarantee via Algorithm 2. For integer -valued objec- tiv es, the algorithm achiev es a Harmonic-based factor , and more generally an O  ln  v ([ n ]) min i ∈ [ n ] v ( i )  - approximation. Under an empirical distribution D with ﬁxed sample size, this becomes a constant appr oximation , only logarithmic in the sample size, yielding a substantially strong guarantee. By contrast, in the local setting — e ven for a single sample point — no bounded approximation exists, marking a sharp gap between global and local cases. Theorem 3. Given a neural network or tree ensemble f and an empirical distribution D over a ﬁxed dataset D , Algorithm 2 yields a constant O  ln  v g con ([ n ]) min i ∈ [ n ] v g con ( { i } )  -appr oximation, bounded by O (ln( | D | )) , for computing a global cardinally minimal δ -contrastive r eason for f , assuming featur e independence. In contrast, unless PTIME = NP , no bounded approximation exists for computing a local car dinally minimal δ -contrastive r eason for any ⟨ f , x ⟩ , even when | D | = 1 . W e also provide a matching lower bound for the bound in Theorem 3 in the speciﬁc case of an empirical distribution without an y independence assumption (see Appendix M). Cardinally minimal sufﬁcient reasons. Unlike the submodular set cover problem — linked to cardi- nally minimal global contrastive reasons and admitting strong approximations — the supermodular variant, tied to global sufﬁcient reasons, is harder to approximate. Still, it offers guarantees when the function has bounded curv ature (Shi et al., 2021). The total curvatur e of a function v : 2 n → R is deﬁned as k f := 1 − min i ∈ [ n ] v ([ n ]) − v ([ n ] \ i ) v ( i ) − v ( ∅ ) . Le veraging results from Shi et al. (2021), we sho w that Algorithm 2 achiev es an O  1 1 − k f + ln  v ([ n ]) min i ∈ [ n ] v ( i )  -approximation. Notably , under a ﬁxed empirical distribution, the approximation becomes constant. While contrastiv e reasons admit a tighter O (ln( | D | )) bound, suf ﬁcient reasons incur an extra 1 1 − k f factor — yet still yield a constant-factor approximation. In sharp contrast, the local v ariant remains inapproximable, lacking any bounded approximation ev en when | D | = 1 . 9 Published as a conference paper at ICLR 2026 Theorem 4. Given a neural network or tr ee ensemble f and an empirical distribution D over a ﬁxed dataset D , Algorithm 2 yields a constant O  1 1 − k f + ln  v g suff ([ n ]) min i ∈ [ n ] v g suff ( { i } )  -appr oximation for computing a global car dinally minimal δ -sufﬁcient r eason for f , assuming featur e independence. In contrast, unless PTIME = NP , ther e is no bounded approximation for computing a local car dinally minimal δ -sufﬁcient r eason for any ⟨ f , x ⟩ , e ven when | D | = 1 . Overall, these ﬁndings strengthen Subsection 7.1, which showed the tractability of computing subset- minimal global e xplanations in stark contrast to local ones. Here, we further show that approximating cardinally minimal global explanations is tractable, unlike their inapproximable local counterparts. 8 L I M I T A T I O N S A N D F U T U R E W O R K While many of our most important ﬁndings — particularly those concerning fundamental properties of value functions — hold generally , we also instantiate them to yield concrete complexity results for speciﬁc model classes (e.g., neural networks), distributional assumptions (e.g., empirical distribu- tions), and explanation deﬁnitions within our framew ork. Naturally , other potential settings remain open for analysis. Nonetheless, we belie ve our ﬁndings of fer compelling insights into foundational aspects of explanations, along with ne w tractability and intractability results, which together lay a strong foundation for in vestigating a broader range of e xplainability scenarios in future work. More speciﬁcally , our study brings forward se veral important open theoretical questions: 1. Firstly , it would be interesting to in vestigate whether some of the tractable complexity results we obtained for global explainability can be tightened for simpler model classes, such as decision trees, linear models, or other models with inherently tractable structure. Although local explanations for such models are often intractable to compute (Arenas et al., 2022; Subercaseaux et al., 2025), the global explanation forms we study in this work exhibit strong structural properties, enabling signiﬁcantly improved complexity guarantees that may be tightened e ven further . Furthermore, tightening the approximation guarantees when w orking with empirical distributions represents a k ey open direction. 2. Secondly , when exact computation of expectations is infeasible, it is natural to ask whether alternativ e techniques — such as Fully Polynomial Randomized Approximation Schemes (FPRAS) for model classes like DNF formulas (Meel et al., 2019) (and thus for tree ensembles), or Monte Carlo–based approximations as the ones used in (Subercaseaux et al., 2025) — can be employed to approximate these expectations. A ke y open challenge is determining ho w to maintain minimality guarantees under approximation error , and what tolerance lev els still allow the guarantees to hold. 3. Finally , we believ e that studying alternative notions of feature importance — e.g., other loss measures such as KL diver gence (Conmy et al., 2023) — may re veal similar patterns of monotonicity , submodularity , or supermodularity when transitioning from local to global explanations. Such properties could enable efﬁcient feature selection under these alternativ e metrics as well. 9 C O N C L U S I O N W e present a uniﬁed frame work for ev aluating di verse explanations and re veal a stark contrast between local and global sufﬁcient and contrasti ve reasons. Notably , while the local explanation variants lack any structural form common in combinatorial optimization — such as monotonicity , submodularity , or supermodularity , we prove that their global counterparts exhibit precisely these crucial properties: (i) monotonicity , (ii) submodularity in the case of contrasti ve reasons, and (iii) supermodularity for sufﬁcient reasons. These proofs form the basis for pro ving a series of surprising complexity results, showing that global explanations with prov able guarantees can be computed ef ﬁciently , ev en for complex model classes such as neural networks. In sharp contrast, we pro ve that computing the corresponding local explanations remains NP-hard — ev en in highly simpliﬁed scenarios. Altogether , our results uncov er foundational properties of explanations and chart both tractable and intractable frontiers, opening new a venues for future research. 10 Published as a conference paper at ICLR 2026 A C K N O W L E D G M E N T S This work was partially funded by the European Union (ERC, V eriDeL, 101112713). V iews and opinions expressed are howe ver those of the author(s) only and do not necessarily reﬂect those of the European Union or the European Research Council Executi ve Agency . Neither the European Union nor the granting authority can be held responsible for them. This research was additionally supported by a grant from the Israeli Science Foundation (grant number 558/24). R E F E R E N C E S Federico Adolﬁ, Martina G V ilas, and T odd W areham. Complexity-Theoretic Limits on the Promises of Artiﬁcial Neural Network Rev erse-Engineering. In Pr oc. 46th Annual Meeting of the Cognitive Science Society (CogSci) , 2024. Federico Adolﬁ, Martina V ilas, and T odd W areham. The Computational Complexity of Circuit Discov ery for Inner Interpretability. In Pr oc. 13th Int. Conf. on Learning Repr esentations (ICLR) , 2025. Gian vincenzo Alfano, Sergio Greco, Domenico Mandaglio, Francesco P arisi, Reza Shahbazian, and Irina T rubitsyna. Even-If Explanations: Formal F oundations, Priorities and Complexity. In Pr oc. 39th AAAI Conf. on Artiﬁcial Intelligence , pp. 15347–15355, 2025. Ibrahim M Almanjahie, Zouaoui Chikr Elmezouar , Ali Laksaci, and Mustapha Rachdi. KNN Local Linear Estimation of the Conditional Cumulativ e Distribution Function: Dependent Functional Data Case. Comptes Rendus. Mathématique , 356(10):1036–1039, 2018. David Alv arez Melis and T ommi Jaakkola. T owards Rob ust Interpretability with Self-Explaining Neural Networks. In Pr oc. 32nd Conf. on Neural Information Pr ocessing Systems (NeurIPS) , 2018. Leila Amgoud and Martin Cooper . Axiomatic Foundations of Counterf actual Explanations. 2026. T echnical Report. . Guy Amir , Shahaf Bassan, and Guy Katz. Hard to Explain: On the Computational Hardness of In-Distribution Model Interpretation. In Pr oc. 27th Eur opean Conf . on Artiﬁcal Intelligence (ECAI) , 2024. Alaa Anani, T obias Lorenz, Mario Fritz, and Bernt Schiele. Pixel-le vel Certiﬁed Explanations via Randomized Smoothing. In Pr oc. 42nd Int. Conf. on Machine Learning (ICML) , 2025. Marcelo Arenas, D. Baez, P . Barceló, J. Pérez, and B. Subercaseaux. Foundations of Symbolic Languages for Model Interpretability. In Pr oc. 34th Int. Conf. on Advances in Neural Information Pr ocessing Systems (NeurIPS) , pp. 11690–11701, 2021. Marcelo Arenas, P . Barceló, M. Romero, and B. Subercaseaux. On Computing Probabilistic Explana- tions for Decision T rees. In Proc. 35th Int. Conf . on Advances in Neural Information Pr ocessing Systems (NeurIPS) , pp. 28695–28707, 2022. Marcelo Arenas, Pablo Barceló, Leopoldo Bertossi, and Mikaël Monet. On the Complexity of SHAP- Score-Based Explanations: Tractability via Kno wledge Compilation and Non-Approximability Results. Journal of Machine Learning Resear ch (JMLR) , 24(63):1–58, 2023. Gilles Audemard, S. Bellart, L. Bounia, F . Koriche, J. Lagniez, and P . Marquis. On the Computa- tional Intelligibility of Boolean Classiﬁers. In Pr oc. 18th Int. Conf. on Principles of Knowledg e Repr esentation and Reasoning (KR) , pp. 74–86, 2021. Gilles Audemard, Stev e Bellart, Louenas Bounia, Frédéric K oriche, Jean-Marie Lagniez, and Pierre Marquis. On Preferred Abductiv e Explanations for Decision Trees and Random Forests. In Pr oc. 31st Int. Joint Conf . on Artiﬁcial Intelligence (IJCAI) , pp. 643–650, 2022a. Gilles Audemard, Stev e Bellart, Louenas Bounia, Frédéric K oriche, Jean-Marie Lagniez, and Pierre Marquis. Trading Complexity for Sparsity in Random F orest Explanations. In Pr oc. 36th AAAI Conf. on Artiﬁcial Intelligence , number 5, pp. 5461–5469, 2022b. 11 Published as a conference paper at ICLR 2026 Gilles Audemard, Jean-Marie Lagniez, Pierre Marquis, and Nicolas Szczepanski. Computing Abductiv e Explanations for Boosted T rees. In Pr oc. 26th Int. Conf. on Artiﬁcial Intelligence and Statistics (AIST ATS) , pp. 4699–4711, 2023. Stev e Azzolin, Sagar Malhorta, Andrea Passerini, and Stefano T eso. Beyond T opological Self- Explainable GNNs: A Formal Explainability Perspective. In Pr oc. 42nd Int. Conf. on Machine Learning (ICML) , 2025. Esma Balkir , Isar Nejadgholi, Kathleen C Fraser , and Svetlana Kiritchenko. Necessity and Sufﬁciency for Explaining T ext Classiﬁers: A Case Study in Hate Speech Detection. In Pr oc. Conf. of the North American Chapter of the Association for Computational Linguistics: Human Language T echnologies (N AACL-HL T) , pp. 2672–2686, 2022. Pablo Barceló, M. Monet, J. Pérez, and B. Subercaseaux. Model Interpretability through the Lens of Computational Complexity. In Pr oc. 33r d Int. Conf. on Advances in Neural Information Pr ocessing Systems (NeurIPS) , pp. 15487–15498, 2020. Pablo Barceló, Alexander Kozachinskiy , Miguel Romero, Bernardo Subsercaseaux, and José V erschae. Explaining k-Nearest Neighbors: Abducti ve and Counterfactual Explanations. In Pr oc. 3r d ACM on Management of Data , pp. 1–26, 2025. Shahaf Bassan and G. Katz. T ow ards Formal XAI: Formally Approximate Minimal Explanations of Neural Networks. In Pr oc. 29th Int. Conf. on T ools and Algorithms for the Construction and Analysis of Systems (T ACAS) , pp. 187–207, 2023. Shahaf Bassan, Guy Amir , Davide Corsi, Idan Ref aeli, and Guy Katz. Formally Explaining Neural Networks W ithin Reactive Systems. In Pr oc. 23r d Int. Conf . on F ormal Methods in Computer-Aided Design (FMCAD) , pp. 1–13, 2023. Shahaf Bassan, Guy Amir , and Guy Katz. Local vs. Global Interpretability: A Computational Complexity Perspecti ve. In Pr oc. 41st Int. Conf. on Mac hine Learning (ICML) , 2024. Shahaf Bassan, Guy Amir, Meirav Zehavi, and Guy Katz. What makes an Ensemble (Un) Inter- pretable? In Proc. 42nd Int. Conf . on Machine Learning (ICML) , 2025a. Shahaf Bassan, Y izhak Y israel Elboher, T obias Ladner , Matthias Althoff, and Guy Katz. Explaining, Fast and Slo w: Abstraction and Reﬁnement of Provable Explanations. In Pr oc. 42nd Int. Conf. on Machine Learning (ICML) , 2025b. Shahaf Bassan, Ron Eliav , and Shlomit Gur . Explain Y ourself, Brieﬂy! Self-Explaining Neural Networks with Concise Suf ﬁcient Reasons. In Pr oc. 13th Int. Conf. on Learning Repr esentations (ICLR) , 2025c. Shahaf Bassan, Shlomit Gur , Serge y Zeltyn, K onstantinos Mavrogior gos, Ron Elia v , and Dimosthenis Kyriazis. Self-Explaining Neural Networks for Business Process Monitoring. 2025d. T echnical Report. . Shahaf Bassan, Michal Moshko vitz, and Guy Katz. Additive Models Explained: A Computational Complexity Approach. In Pr oc. 39th Conf. on Neural Information Pr ocessing Systems (NeurIPS) , 2025e. Shahaf Bassan, Y izhak Y israel Elboher , T obias Ladner , V olkan ¸ Sahin, Jan Kretinsky , Matthias Althoff, and Guy Katz. Prov ably Explaining Neural Additiv e Models. In Pr oc. 14th Int. Conf. on Learning Repr esentations (ICLR) , 2026. Robi Bhattacharjee and Ulrike Luxbur g. Auditing Local Explanations is Hard. In Pr oc. 38th Conf. on Neural Information Pr ocessing Systems (NeurIPS) , pp. 18593–18632, 2024. Blair Bilodeau, Natasha Jaques, Pang W ei K oh, and Been Kim. Impossibility Theorems for Feature Attribution. Proc. of the National Academy of Sciences (PN AS) , 121(2):e2304406120, 2024. Guy Blanc, Jane Lange, and Li-Y ang T an. Prov ably Efﬁcient, Succinct, and Precise Explanations. In Proc. 35th Int. Conf. on Advances in Neural Information Pr ocessing Systems (NeurIPS) , pp. 6129–6141, 2021. 12 Published as a conference paper at ICLR 2026 Guy Blanc, Caleb Koch, Jane Lange, and Li-Y ang T an. A Query-Optimal Algorithm for Finding Counterfactuals. In Pr oc. 39th Int. Conf. on Machine Learning (ICML) , pp. 2075–2090, 2022. Ryma Boumazouza, Fahima Cheikh-Alili, Bertrand Mazure, and Karim T abia. ASTER YX: A model- Agnostic SaT -basEd appRoach for SYmbolic and Score-Based EXplanations. In Proc. 30th A CM Int. Conf. on Information & Knowledge Mana gement (CIKM) , pp. 120–129, 2021. Ryma Boumazouza, Raya Elsaleh, Melanie Ducoffe, Shahaf Bassan, and Guy Katz. F AME: For- mal Abstract Minimal Explanation for Neural Networks. In Proc. 14th Int. Conf. on Learning Repr esentations (ICLR) , 2026. Louenas Bounia. Enhancing the Intelligibility of Boolean Decision T rees with Concise and Reliable Probabilistic Explanations. In Pr oc. 20th Int. Conf. on Information Pr ocessing and Management of Uncertainty in Knowledge-Based Systems (IPMU) , pp. 205–218, 2024. Louenas Bounia. Using Submodular Optimization to Approximate Minimum-Size Abductiv e Path Explanations for T ree-Based Models. In Pr oc. 41st Conf. on Uncertainty in Artiﬁcial Intelligence (U AI) , 2025. Louenas Bounia and Frederic Koriche. Approximating Probabilistic Explanations via Supermodular Minimization. In Pr oc. 39th Conf. on Uncertainty in Artiﬁcial Intelligence (U AI) , pp. 216–225, 2023. Marco Calautti, Enrico Malizia, and Cristian Molinaro. On the Comple xity of Global Necessary Reasons to Explain Classiﬁcation. 2025. T echnical Report. 2501.06766 . Emilio Carrizosa, Jasone Ramírez-A yerbe, and Dolores Romero Morales. Mathematical Optimization Modelling for Group Counterfactual Explanations. Eur opean Journal of Oper ational Resear ch (EJOR) , 319(2):399–412, 2024. Brandon Carter , Jonas Mueller , Siddhartha Jain, and David Gif ford. What Made Y ou Do This? Understanding Black-Box Decisions with Sufﬁcient Input Subsets. In Pr oc. 22nd Int. Conf. on Artiﬁcial Intelligence and Statistics (AIST A TS) , pp. 567–576, 2019. Brandon Carter , Siddhartha Jain, Jonas W Mueller , and David Gif ford. Overinterpretation Rev eals Image Classiﬁcation Model P athologies. In Pr oc. 35th Conf. on Neur al Information Pr ocessing Systems (NeurIPS) , pp. 15395–15407, 2021. W enjing Chen and V ictoria Crawford. Bicriteria Approximation Algorithms for the Submodular Cov er Problem. In Pr oc. 37th Conf. on Neural Information Pr ocesing Systems (NeurIPS) , pp. 72705–72716, 2023. Arthur Conmy , Augustine Mav or-Park er , Aengus L ynch, Stefan Heimersheim, and Adrià Garriga- Alonso. T ow ards Automated Circuit Discovery for Mechanistic Interpretability. Pr oc. 37th Conf. on Neural Information Pr ocessing Systems (NeurIPS) , pp. 16318–16352, 2023. Yves Crama and Peter L Hammer . Boolean Functions: Theory , Algorithms, and Applications . Cambridge Univ ersity Press, 2011. Adnan Darwiche. Logic for Explainable AI. In Pr oc. 38th ACM/IEEE Symposium on Logic in Computer Science (LICS) , pp. 1–11, 2023. Adnan Darwiche and Auguste Hirth. On the Reasons Behind Decisions. In Pr oc. 24th Eur opean Conf. on Artiﬁcial Intelligence (ECAI) , pp. 712–720, 2020. Adnan Darwiche and Chunxi Ji. On the Computation of Necessary and Sufﬁcient Explanations. In Pr oc. 36th AAAI Conf. on Artiﬁcial Intelligence , pp. 5582–5591, 2022. Sanjoy Dasgupta, Na ve Frost, and Michal Moshko vitz. Framework for Evaluating Faithfulness of Local Explanations. In Pr oc. 39th Int. Conf. on Machine Learning (ICML) , pp. 4794–4815, 2022. Ian Davidson, Nicolás Kennedy , and SS Ravi. CXAD: Contrastive Explanations for Anomaly Detection: Algorithms, Complexity Results and Experiments. T ransactions on Machine Learning Resear ch (TMLR) , 2025. 13 Published as a conference paper at ICLR 2026 Alessandro De Palma, Greta Dolcetti, and Caterina Urban. Faster V eriﬁed Explanations for Neural Networks. 2025. T echnical Report. . Amit Dhurandhar, Pin-Y u Chen, Ronny Luss, Chun-Chen T u, P aishun Ting, Karthikeyan Shanmugam, and Payel Das. Explanations Based on the Missing: T ow ards Contrastiv e Explanations with Pertinent Negati ves. Pr oc. 32nd Conf. on Neur al Information Pr ocessing Systems (NeurIPS) , 2018. Hidde Fokkema, Rianne De Heide, and T im V an Erven. Attribution-Based Explanations that Provide Recourse Cannot be Robust. Journal of Machine Learning Resear ch (JMLR) , 24(360):1–37, 2023. Hidde Fokkema, Damien Garreau, and Tim van Erven. The Risks of Recourse in Binary Classiﬁcation. In Pr oc. 27th Int. Conf. on Artiﬁcial Intelligence and Statistics (AIST ATS) , pp. 550–558, 2024. T obias Geibinger, Reijo Jaakk ola, Antti Kuusisto, Xinghan Liu, and Miikka V ilander . Why This and Not That? A Logic-based Framew ork for Contrastive Explanations. In Pr oc. 19th Eur opean Conf. on Logics in Artiﬁcial Intelligence (JELIA) , pp. 45–60. Springer , 2025. Joshua Größer and Ostap Okhrin. Copulae: An Overvie w and Recent Developments. W iley Inter dis- ciplinary Revie ws: Computational Statistics , 14(3):e1557, 2022. Riccardo Guidotti. Counterfactual Explanations and Ho w to Find Them: Literature Revie w and Benchmarking. Data Mining and Knowledge Discovery (DMKD) , 38(5):2770–2824, 2024. Hyukjun Gweon, Matthias Schonlau, and Stefan H Steiner . The K Conditional Nearest Neighbor Algorithm for Classiﬁcation and Class Probability Estimation. P eerJ Computer Science , 5:e194, 2019. Itamar Hadad, Shahaf Bassan, and Guy Katz. Formal Mechanistic Interpretability: Automated Circuit Discov ery with Prov able Guarantees. In Proc. 14th Int. Conf. on Learning Repr esentations (ICLR) , 2026. Ouns El Harzli, Bernardo Cuenca Grau, and Ian Horrocks. Cardinality-Minimal Explanations for Monotonic Neural Networks. In Pr oc. 32nd Int. Joint Conf . on Artiﬁcial Intelligence (IJCAI) , pp. 3677–3685, 2023. W Keith Hastings. Monte Carlo Sampling Methods using Markov Chains and their Applications. Biometrika , 57:97–109, 1970. Rongjie Huang, Max W . Y . Lam, Jun W ang, Dan Su, Dong Y u, Y i Ren, and Zhou Zhao. FastDif f: A Fast Conditional Dif fusion Model for High-Quality Speech Synthesis. In Pr oc. 31st Int. Joint Conf. on Artiﬁcial Intelligence (IJCAI) , pp. 4157–4163, 2022. Xuanxiang Huang, Y acine Izza, Alex ey Ignatiev , and Joao Marques-Silva. On Efﬁciently Explaining Graph-Based Classiﬁers. In Proc. 18th Int. Conf . on Principles of Knowledge Repr esentation and Reasoning (KR) , 2021. Alex ey Ignatie v , Nina Narodytska, and Joao Marques-Silva. Abduction-Based Explanations for Machine Learning Models. In Pr oc. 33rd AAAI Confer ence on Artiﬁcial Intelligence , pp. 1511– 1519, 2019. Alex ey Ignatiev , Martin Cooper , Mohamed Siala, Emmanuel Hebrard, and Joao Marques-Silva. T owards Formal F airness in Machine Learning. In Pr oc. 26th Int. Conf. on Principles and Pr actice of Constraint Pr ogramming (CP) , pp. 846–867, 2020a. Alex ey Ignatie v , Nina Narodytska, Nicholas Asher , and Joao Marques-Silv a. From Contrasti ve to Abducti ve Explanations and Back Again. In Pr oc. Int. Conf. of the Italian Association for Artiﬁcial Intelligence (AIxAI) , pp. 335–355, 2020b. Alex ey Ignatie v , Y acine Izza, Peter J Stuckey , and Joao Marques-Silva. Using MaxSA T for Efﬁcient Explanations of Tree Ensembles. In Proc. 36th AAAI Conference on Artiﬁcial Intelligence , pp. 3776–3785, 2022. Rishabh K Iyer and Jeff A Bilmes. Submodular Optimization with Submodular Cover and Submodular Knapsack Constraints. Pr oc. 27th Conf. on Neural Information Processing Systems (NeurIPS) , 2013. 14 Published as a conference paper at ICLR 2026 Y acine Izza and Joao Marques-Silv a. On Explaining Random Forests with SA T. In Pr oc. 30th Int. Joint Conf . on Artiﬁcial Intelligence (IJCAI) , 2021. Y acine Izza, Xuanxiang Huang, Ale xey Ignatiev , Nina Narodytska, Martin Cooper , and Joao Marques- Silv a. On Computing Probabilistic Abductiv e Explanations. Int. J ournal of Approximate Reasoning (IJ AR) , 159:108939, 2023. Y acine Izza, Xuanxiang Huang, Antonio Morgado, Jordi Planes, Alex ey Ignatie v , and Joao Marques- Silva. Distance-Restricted Explanations: Theoretical Underpinnings & Efﬁcient implementation. In Proc. 21st Int. Conf. on Principles of Knowledge Repr esentation and Reasoning (KR) , pp. 475–486, 2024a. Y acine Izza, Kuldeep S Meel, and Joao Marques-Silva. Locally-Minimal Probabilistic Explanations. In Pr oc. 27th European Conf . on Artiﬁcial Intelligence (ECAI) , 2024b. Reijo Jaakkola, T omi Janhunen, Antti Kuusisto, Masood Feyzbakhsh Rankooh, and Miikka V ilander . Explainability via Short Formulas: the Case of Propositional Logic with Implementation. Journal of Artiﬁcial Intelligence Resear ch (J AIR) , 83, 2025. Alon Jacovi, Sw abha Swayamdipta, Shauli Ravfogel, Y anai Elazar , Y ejin Choi, and Y oav Goldber g. Contrastiv e Explanations for Model Interpretability. In Pr oc. Conf.Confer ence on Empirical Methods in Natural Languag e Pr ocessing (EMNLP) , pp. 1597–1611, 2021. Helen Jin, Anton Xue, W eiqiu Y ou, Surbhi Goel, and Eric W ong. Probabilistic Stability Guarantees for Feature Attrib utions. In Pr oc. 39th Conf. on Neural Information Pr ocessing Systems (NeurIPS) , 2025. Amir-Hossein Karimi, Bernhard Schölk opf, and Isabel V alera. Algorithmic recourse: from counter- factual explanations to interventions. In Pr oc. 4th ACM Conf. on F airness, Accountability , and T ranspar ency (F AccT) , pp. 353–362, 2021. Eoin M Kenny and Mark T Keane. On Generating Plausible Counterfactual and Semi-Factual Explanations for Deep Learning. In Pr oc. 35th AAAI Conf. on Artiﬁcial Intelligence , number 13, pp. 11575–11585, 2021. Alexander K ozachinskiy . Inapproximability of Suf ﬁcient Reasons for Decision Trees. 2023. T echnical Report. . I Elizabeth Kumar , Suresh V enkatasubramanian, Carlos Scheidegger , and Sorelle Friedler . Problems with Shapley-V alue-Based Explanations as Feature Importance Measures. In Proc. 37th Int. Conf . on Machine Learning (ICML) , pp. 5491–5500, 2020. Emanuele La Malfa, Agnieszka Zbrzezny , Rhiannon Michelmore, Nicola Paoletti, and Marta Kwiatko wska. On Guaranteed Optimal Robust Explanations for NLP Models. In Proc. Int. Joint Conf . on Artiﬁcial Intelligence (IJCAI) , pp. 2658–2665, 2021. Eduardo Sany Laber . The Computational Complexity of some Explainable Clustering Problems. Information Pr ocessing Letters , 184:106437, 2024. Shuai Li, Y ingjie Zhang, Hongtu Zhu, Christina W ang, Hai Shu, Ziqi Chen, Zhuoran Sun, and Y anfeng Y ang. K-Nearest-Neighbor Local Sampling Based Conditional Independence T esting. Pr oc. 37th Conf. on Neural Information Pr ocessing Systems (NeurIPS) , pp. 23321–23344, 2023. Gianluigi Lopardo, Frédéric Precioso, and Damien Garreau. A Sea of W ords: An In-Depth Analysis of Anchors for T ext Data. In Pr oc. 26th Int. Conf. on Artiﬁcial Intelligence and Statistics (AIST A TS) 2023 , 2023. Scott M Lundberg and Su-In Lee. A Uniﬁed Approach to Interpreting Model Predictions. Pr oc. 31st Conf. on Neur al Information Pr ocessing Systems (NeurIPS) , 2017. Joao Marques-Silv a and Alexe y Ignatiev . Delivering T rustworthy AI through formal XAI. In Pr oc. 36th AAAI Conf. on Artiﬁcial Intelligence , pp. 12342–12350, 2022. 15 Published as a conference paper at ICLR 2026 Joao Marques-Silva and Alexe y Ignatiev . No Silver Bullet: Interpretable ML Models Must be Explained. F r ontiers in Artiﬁcial Intelligence , 6:1128212, 2023. Joao Marques-Silva, Thomas Gerspacher , Martin Cooper , Alexey Ignatiev , and Nina Narodytska. Explaining Nai ve Bayes and Other Linear Classiﬁers with Polynomial T ime and Delay. In Proc. 33r d Int. Conf. on Advances in Neural Information Processing Systems (NeurIPS) , pp. 20590– 20600, 2020. Joao Marques-Silva, Thomas Gerspacher , Martin Cooper , Alexey Ignatiev , and Nina Narodytska. Explanations for Monotonic Classiﬁers. In Pr oc. Int. Conf. on Mac hine Learning (ICML) , pp. 7469–7479, 2021. Reda Marzouk and Colin De La Higuera. On the Tractability of SHAP Explanations under Mark ovian Distributions. In Proc. 41st Int. Conf . on Machine Learning (ICML) , pp. 34961–34986, 2024. Reda Marzouk, Shahaf Bassan, and Guy Katz. SHAP Meets T ensor Networks: Prov ably T ractable Explanations with Parallelism. In Pr oc. 39th Conf. on Neural Information Processing Systems (NeurIPS) , 2025a. Reda Marzouk, Shahaf Bassan, Guy Katz, and De la Higuera. On the Computational Tractability of the (Many) Shapley V alues. In Pr oc. 28th Int. Conf. on Artiﬁcial Intelligence and Statistics (AIST ATS) , 2025b. Kuldeep S Meel, Aditya A Shrotri, and Moshe Y V ardi. Not all FPRASs are Equal: Demystifying FPRASs for DNF-Counting. Constraints , 24(3):211–233, 2019. Ramaravind K Mothilal, Amit Sharma, and Chenhao T an. Explaining Machine Learning Classiﬁers Through Div erse Counterfactual Explanations. In Proc. 3r d Conf. on F airness, Accountability , and T ranspar ency (F AccT) , pp. 607–617, 2020. V idya Muthukumar, T ejaswini Pedapati, Nalini Ratha, Prasanna Sattigeri, Chai-W ah W u, Brian Kingsbury , Abhishek Kumar , Samuel Thomas, Aleksandra Mojsilovic, and Kush R V arshney . Understanding Unequal Gender Classiﬁcation Accuracy from Face Images. 2018. T echnical Report. . Sebastian Ordyniak, Giacomo P aesani, and Stef an Szeider . The P arameterized Complexity of Finding Concise Local Explanations. In Pr oc. 32nd Int. J oint Conf. on Artiﬁcial Intelligence (IJCAI) , 2023. Sebastian Ordyniak, Giacomo Paesani, Mateusz Rychlicki, and Stefan Szeider . Explaining Decisions in ML Models: A Parameterized Complexity Analysis. In Pr oc. 21st Int. Conf. on Principles of Knowledge Repr esentation and Reasoning (KR) , pp. 563–573, 2024. Marco T ulio Ribeiro, Sameer Singh, and Carlos Guestrin. Anchors: High-Precision Model-Agnostic Explanations. In Pr oc. 32nd AAAI Conf. on Artiﬁcial Intelligence , 2018. Majun Shi, Zishen Y ang, and W ei W ang. Minimum Non-Submodular Cover Problem with Applica- tions. Applied Mathematics and Computation , 410:126442, 2021. Andy Shih, Arthur Choi, and Adnan Darwiche. A Symbolic Approach to Explaining Bayesian Network Classiﬁers. In Proc. 27th Int. Joint Conf. on Artiﬁcial Intelligence (IJCAI) , pp. 5103– 5111, 2018. Bernardo Subercaseaux, Marcelo Arenas, and Kuldeep S Meel. Probabilistic Explanations for Linear Models. In Pr oc. 39th AAAI Conf. on Artiﬁcial Intelligence , pp. 20655–20662, 2025. Guy V an den Broeck, Anton L ykov , Maximilian Schleich, and Dan Suciu. On the Tractability of SHAP Explanations. Journal of Artiﬁcial Intelligence Resear ch (J AIR) , 74:851–886, 2022. Gal V ardi, Gilad Y ehudai, and Ohad Shamir . On the Optimal Memorization Power of ReLU Neural Networks. In Pr oc. 10th Int. Conf. on Learning Repr esentations, (ICLR) , 2022. V ijay V V azirani. Appr oximation Algorithms . Springer , 2001. 16 Published as a conference paper at ICLR 2026 Sahil V erma, V arich Boonsanong, Minh Hoang, Keeg an Hines, John Dickerson, and Chirag Shah. Counterfactual Explanations and Algorithmic Recourses for Machine Learning: A Revie w. ACM Computing Surve ys , 56(12):1–42, 2024. Stephan Wäldchen, Jan Macdonald, Sascha Hauch, and Gitta Kutyniok. The Computational Com- plexity of Understanding Binary Classiﬁer Decisions. Journal of Artiﬁcial Intellig ence Researc h (J AIR) , 70:351–387, 2021. E. W ang, P . Khosravi, and G. V an den Broeck. Probabilistic Sufﬁcient Explanations. In Pr oc. 30th Int. Joint Conf . on Artiﬁcial Intelligence (IJCAI) , 2021a. S. W ang, H. Zhang, K. Xu, X. Lin, S. Jana, C. Hsieh, and Z. K olter . Beta-Crown: Efﬁcient Bound Propagation with Per-Neuron Split Constraints for Neural Netw ork Robustness V eriﬁcation. In Pr oc. 35th Int. Conf. on Advances in Neural Information Pr ocessing Systems (NeurIPS) , pp. 29909–29921, 2021b. Greta W arren, Eoin Delaney , Christophe Guéret, and Mark T Keane. Explaining Multiple Instances Counterfactually: User T ests of Group-Counterfactuals for XAI. In Pr oc. 32nd Int. Conf. on Case-Based Reasoning (ICCBR) , pp. 206–222, 2024. Ingo W egener . The Complexity of Symmetric Boolean Functions. Computation Theory and Logic , pp. 433–442, 2005. Laurence A W olsey . An Analysis of the Greedy Algorithm for the Submodular Set Co vering Problem. Combinatorica , 2(4):385–393, 1982. Haoze W u, Omri Isac, Aleksandar Zelji ´ c, T eruhiro T agomori, Matthew Daggitt, W en Kokk e, Idan Refaeli, Guy Amir , Kyle Julian, Shahaf Bassan, et al. Marabou 2.0: A V ersatile Formal Analyzer of Neural Networks. In Pr oc. 36th Int. Conf. on Computer Aided V eriﬁcation (CA V) , pp. 249–264, 2024a. Min W u, Haoze W u, and Clark Barrett. V erix: T owards V eriﬁed Explainability of Deep Neural Networks. In Pr oc. 36th Int. Conf. on Advances in Neural Information Pr ocessing Systems (NeurIPS) , 2024b. Lei Xu, Maria Skoularidou, Alfredo Cuesta-Infante, and Kalyan V eeramachaneni. Modeling T abular Data Using Conditional GAN. Pr oc. 33rd Cond. on Neural Information Pr ocessing Systems (NeurIPS) , 2019. Anton Xue, Rajee v Alur , and Eric W ong. Stability Guarantees for Feature Attributions with Multi- plicativ e Smoothing. Proc. 37th Conf . on Neural Information Pr ocessing Systems (NeurIPS) , pp. 62388–62413, 2023. Kayo Y in and Graham Neubig. Interpreting Language Models with Contrastive Explanations. In Pr oc. Conf. on Empirical Methods in Natural Languag e Pr ocessing (EMNLP) , pp. 184–198, 2022. Jinqiang Y u, Alexe y Ignatiev , Peter Stuckey , Nina Narodytska, and Joao Marques-Silva. Eliminating the Impossible, Whatev er Remains Must Be T rue: On Extracting and Applying Background Knowledge in the Context of Formal Explanations. In Pr oc. 37th AAAI Conf. on Artiﬁcial Intelligence , pp. 4123–4131, 2023. 17 Published as a conference paper at ICLR 2026 Appendix The appendix contains formalizations and proofs that were mentioned throughout the paper: Appendix A contains e xtended related work on Formal XAI. Appendix B contains the formalizations of the models and distrib utions used in this work. Appendix C contains the proof of Proposition 1. Appendix D contains the proof of Proposition 2. Appendix E contains the proof of Proposition 3. Appendix F contains the proof of Proposition 4. Appendix G contains the proof of Proposition 5. Appendix H contains the proof of Theorem 1. Appendix I contains the proof of Proposition 6. Appendix J contains the proof of Theorem 2. Appendix K contains the proof of Theorem 3. Appendix L contains the proof of Theorem 4. Appendix O contains an LLM usage disclosure. A E X T E N D E D R E L A T E D W O R K O N F O R M A L X A I Our work falls within the line of research on formal explainable AI (formal XAI) (Marques-Silva & Ignatie v, 2022), which studies explanations equipped with prov able guarantees (Y u et al., 2023; Darwiche & Ji, 2022; Darwiche, 2023; Shih et al., 2018; Azzolin et al., 2025; Audemard et al., 2021; Calautti et al., 2025). A central theme in this area concerns the computational comple xity of generating such guaranteed explanations (Barceló et al., 2020; ? ; Marzouk et al., 2025a;b; Marzouk & De La Higuera, 2024; Bassan et al., 2025e; Amir et al., 2024; Blanc et al., 2021; 2022; Jaakk ola et al., 2025; Amgoud & Cooper, 2026; Geibinger et al., 2025). Since obtaining formal explanations is often computationally intractable for expressi ve models such as neural networks (Adolﬁ et al., 2024; Barceló et al., 2020) and tree ensembles (Ordyniak et al., 2024; Bassan et al., 2025a), much of the literature has focused on more tractable model classes (Marques-Silva & Ignatie v, 2023), including decision trees (Bounia, 2025; Arenas et al., 2022; Bounia, 2024; Bounia & K oriche, 2023), monotonic models (Marques-Silv a et al., 2021; Harzli et al., 2023), linear models (Marques-Silva et al., 2020; Subercaseaux et al., 2025), and additiv e models (Bassan et al., 2026). Beyond comple xity-theoretic in vestigations, substantial effort has also been de voted to computing formal explanations in practice using automated reasoning tools, such as MaxSA T and MILP solvers for tree ensembles (Audemard et al., 2023; 2022a; Ignatiev et al., 2022; Boumazouza et al., 2021), and neural network veriﬁcation frame works (W ang et al., 2021b; W u et al., 2024a) to derive certiﬁed explanations for neural netw orks (W u et al., 2024b; Bassan & Katz, 2023; Bassan et al., 2023; Izza et al., 2024a; Hadad et al., 2026; Ignatie v et al., 2019). Additional approaches seek to alle viate the inherent computational burden through abstractions (De Palma et al., 2025; Bassan et al., 2025b; Boumazouza et al., 2026), relaxations such as smoothing (Xue et al., 2023; Anani et al., 2025; Jin et al., 2025), or training-time interventions (Alv arez Melis & Jaakkola, 2018; Bassan et al., 2025c;d). Finally , within formal XAI, a related line of w ork studies methods that le verage combinatorial and submodular optimization techniques to improve the computation of local probabilistic sufﬁcient reasons (Bounia, 2025; Bounia & K oriche, 2023), particularly for decision tree classiﬁers. As noted in that literature, as well as in this w ork, the value-function objecti ve underlying these local explanation v ariants is not submodular in practice. Ne vertheless, greedy algorithms inspired by submodular optimization often serve as effecti ve heuristics and can yield high-quality explanations empirically . In contrast, our work establishes a rigorous theoretical connection between combinatorial and submodular optimization objectiv es and the global variants of sufﬁcient and contrasti ve reasons. This connection provides formal guarantees and opens the door to a broader range of principled algorithmic implementations. 18 Published as a conference paper at ICLR 2026 B M O D E L A N D D I S T R I B U T I O N F O R M A L I Z A T I O N S In this section, we formalize the models and distributions referenced throughout the paper . Speciﬁcally , Subsection B.1 deﬁnes the model families, while Subsection B.2 formalizes the distrib utions. B . 1 M O D E L F O R M A L I Z AT I O N S In this subsection, we formalize the three base-model types that were analyzed throughout the paper: (i) (axis-aligned) decision trees, (ii) linear classiﬁers, and (iii) neural networks with ReLU acti vations. Decision T rees. W e deﬁne a decision tree (DT) as a directed acyclic graph that represents a function f : F → [ c ] , where c ∈ N denotes the number of classes. The graph encodes the function as follows: (i) Each internal node v is assigned a distinct binary input feature from the set { 1 , . . . , n } ; (ii) Every internal node v has at most k outgoing edges, each corresponding to a v alue in [ k ] assigned to v ; (iii) Along an y path α in the decision tree, each variable appears at most once; (i v) Each leaf node is labeled with a class from [ c ] . Thus, assigning values to the inputs x ∈ F uniquely determines a path α from the root to a leaf in the DT , where the function output f ( x ) corresponds to a class label i ∈ [ c ] . The size of the DT , denoted | f | , is deﬁned as the total number of edges in the graph. T o allo w for ﬂexible modeling, the ordering of input v ariables { 1 , . . . , n } may dif fer across distinct paths α and α ′ , ensuring that no variable is repeated along a single path. Neural Networks. W e present our hardness proofs for neural networks with ReLU activ ations, though our tractability r esults (i.e., polynomial-time algorithms) apply to any architecture that allows for polynomial-time inference — a standard assumption. Thus, all separation results between tractable and intractable cases we prov e carry over to any neural ar chitectur e at least as expressi ve as a standard feed-forward ReLU network, encompassing man y widely used models such as ResNets, CNNs, T ransformers, Diff usion models, and more. Moreov er , note that any ReLU network can be represented as a fully connected network by assigning zero weights and biases to missing connections. Follo wing standard conv entions (Barceló et al., 2020; Bassan et al., 2024; Adolﬁ et al., 2025), we thus assume the network is fully connected. Speciﬁcally , our analysis applies to multi-layer perceptrons (MLPs). Formally , an MLP f consists of t − 1 hidden layers ( g ( j ) for j = 1 , . . . , t − 1 ) and one output layer g ( t ) , where each layer is deﬁned recursiv ely as: g ( j ) := σ ( j ) ( g ( j − 1) W ( j ) + b ( j ) ) (9) where W ( j ) denotes the weight matrix, b ( j ) the bias vector , and σ ( j ) the activ ation function of the j -th layer . Accordingly , the model comprises t weight matrices ( W (1) , . . . , W ( t ) ), t bias vectors ( b (1) , . . . , b ( t ) ), and t activ ation functions ( σ (1) , . . . , σ ( t ) ). The input layer is deﬁned as g (0) = x ∈ { 0 , 1 } n , representing the binary input vector . The dimensions of the network are gov erned by a sequence of positiv e integers d 0 , . . . , d t , with weight matrices and bias vectors gi ven by W ( j ) ∈ Q d j − 1 × d j and b ( j ) ∈ Q d j , respectiv ely . These parameters are learned during training. Since the model functions as a binary classiﬁer ov er n features, we set d 0 = n and d t = 1 . The hidden layers use the ReLU activ ation function, deﬁned by ReLU ( x ) = max(0 , x ) . Although a sigmoid acti vation is typically used during training, for interpretability purposes, we assume the output layer applies a threshold-based step function, deﬁned as step ( z ) = 1 if z ≥ 0 and step ( z ) = 0 otherwise. T ree Ensembles. Many popular ensemble methods exist, b ut since our goal is to provide post-hoc explanations, we focus on the infer ence phase rather than the training process. Our analysis centers on ensemble families that rely on either majority voting (e.g., Random Forests) or weighted voting (e.g., XGBoost) during inference. Howe ver , as with our treatment of neural networks, our tractability results — namely , polynomial-time algorithms — extend to any possible ensemble conﬁguration with polynomial-time inference, encompassing an ev en broader range of ensemble conﬁgurations. Majority V oting Inference. In majority voting inference, the ﬁnal prediction f ( x ) is assigned to the class j ∈ [ c ] that recei ves the majority of v otes among the indi vidual predictions f i ( x ) from all i ∈ [ k ] (i.e., from each tree in the ensemble). 19 Published as a conference paper at ICLR 2026 f ( x ) := ma jority( { f i ( x ) | i ∈ [ k ] } ) (10) where ma jority( S ) denotes the most frequent label in the multiset S . If there is a tie, it is resolv ed by a ﬁxed tie-breaking rule (e.g., le xicographic order or predeﬁned priority). W eighted V oting Inference. In weighted voting inference, each model in the ensemble is assigned a weight ϕ i ∈ Q representing its relati ve importance. The predicted class is the one with the highest total weight across all models. Formally , for any x ∈ F , we deﬁne f as: f ( x ) := arg max j ∈ [ c ] k X i =1 ϕ i · I [ f i ( x ) = j ] (11) where I denotes the identity function. B . 2 D I S T R I B U T I O N F O R M A L I Z A T I O N S This subsection formalizes the distribution deﬁnitions discussed in the main paper . Empirical Distributions. The distribution D ov er the input features will be deﬁned based on a dataset D comprising various inputs z 1 , z 2 , . . . , z | D | . Here, the distribution of an y giv en input x ∈ F is deﬁned by the frequency of occurrence of x within D , speciﬁcally by: Pr ( x ) := 1 | D | | D | X i =1 I ( z i = x ) (12) Independent Distributions. Formally , given a probability value p ( x i ) ∈ [0 , 1] deﬁned for each individual input feature, we say that the distribution D is independent if the joint probability over inputs is giv en by Pr ( x ) := Q i ∈ [ n ] p ( x i ) . W e observe that when p ( x i ) = p ( x j ) holds for all i, j ∈ [ n ] , the distrib ution reduces to the uniform distribution , a special case of independent distrib utions. General Distrib utions. W e note that many of the proofs in this work — particularly those concerning fundamental properties of v alue functions — apply broadly o ver general distrib ution assumptions. While empirical distributions (using the training dataset as a proxy) are common in XAI, alterna- tiv e framew orks for approximating distributions include k-NN resampling from nearby points (Li et al., 2023; Almanjahie et al., 2018; Gweon et al., 2019), copulas for modeling tabular dependen- cies (Größer & Okhrin, 2022), or more advanced conditional generati ve models such as CTGAN (Xu et al., 2019) and conditional diffusion models (Huang et al., 2022). Such choices are common in the counterfactual e xplanation and algorithmic recourse literature (Karimi et al., 2021; F okkema et al., 2024; 2023; V erma et al., 2024), which are related to contrastiv e explanations. Furthermore, when the v alue function is not computed directly from a structural property of the model (e.g., leaf enumeration in decision trees) or from empirical distributions, one may instead approximate it using methods such as Monte Carlo sampling (Hastings, 1970), follo wing approaches similar to (Subercaseaux et al., 2025; Lopardo et al., 2023). B . 3 S U B S E T V S . C A R D I NA L M I N I M A L I T Y In this subsection, we provide a more detailed discussion of the distinction between cardinal and subset minimality . Cardinal minimality offers a substantially stronger guarantee, as it corresponds to a globally minimal explanation size, whereas subset minimality only ensures a local form of minimality . T o see why cardinal minimality is stronger , note that if S ⊆ [ n ] is a cardinally minimal explanation, then no smaller set S ′ with | S ′ | < | S | can qualify as an explanation. Hence, no strict subset S ′ ⊊ S is an explanation, which means S is also subset minimal. Howe ver , the rev erse does not hold. Consider the function: 20 Published as a conference paper at ICLR 2026 f := x 1 ∨ ( x 2 ∧ x 3 ) (13) and the assignment x := (1 , 1 , 1) , which giv es f (1 , 1 , 1) = 1 . Fixing feature x 1 = 1 yields a cardinally minimal (and subset minimal) sufﬁcient reason, since the prediction remains 1 regardless of x 2 , x 3 . But ﬁxing both x 2 = 1 and x 3 = 1 also giv es a subset minimal explanation—yet not a cardinally minimal one. Thus, while ev ery cardinally minimal explanation is subset minimal, not ev ery subset minimal explanation is cardinally minimal. C P RO O F O F P RO P O S I T I O N 1 Proposition 1. While the non-pr obabilistic case ( δ = 1 ) admits a unique subset-minimal global sufﬁcient r eason (Bassan et al., 2024), in the gener al pr obabilistic setting (for arbitrary δ ), ther e e xist functions f that have Θ( 2 n √ n ) subset-minimal global sufﬁcient r easons. Pr oof. It is kno wn that certain Boolean functions admit an exponential number of subset-minimal local (non-probabilistic) suf ﬁcient reasons for some input x Bassan et al. (2024). In particular, this phenomenon was sho wn to occur in functions of the follo wing form. W e demonstrate that the same function admits an e xponential number of global and pr obabilistic sufﬁcient reasons. Notably , the mentioned function is a special case of a broader class of thr eshold Boolean functions described in W egener (2005). Speciﬁcally , for n = 2 k + 1 with k ∈ N , this function is deﬁned as: f ( x ) :=  1 if P n i =1 x i ≥ k + 1 0 otherwise (14) where x is drawn from a uniform distribution D . Notably , the function f is symmetric , in the sense of symmetric threshold Boolean functions W egener (2005): its output depends solely on the number of 1’ s (or , equi valently , 0’ s) in the input and is in variant under any permutation of input bits. Furthermore, the two e xtreme inputs — the all-zeros vector (0 , . . . , 0) and the all-ones vector (1 , . . . , 1) — each admit an exponential number of subset-minimal local (non-probabilistic) suf ﬁcient reasons. Since the function f is symmetric, the condition v g suff ( S 1 ) = v g suff ( S 2 ) holds for any two subsets S 1 , S 2 ⊆ [ n ] . Moreov er , it can be veriﬁed that for any subset S ⊆ [ n ] of size k + 1 , we hav e: ( ∀ i ∈ S ) . v g suff ( S ) > v g suff ( S \ { i } ) (15) W e also deliberately set δ to satisfy: v g suff ( S \ { i } ) < δ ≤ v g suff ( S ) , (16) and also set k := ⌊ n 2 ⌋ . Each of these subsets is subset-minimal because any subset of size at most ⌊ n 2 ⌋ − 1 f ails to be a sufﬁcient reason for ⟨ f , x ⟩ (with respect to the δ threshold). Therefore, we can directly apply the same analysis as in (Bassan et al., 2024). In particular , there are exactly  n ⌊ n 2 ⌋  subset-minimal local sufﬁcient reasons for ⟨ f , x ⟩ . Using Stirling’ s approximation, we obtain: lim n →∞ 2 √ 2 π e 2 · 2 n √ n ≤  n ⌊ n 2 ⌋  ≤ e π · 2 n √ n (17) This yields the corresponding bound on the number of subset-minimal global δ -sufﬁcient reasons. D P RO O F O F P RO P O S I T I O N 2 Proposition 2. While the local pr obabilistic setting (for any δ ) lacks monotonicity — i.e ., the value functions v ℓ con and v ℓ suff ar e non-monotonic (Arenas et al., 2022; Izza et al., 2023; Subercaseaux et al., 2025; Izza et al., 2024b) — in the global pr obabilistic setting (also for any δ ), both value functions v g con and v g suff ar e monotonic non-decr easing. 21 Published as a conference paper at ICLR 2026 Pr oof. In this section, we pro ve the monotonicity property for both types of global e xplanations — sufﬁcient and contrasti ve. W e be gin by establishing the property for global suf ﬁciency . T o b uild intuition, we ﬁrst focus on the simpler case of Boolean functions, then generalize the result to functions with discrete multi-valued input domains and multiple output classes. W e further e xtend the result to continuous input domains and sho w that the monotonicity property holds for any well-deﬁned classiﬁcation function. Finally , we demonstrate how the same monotonicity guarantees apply to global contrastiv e explanations. Importantly , none of our proofs rely on the assumption of feature independence — the monotonicity property holds strongly for any underlying distribution. Lastly , we present a simple example sho wing that the monotonicity property does not hold for the local probabilistic sufﬁcient or contrasti ve value functions. Lemma 1. The global sufﬁcient value function v g suff is monotonic non-decr easing for Boolean functions, under any data distribution. Pr oof. Given an arbitrary set S ∈ [ n ] , and a feature i ∈ S , we have that: Pr ( x ) = Pr ( x S , x ¯ S ) = Pr ( x ¯ S | x S ) · Pr ( x S ) (18) Consequently , this implies that: Pr ( x ) = Pr ( x S ∪{ i } | x S ∪{ i } ) · Pr ( x S ∪{ i } ) = Pr ( x S ∪{ i } | x S ∪{ i } ) · Pr ( x S , x i ) = Pr ( x S ∪{ i } | x S ∪{ i } ) · Pr ( x i | x S ) · Pr ( x S ) (19) Moreov er , for any two points x , x ′ ∼ D such that x S = x ′ S and f ( x ) = 1 − f ( x ′ ) , it holds that: Pr z ∼D ( f ( z ) = f ( x ) | z S = x S ) = 1 − Pr z ∼D ( f ( z ) = f ( x ′ ) | z S = x ′ S ) (20) T o simplify notation, we occasionally omit the distribution notation ∼ D in the local probability expression Pr z ∼D ( f ( z ) = f ( x ) | z S = x S ) , and use f + and f − to denote the e vents f ( z ) = 1 and f ( z ) = 0 , respectively . W e begin with a technical simpliﬁcation of the probability term to facilitate the proof, and then proceed to establish monotonicity . Simplifying the probability term. Let D S denote the distribution restricted to the set S . Giv en a point x ∼ D and a set S , consider all points z ∼ D such that z S = x S and f ( z ) = b , where b ∈ { 0 , 1 } — they all share the same local probability Pr z ∼D ( f ( z ) = b | z S = x S ) . For every x ∼ D that shares the same x S and the same output b , we marginalize over x ¯ S . This yields Pr ( f ( x ) = b, x S ) = Pr ( f ( x ) = b | x S ) · Pr ( x S ) , from which we can infer: Pr z ∼D ( f ( z ) = b | z S = x S ) = Pr ( f ( x ) = b | x S ) . (21) Thus, for E x ∼D [ Pr z ∼D ( f ( z ) = f ( x ) | z S = x S )] , we get that: E x ∼D [ Pr z ∼D ( f ( z ) = f ( x ) | z S = x S )] = X x S ∼D S ,f + Pr ( f + , x S ) · Pr z ∼D ( f + | z S = x S ) + X x S ∼D S ,f − Pr ( f − , x S ) · Pr z ∼D ( f − | z S = x S ) = X x S ∼D S Pr ( x S ) ·  Pr z ∼D ( f + | z S = x S )  2 +  Pr z ∼D ( f − | z S = x S )  2  = X x S ∼D S Pr ( x S ) ·  Pr z ∼D ( f + | z S = x S )  2 +  1 − Pr z ∼D ( f + | z S = x S )  2  , (22) From which it follows that: 22 Published as a conference paper at ICLR 2026 E x ∼D [ Pr z ∼D ( f ( z ) = f ( x ) | z S ∪{ t } = x S ∪{ t } )] = X x S ∪{ t } ∼D S ∪{ t } Pr ( x S ∪{ t } ) ·  Pr z ( f + | z S ∪{ t } = x S ∪{ t } )  2 +  1 − Pr z ( f + | z S ∪{ t } = x S ∪{ t } )  2  = X x S ∼D S x t =1 Pr ( x t = 1 , x S ) ·  Pr z ( f + | z S = x S , z t = 1)  2 +  1 − Pr z ( f + | z S = x S , z t = 1)  2  + X x S ∼D S x t =0 Pr ( x t = 0 , x S ) ·  Pr z ( f + | z S = x S , z t = 0)  2 +  1 − Pr z ( f + | z S = x S , z t = 0)  2  (23) Pro ving monotonicity . W e now proceed to prov e the monotonicity claim, building on the pre vious simpliﬁcation. W e begin by introducing a few additional notations. Let g + 1 := Pr z ∼D ( f ( z ) = 1 | z S = x S , z t = 1) and g + 0 := Pr z ∼D ( f ( z ) = 1 | z S = x S , z t = 0) . Similarly , deﬁne P 1 := Pr ( z t = 1 | x S ) and P 0 := Pr ( z t = 0 | x S ) . Then, the follo wing expectations E x ∼D [ Pr z ∼D ( f ( z ) = f ( x ) | z S = x S )] and E x ∼D [ Pr z ∼D ( f ( z ) = f ( x ) | z S ∪{ t } = x S ∪{ t } )] can be simpliﬁed accordingly 1 : E x ∼D [ Pr z ∼D ( f ( z ) = f ( x ) | z S = x S )] = X x S ∼D S Pr ( x S ) · [( P 1 · g + 1 + P 0 · g + 0 ) 2 + (1 − ( P 1 · g + 1 + P 0 · g + 0 )) 2 ] = X x S ∼D S Pr ( x S ) · [2( P 1 · g + 1 + P 0 · g + 0 ) 2 − 2( P 1 · g + 1 + P 0 · g + 0 ) + 1] . (24) As a result, we obtain the following: E x ∼D [ Pr z ∼D ( f ( z ) = f ( x ) | z S ∪{ t } = x S ∪{ t } )] = X x S ∼D S x t =1 Pr ( x S ) · P 1 · (( g + 1 ) 2 + (1 − g + 1 ) 2 ) + X x S ∼D S x t =0 Pr ( x S ) · P 0 · (( g + 0 ) 2 + (1 − g + 0 ) 2 ) = X x S ∼D S x t =1 Pr ( x S ) · P 1 · (2( g + 1 ) 2 − 2 g + 1 + 1) + X x S ∼D S x t =0 Pr ( x S ) · P 0 · (2( g + 0 ) 2 − 2 g + 0 + 1) = X x S ∼D S Pr ( x S ) · [ P 1 · (2( g + 1 ) 2 − 2 g + 1 + 1) + P 0 · (2( g + 0 ) 2 − 2 g + 0 + 1)] = X x S ∼D S Pr ( x S ) · [2 P 1 · (( g + 1 ) 2 − g + 1 ) + 2 P 0 · (( g + 0 ) 2 − g + 0 ) + 1] = X x S ∼D S Pr ( x S ) · [2( P 1 · ( g + 1 ) 2 + P 0 · ( g + 0 ) 2 ) − 2( P 1 · g + 1 + P 0 · g + 0 ) + 1] . (25) Let ∆ t ( S, f ) denote E x ∼D [ Pr z ∼D ( f ( z ) = f ( x ) | z S ∪{ t } = x S ∪{ t } )] − E x ∼D [ Pr z ∼D ( f ( z ) = f ( x ) | z S = x S )] . W ithout loss of generality , we assume that both P 1 and P 0 are positiv e. Otherwise, it can be veriﬁed that ∆ t ( S, f ) = 0 . Then, we get: 1 Note that the notations P 1 , P 0 , g + 1 , and g + 0 —as well as all others used throughout the paper—are not ﬁxed constants, but rather depend on the speciﬁc input x S . Formally , they should be written as P 1 ( x S ) , P 0 ( x S ) , and so on. Howe ver , for readability , we omit the explicit dependence on x S . 23 Published as a conference paper at ICLR 2026 ∆ t ( S, f ) = X x S ∼D S Pr ( x S ) · [2( P 1 · ( g + 1 ) 2 + P 0 · ( g + 0 ) 2 ) − 2( P 1 · g + 1 + P 0 · g + 0 ) 2 ] = X x S ∼D S 2 · Pr ( x S ) · [( P 1 − P 2 1 )( g + 1 ) 2 + ( P 0 − P 2 0 )( g + 0 ) 2 − 2 · P 1 · P 0 · g + 1 · g + 0 ] = X x S ∼D S 2 · Pr ( x S ) · P 1 · P 0 · ( g + 1 − g + 0 ) 2 ≥ 0 . (26) Note that P 1 − P 2 1 = P 1 (1 − P 1 ) = P 1 P 0 and similarly: P 0 − P 2 0 = P 0 (1 − P 0 ) = P 0 P 1 ). Using these identities along with the previous result, we conclude that E x ∼D [ Pr z ∼D ( f ( z ) = f ( x ) | z S = x S )] is monotone, thereby concluding our proof. Lemma 2. The global sufﬁcient value function v g suff is monotonic non-decr easing for functions with discr ete multi-valued input domains and multiple output classes, under any data distribution. Pr oof. Extending the proof from the simpliﬁed Boolean case, we now generalize the result to functions whose input domains and output ranges each consist of a ﬁnite set of values. Let K represent the set of output classes. W e can generalize equation 22 as follows: E x ∼D [ Pr z ∼D ( f ( z ) = f ( x ) | z S = x S )] = X x S ∼D S Pr ( x S ) ·  X j ∈K  Pr z ∼D ( f ( z ) = j | z S = x S )  2  (27) Giv en the condition z S = x S , suppose feature t can take r possible values { v 1 , . . . , v r } . Then, equation 23 can be generalized as follows: E x ∼D [ Pr z ∼D ( f ( z ) = f ( x ) | z S ∪{ t } = x S ∪{ t } )] = r X k =1 h X x S ∼D S x t = v k Pr ( x S ) · Pr ( z t = v k | x S ) ·  X j ∈K  Pr z ∼D ( f ( z ) = j | z S = x S , z t = v k )  2  i . (28) Let P k denote Pr ( z t = v k | x S ) , Let g j k denote Pr z ∼D ( f ( z ) = j | z S = x S , z t = v k ) , that is, z t takes the v alue v k and the output class is j . Then, we ha ve E x ∼D [ Pr z ∼D ( f ( z ) = f ( x ) | z S = x S )] = X x S ∼D S Pr ( x S ) · h X j ∈K  r X k =1 P k · g j k  2 i (29) and E x ∼D [ Pr z ∼D ( f ( z ) = f ( x ) | z S ∪{ t } = x S ∪{ t } )] = X x S ∼D S Pr ( x S ) · h r X k =1 P k ·  X j ∈K ( g j k ) 2  i Combining these two implications, we can conclude that: 24 Published as a conference paper at ICLR 2026 ∆ t ( S, f ) = X x S ∼D S Pr ( x S ) · h r X k =1 P k ·  X j ∈K ( g j k ) 2  − X j ∈K  r X k =1 P k · g j k  2 i = X x S ∼D S Pr ( x S ) · h X j ∈K  r X k =1 P k · ( g j k ) 2  − X j ∈K  r X k =1 ( P k · g j k ) 2 + X k ∆ t (1 , f ) , indicating that — unlike in the monotonicity setting where it was not required — feature independence is a necessary condition for supermodularity . 32 Published as a conference paper at ICLR 2026 G P RO O F O F P RO P O S I T I O N 5 Proposition 5. Computing Algorithm 1 with the local value functions v ℓ con and v ℓ suff does not always con ver ge to a subset minimal δ -sufﬁcient/contr astive r eason. However , computing it with the global value functions v g con or v g suff necessarily pr oduces subset minimal δ -sufﬁcient/contr astive r easons. Pr oof. W e note that this proposition follows directly from our proof that both v g suff and v g con are monotone non-decreasing (as established in Proposition 2), along with other relev ant conclusions drawn in prior work. In the remainder of this section, we elaborate on how this monotonicity property leads to the stated corollary . The con vergence of Algorithm 1 to a subset-minimal δ global suf ﬁcient or contrastiv e reason with respect to the value functions v g con and v g suff follows directly from the monotonicity property , for which prior work (Ignatiev et al., 2019; W u et al., 2024b; Arenas et al., 2022) showed that this type of greedy algorithm yields a subset-minimal e xplanation when the underlying v alue function is monotone non-decreasing. The algorithm halts when, for all i ∈ S , the value v ( S \ { i } ) drops below δ , ensuring that although v ( S ) ≥ δ (a maintained in variant), remo ving any element causes the condition to fail. Due to monotonicity , this implies that no proper subset of S satisﬁes the condition, guaranteeing that S is subset-minimal. W e now turn to explain why Algorithm 1 does not con verge to a subset-minimal solution in the local setting. W e note that it is well known that the local probabilistic suf ﬁcient value function (and like wise the local probabilistic contrastive v alue function) is not monotone Arenas et al. (2022); Izza et al. (2023); Subercaseaux et al. (2025); Izza et al. (2024b). As a simple illustration, ﬁx the uniform distribution on the tw o binary variables { x 1 , x 2 } and deﬁne a Boolean function as follows: f ( x 1 , x 2 ) := x 1 ∨ x 2 . (59) Consider the input x = (0 , 1) which is classiﬁed as 1 , then: v ℓ suff ( ∅ ) = 3 4 , v ℓ suff ( { 1 } ) = 1 2 , v ℓ suff ( { 2 } ) = 1 , v ℓ suff ( { 1 , 2 } ) = 1 . (60) Because v ℓ suff ( ∅ ) > v ℓ suff ( { 1 } ) < v ℓ suff ( { 1 , 2 } ) , the function v ℓ suff is not monotone. Like wise, it can be computed that: v ℓ con ( ∅ ) = 1 , v ℓ con ( { 1 } ) = 1 , v ℓ con ( { 2 } ) = 1 2 , v ℓ con ( { 1 , 2 } ) = 3 4 . (61) (For v ℓ con , the S denotes the set of features that are allowed to v ary .) Because v ℓ con ( ∅ ) > v ℓ con ( { 2 } ) < v ℓ con ( { 1 , 2 } ) , the function v ℓ con is also non-monotone. H P RO O F O F T H E O R E M 1 Theorem 1. If f is a decision tr ee and the pr obability term v g suff can be computed in polynomial time given the distrib ution D (which holds for independent distributions, among other s), then obtaining a subset-minimal global δ -sufﬁcient r eason can be obtained in polynomial time . However , unless PTIME = NP , no polynomial-time algorithm exists for computing a local δ -sufﬁcient r eason for decision tr ees even under independent distrib utions. Pr oof. W e begin by noting that for decision trees, assuming that D represents independent distribu- tions, the computation of the value function: v g suff ( S ) = E x ∼D [ Pr z ∼D ( f ( x ) = f ( z ) | x S = z S )] (62) Can be carried out in polynomial time using the follo wing procedure. Thanks to the tractability of decision trees, we can iterate ov er all pairs of leaf nodes, each corresponding to partial assignments 33 Published as a conference paper at ICLR 2026 x S and z S ′ . For each such pair , if both leaf nodes yield the same prediction under f , we compute the corresponding term in the expected v alue E x ∼D [ Pr z ∼D ( f ( x ) = f ( z ) | x S = z S )] by multiplying the respective feature-wise probabilities ov er the shared features of the two vectors. Under the feature independence assumption, these probabilities decompose, allowing the full expectation to be computed by summing the contributions of all such matching leaf pairs. Since all probabilities in volv ed are provided as part of the input and each step in volv es only polynomial-time operations, the entire procedure runs in polynomial time. This establishes that every step in Algorithm 1 is ef ﬁcient, and thus the algorithm as a whole runs in polynomial time. Finally , combining this with Lemma 2, which proves that v g suff is monotone non-decreasing, we conclude that Algorithm 1 con ver ges to a subset-minimal explanation. This is because the algorithm halts when, for all i ∈ S , the v alue v ( S \ { i } ) falls belo w δ , while v ( S ) ≥ δ is preserved throughout. By monotonicity , this guarantees that no strict subset of S satisﬁes the condition, ensuring S is indeed subset-minimal. For the remaining part of the claim, the result follows from (Arenas et al., 2022), which showed that, assuming P  = NP , there is no polynomial-time algorithm for computing a subset-minimal δ -sufﬁcient reason for decision trees. While this was proven speciﬁcally under the uniform distribution, the hardness clearly extends to independent distributions, which include this case. Combined, these results establish the complexity separation between the local and global variants, thus concluding our proof. I P RO O F O F P RO P O S I T I O N 6 Proposition 6. F or any model f , and empirical distribution D — computing a subset-minimal global δ -sufﬁcient or δ -contrastive r eason for f can be done in polynomial time. Pr oof. The computation of the global sufﬁcient probability function: v g suff ( S ) = E x ∼D [ Pr z ∼D ( f ( x ) = f ( z ) | x S = z S )] (63) or the computation of the global contrastiv e probability function v g con ( S ) = E x ∼D [ Pr z ∼D ( f ( x )  = f ( z ) | x ¯ S = z ¯ S )] (64) Can be performed in polynomial time when D is selected from the class of empirical distrib utions. This is achie ved by iterating ov er pairs of instances x , z within the dataset and running an inference through f ( x ) and f ( z ) to compute the expected v alues at each step. Consequently , determining whether the probability function exceeds or falls below a giv en threshold δ can also be accomplished in polynomial time. Furthermore, lev eraging the proof of non-decreasing monotonicity (Proposition D) of the global probability function, we can utilize algorithm 1 for contrasti ve reason search or for sufﬁcient reason search. By executing a linear number of polynomial queries, a subset-minimal explanation (whether a subset minimal contrasti ve reason or a subset minimal suf ﬁcient reason) can thus be obtained in polynomial time. J P RO O F O F T H E O R E M 2 Theorem 2. Assuming f is a neural network or a tr ee ensemble, and D is an empirical distribution — ther e exist polynomial-time algorithms for obtaining subset minimal global δ -sufﬁcient and con- trastive r easons. However , unless PTIME = NP , there is no polynomial time algorithm for computing a subset minimal local δ -sufﬁcient r eason or a subset minimal local δ -contrastive r eason. Pr oof. The ﬁrst part of the proof follo ws directly from Proposition 6, which established that this holds for any model, since both v g suff ( S ) = E x ∼D [ Pr z ∼D ( f ( x ) = f ( z ) | x S = z S )] and v g con ( S ) = E x ∼D [ Pr z ∼D ( f ( x )  = f ( z ) | x ¯ S = z ¯ S )] can be computed in polynomial time from 34 Published as a conference paper at ICLR 2026 the empirical distribution. For the second part, we proceed by proving two lemmas: one establishing the intractability of computing the sufﬁcient case, and the other addressing the contrasti ve case. Lemma 6. Unless PTIME = NP , then there is no polynomial time algorithm for computing a subset minimal local δ -sufﬁcient r eason for either a tree ensemble or a neural network, under empirical distributions. Pr oof. W e will ﬁrst prov e this claim for neural networks and then extend the result to tree ensembles. W e will establish this claim by demonstrating that if a polynomial-time algorithm exists for computing a subset-minimal local δ sufﬁcient reason for neural networks, then it would enable us to solve the classic NP-complete CNF-SA T problem in polynomial time. The CNF-SA T problem is deﬁned as follows: CNF-SA T : Input : A formula in conjuncti ve normal form (CNF): ϕ . Output : Y es , if there exists an assignment to the n literals of ϕ such that ϕ is ev aluated to True, and No otherwise Our proof will also utilize the following lemma (with its proof pro vided in Barceló et al. (2020)): Lemma 7. Any boolean cir cuit ϕ can be encoded into an equivalent MLP o ver the binary domain { 0 , 1 } n → { 0 , 1 } in polynomial time. Pr oof. W e will actually establish hardness for a simpler , more speciﬁc case, which will consequently imply hardness for the more general setting. In this case, we assume that the empirical distrib ution D consists of a single element, which, for simplicity , we take to be the zero vector 0 n . Since the sufﬁcienc y of S in this scenario depends only on the local instance x and the single instance 0 n in our empirical distribution, the nature of the input format (whether discrete, continuous, etc.) does not affect the result. Therefore, the hardness results hold univ ersally across all these settings. Similarly to 0 n , we deﬁne 1 n as an n -dimensional vector consisting entirely of ones. Giv en an input ϕ , we initially assign 1 n to ϕ , ef fectiv ely setting all variables to T rue. If ϕ ev aluates to True under this assignment, then a satisfying truth assignment exists. Therefore, we assume that ϕ ( 1 n ) = 0 . W e now introduce the follo wing deﬁnitions: ϕ 2 := ( x 1 ∧ x 2 ∧ . . . x n ) , ϕ ′ := ϕ ∨ ϕ 2 (65) ϕ ′ , while no longer a CNF , can still be transformed into an MLP f using Lemma 8, ensuring that f behav es equiv alently to ϕ ′ ov er the domain { 0 , 1 } n . Given our assumption that computing a subset-minimal local δ -sufﬁcient reason is feasible in polynomial time, we can determine one for the instance ⟨ f , x := 1 n ⟩ , δ := 1 , noting that the empirical distribution D is simply deﬁned ov er the single data point 0 n . W e now assert that the subset-minimal δ -suf ﬁcient reason generated for ⟨ f , x ⟩ encompasses the entire input space, i.e., S = { 1 , . . . , n } , if and only if ⟨ ϕ ⟩ ∈ CNF-SAT . Let us assume that S = { 1 , . . . , n } . Since S is a δ -suf ﬁcient reason for ⟨ f , x ⟩ , this simply means that setting the complementary set to an y v alue maintains the prediction. Since the complementary set is ∅ in this case, this tri vially holds. The fact that S is subset-minimal means that any other subset S ′ ⊆ S satisﬁes v ( S ′ ) < δ = 1 . Since the probability function Pr z ∼D ( f ( x ) = f ( z ) | x S = z S ) is determined by a single point (the distribution contains only the point 0 n ), the probability function can only take values of 1 or 0 . Hence, we also know that v ( S ′ ) = 0 . This tells us that, aside from the subset S = { 1 , . . . , n } , for any subset S ′ ⊆ S , ﬁxing the features in S ′ to 1 and the rest to 0 does not result in a classiﬁcation outcome of 1 . Since the ϕ 2 component within ϕ ′ is T rue only if all features are assigned 1 , this directly implies that ϕ is assigned False for any of these inputs. Since we already kno w that ϕ does not return a T rue answer for the vector assignment 1 n (as veriﬁed at the beginning), and no w we have established that the same holds for all other input vectors, we conclude that ϕ ∈ CNF-SA T . Now , suppose that S is a subset that is strictly contained within { 1 , . . . , n } . Giv en that S is sufﬁcient under our deﬁnitions of the distribution D with δ = 1 , we can apply the same reasoning as before to 35 Published as a conference paper at ICLR 2026 conclude that v ( S ) = 1 . This implies that setting the features in S to 1 while setting the remaining features to 0 ensures that the function f ev aluates to 1 . Since this assignment is necessarily not a vector consisting entirely of ones, it follows that the ϕ 2 component within ϕ ′ must be False. Consequently , the ϕ component must be T rue, which implies that ⟨ ϕ ⟩ ∈ CNF-SA T . This completes the proof. W e ha ve established that the following claim holds for neural networks. Howe ver , extending the proof to tree ensembles requires a minor and straightforward adaptation. T o do so, we will utilize the following lemma, which has been noted in se veral pre vious works (Ordyniak et al., 2024; Audemard et al., 2022b): Lemma 8. Any CNF or DNF ϕ can be encoded into an equivalent random for est classiﬁer over the binary domain { 0 , 1 } n → { 0 , 1 } in polynomial time. Pr oof. W e observe that we can apply the same process used in our proof for neural networks, where we encoded ϕ ′ into an equi valent neural network. Howe ver , ϕ ′ is no longer a v alid CNF due to our construction (though encoding it into an MLP was not an issue, as any Boolean circuit can be transformed into an MLP). Ne vertheless, since ϕ ′ consists of a conjunction of only two terms, we can easily represent it as an equiv alent CNF: ϕ ′ := ( c 1 ∨ ϕ 2 ) ∧ ( c 2 ∨ ϕ 2 ) ∧ . . . ( c m ∨ ϕ 2 ) (66) Where each c i is a disjunction of a few terms. Consequently , ϕ ′ is a valid CNF , allowing us to transform it into an equiv alent random forest classiﬁer . The reduction we outlined for MLPs applies directly to these models as well, thereby completing the proof for both model families. Lemma 9. Unless PTIME = NP , then there is no polynomial time algorithm for computing a subset minimal local δ -contrastive r eason for either a tr ee ensemble or a neural network, under empirical distributions. Pr oof. W e will present a proof analogous to the one in Lemma 6. Speciﬁcally , we will once again utilize the classical NP-hard CNF-SA T problem deﬁned in Lemma 6. In particular, gi ven a Boolean formula ϕ , we will demonstrate that determining a subset-minimal contrastive reason — whether for a neural network or a tree ensemble — allo ws us to decide the satisﬁability of ϕ . First, we check whether assigning all variables in ϕ to 1 ev aluates ϕ to True. If so, the formula is satisﬁable, and we have determined its satisﬁability . Otherwise, we use Lemma 8 to encode the CNF formula as a neural network f . Next, we compute a subset-minimal δ -sufﬁcient reason by setting D as the empirical distribution containing only a single data point 0 n , following a similar procedure to Lemma 6. Additionally , we set δ = 1 and compute a subset-minimal δ -contrastiv e reason concerning ⟨ f , x ⟩ . W e will now demonstrate that if any subset-minimal δ contrastiv e reason obtained for ϕ is satisﬁable, then ϕ itself is satisﬁable. Con versely , if no subset-minimal δ contrastiv e reason is obtained, then ϕ is unsatisﬁable. The v alidity of this claim follo ws a reasoning similar to that pro vided in Lemma 6. Speciﬁcally , the term Pr z ∼D ( f ( x )  = f ( z ) | x ¯ S = z ¯ S ) , where the distribution D considers sampling from a single datapoint, can set the probability to either 0 or 1. Furthermore, since we are searching for a δ = 1 contrastiv e reason, this is equiv alent to asking whether there exists an assignment that changes the classiﬁcation of f , which corresponds to modifying the assignment of ϕ from False to T rue. If such an assignment e xists, then ϕ is satisﬁable. Ho wev er , if no subset-minimal contrastiv e reason exists, then no subset of features ﬁxed to zero — when the complementary set is set to ones — ev aluates to true. This is equiv alent to stating that no assignment ev aluates f (and consequently ϕ ) to T rue, implying that ϕ is unsatisﬁable. T o extend the proof from neural networks to tree ensembles, we can follow the same procedure outlined in Lemma 6, encoding the CNF formula into an equiv alent random forest classiﬁer . Conse- quently , the proof remains valid for tree ensembles, thereby concluding the proof. 36 Published as a conference paper at ICLR 2026 K P RO O F O F T H E O R E M 3 Theorem 3. Given a neural network or tree ensemble f and an empirical distribution D over a ﬁxed dataset D , Algorithm 2 yields a constant O  ln  v g con ([ n ]) min i ∈ [ n ] v g con ( { i } )  -appr oximation, bounded by O (ln( | D | )) , for computing a global cardinally minimal δ -contrastive r eason for f , assuming featur e independence. In contrast, unless PTIME = NP , no bounded approximation exists for computing a local car dinally minimal δ -contrastive r eason for any ⟨ f , x ⟩ , even when | D | = 1 . Pr oof. W e divide the proof of the theorem into two lemmas, covering both the approximation guarantee and the result on the absence of a bounded approximation. Lemma 10. Given a neural network or tree ensemble f and an empirical distribution D over a ﬁxed dataset D , Algorithm 2 yields a constant O  ln  v g con ([ n ]) min i ∈ [ n ] v g con ( { i } )  -appr oximation, bounded by O (ln( | D | )) , for computing a global cardinally minimal δ -contrastive r eason for f , assuming featur e independence. Pr oof. W e will in fact prove a stronger claim, sho wing that this holds for any model, provided the trivial condition that its inference time is computable in polynomial time, along with one additional mild condition that we will detail later — both of which apply to both our neural netw ork and tree ensemble formalizations. W e begin by noting that, since we are w orking with empirical distributions, the computation of the global contrastiv e probability value function: v g con ( S ) = E x ∼D [ Pr z ∼D ( f ( x )  = f ( z ) | x ¯ S = z ¯ S )] (67) Can be computed in polynomial time by iterating ov er all pairs x , z in the dataset D, as pre viously established in Proposition 2. Since Algorithm 2 performs only a linear number of these polynomial- time queries, its total runtime is therefore polynomial. Regarding the approximation, the classical work by W olsey et al. (W olsey, 1982) established a harmonic-series-based approximation guarantee for monotone non-decreasing submodular func- tions v with integer values. More generally , their result yields an approximation factor of O  ln  v ([ n ]) min i ∈ [ n ] v ( { i } )  . W e showed in Proposition 2 that the value function v g con is monotone non-decreasing, and under feature independence, Proposition 4 establishes its submodularity . Com- bined with the fact that Algorithm 2 runs in polynomial time, this directly yields an approximation guarantee of O  ln  v g con ([ n ]) min i ∈ [ n ] v g con ( { i } )  . Ho wever , we must also show that the expression O  ln  v g con ([ n ]) min i ∈ [ n ] v g con ( { i } )  is both ﬁnite and bounded by O (ln( | D | )) , implying that it is ef fectiv ely constant, due to the assumption of a ﬁx ed dataset | D | . T o ensure this, we begin by conﬁrming that the expression is well-deﬁned and ﬁnite. W e then proceed to establish the desired bounds. T o do so, we introduce a preprocessing step in which we eliminate “redundant” elements—those that could theoretically cause the denominator min i ∈ [ n ] v g con ( { i } ) to be zero. W e begin by formally deﬁning what we mean by redundancy: Deﬁnition 1. Let D denote some empirical distribution over a dataset D . Then we say that some featur e i ∈ [ n ] is redundant with respect to D if for any pair x , z ∈ D it holds that f ( x [ n ] \{ i } ; z { i } ) = f ( x ) . Here, the notation f ( x [ n ] \{ i } ; z { i } ) indicates that all features in [ n ] \ { i } are ﬁxed to their v alues in x , while feature i is set to its value in z . Notably , this is equiv alent to deﬁning: ∀ S ⊆ [ n ] , z ∈ D f ( x S ; z ¯ S ) = f ( x S \{ i } ; z ¯ S ∪{ i } ) (68) As before, the notation ( x S ; z ¯ S ) indicates that the features in S are ﬁx ed to their v alues in x , while the features in S are ﬁxed to their v alues in z . Thus, in this sense, if we “remo ve” feature i from the input space and deﬁne a ne w function $f ’$ ov er the reduced space [ n ′ ] := [ n ] \ { i } , then for any input of size n ′ , the output of f ′ will e xactly match the output of f when applied to the same features [ n ] \ { i } . 37 Published as a conference paper at ICLR 2026 This remov al can be done in polynomial time for both tree ensembles and neural netw orks: for neural networks, it in volves detaching the corresponding input neuron from the network; for tree ensembles, it in volv es removing any splits on that feature from all decision trees. Giv en the empirical nature of the contrasti ve v alue function v g con ( S ) — as pre viously discussed in this proof and in Proposition 2 — we can compute each v g con ( { i } ) for all i ∈ [ n ] by iterating over all pairs x , z ∈ D and checking whether: f ( x [ n ] \{ i } ; z { i } )  = f ( x ) (69) If this condition holds, it indicates that feature i is not redundant. Conv ersely , if the condition fails for all pairs considered during iteration, then i is deemed redundant. Once all redundant features have been identiﬁed with respect to the empirical distrib ution D , we can remov e them and construct an equiv alent model f ′ . As discussed abo ve, this transformation can be performed in polynomial time for both neural networks and decision trees. T o conclude our proof, we observe that v g con ( { i } ) equals 0 for some empirical distribution D if and only if feature i is redundant with respect to D . Therefore, once the preprocessing step re- mov es all redundant features from f and we construct the resulting function f ′ , we ensure that min i ∈ [ n ] v g con ( { i } ) > 0 . W e now present the more precise bounds referenced in our proof. Speciﬁcally , we demonstrate that the approximation factor is O (ln( | D | )) . Gi ven that the empirical distribution D is ﬁxed, this yields a constant approximation. While tighter bounds may be achievable, our goal here is solely to establish that the bound is constant — a property that will later sharply contrast with the local e xplainability setting, where no bounded approximation exists. More speciﬁcally , we know that the probability function Pr z ∼D ( f ( x )  = f ( z ) | x ¯ S = z ¯ S ) is, by the deﬁnition of empirical distrib utions, at least 1 | D | . Therefore, the deﬁnition of E x ∼D [ Pr z ∼D ( f ( x )  = f ( z ) | x ¯ S = z ¯ S )] (i.e., the value function v g con ) is at least 1 | D | 2 . In particular , this also implies that: min i ∈ [ n ] v g con ( { i } ) ≥ 1 | D | 2 (70) Consequently , we obtain that: ln  v g con ([ n ]) min i ∈ [ n ] v g con ( { i } )  ≤ ln( | D | 2 ) = O (ln( | D | ) (71) Which concludes our proof. Lemma 11. Unless PTIME=NP , ther e is no bounded appr oximation for computing a car dinally minimal local δ -contrastive r eason for any ⟨ f , x ⟩ wher e f is either a neural network or a tree ensemble, even when | D | = 1 . Pr oof. Assume, for contradiction, that there exists a bounded approximation algorithm for computing a cardinally minimal local δ -contrastiv e reason for some ⟨ f , x ⟩ , where f is a neural network or a tree ensemble—ev en when | D | = 1 . Ho wev er , Lemma 9 establishes that even with a single baseline z = 0 n (i.e., when the entire dataset is just one instance), deciding whether a contrasti ve reason exists is NP-hard unless PTIME = NP . Therefore, if a polynomial-time algorithm could yield a bounded approximation for a cardinally minimal contrastiv e reason, it would contradict this result, as such an approximation would implicitly decide the existence of a contrasti ve explanation. L P RO O F O F T H E O R E M 4 Theorem 4. Given a neural network or tr ee ensemble f and an empirical distribution D over a ﬁxed dataset D , Algorithm 2 yields a constant O  1 1 − k f + ln  v g suff ([ n ]) min i ∈ [ n ] v g suff ( { i } )  -appr oximation for 38 Published as a conference paper at ICLR 2026 computing a global car dinally minimal δ -sufﬁcient r eason for f , assuming featur e independence. In contrast, unless PTIME = NP , there is no bounded approximation for computing a local car dinally minimal δ -sufﬁcient r eason for any ⟨ f , x ⟩ , e ven when | D | = 1 . Pr oof. W e divide the proof into two lemmas: one establishing the approximation guarantee for the global case, and the other demonstrating the intractability of the local case. Lemma 12. Given a neural network or tree ensemble f and an empirical distribution D over a ﬁxed dataset D , Algorithm 2 yields a constant O  1 1 − k f + ln  v g suff ([ n ]) min i ∈ [ n ] v g suff ( { i } )  -appr oximation for computing a global car dinally minimal δ -sufﬁcient r eason for f , assuming feature independence . Pr oof. The proof will follow a similar approach to that of Lemma 10, where we showed that for both neural networks and tree ensembles, Algorithm 2 achieves an approximation factor of ln  v g con ([ n ]) min i ∈ [ n ] v g con ( { i } )  . After applying the preprocessing step, this ratio is guaranteed to be ﬁnite and bounded by O (ln( | D | )) . Similar to Lemma 10, we will again prov e a stronger claim — namely , that the result holds for any model, assuming the trivial condition that its inference time is polynomially computable, and the additional condition concerning the remo val of redundant features, as described in Lemma 10. As discussed there, both conditions are satisﬁed by our neural network and tree ensemble formalizations. Here as well, since we are working with empirical distributions, the computation of the global sufﬁcient probability v alue function: v g suff ( S ) = E x ∼D [ Pr z ∼D ( f ( x ) = f ( z ) | x S = z S )] (72) can be computed in polynomial time by iterating over all pairs x , z in the dataset D , as previously shown in both Lemma 10 and Proposition 2. Since Algorithm 2 makes only a linear number of such polynomial-time queries, its ov erall runtime is polynomial. Unlike Lemma 10, where the approximation relied on the result by W olsey et al. (W olsey, 1982) for monotone non-decreasing submodular functions, the setting here requires a different condition due to the supermodular nature of the function. Speciﬁcally , Shi et al.(Shi et al., 2021) provided an approx- imation guarantee of O  1 1 − k f + ln  v g suff ([ n ]) min i ∈ [ n ] v g suff ( { i } )  , where k f := 1 − min i ∈ [ n ] v ([ n ]) − v ([ n ] \ i ) v ( i ) − v ( ∅ ) . W e established in Proposition 2 that the v alue function v g suff is monotone non-decreasing, and under the assumption of feature independence, Proposition 3 further sho ws that it is supermodular . Giv en that Algorithm 2 runs in polynomial time, this leads directly to an approximation guarantee of O  1 1 − k f + ln  v g suff ([ n ]) min i ∈ [ n ] v g suff ( { i } )  , where k f := 1 − min i ∈ [ n ] v g suff ([ n ]) − v g suff (([ n ] \{ i } ) v g suff ( { i } ) − v g suff ( ∅ ) . T o show that this expression is both bounded and constant, we follo w the same preprocessing step as in Lemma 10, where we remove all redundant features from f — a process that, as previously explained, can be carried out in polynomial time for both neural networks and tree ensembles. As before, a feature i is strictly redundant if and only if v ( { i } ) − v ( ∅ ) = 0 . This preprocessing yields a ne w function f ′ that behav es identically to f ov er the remaining features and ensures that, for e very i , both v g suff ( { i } ) > 0 and v g suff ( { i } ) − v g suff ( ∅ ) > 0 hold. In order for us to show that this e xpression is both bounded and constant, similarly to Lemma 10 we will perform the exact same preprocessing phase (which, as wel explained there, can be performed in polynomial time both for neural networks as well as tree ensembles) as before where we remov e all redundant features from f . W e note that here too, it holds that a feature i is strictly redundant if and only if v ( { i } ) − v ( ∅ ) = 0 . Hence, this preprocessing phase will giv e us a new function f ′ for which it both holds that it is equi valent to the beha viour of f for all remaining feautres and also satisﬁes that for any i it holds that both v g suff ( { i } ) > 0 and v g suff ( { i } ) − v g suff ( ∅ ) > 0 hold. Now , following the same reasoning as in Lemma 10, where the probability term is lower bounded by 1 | D | and its e xpected value by 1 | D | 2 , we obtain that min i ∈ [ n ] v g suff ( i ) ≥ 1 | D | 2 and v g suff ( i ) − v g suff ( ∅ ) ≥ 1 | D | 2 . This implies, as in Lemma 10, that the ﬁrst term is lower bounded by O (ln( | D | )) , and for the second term we deriv e the following: 39 Published as a conference paper at ICLR 2026 Since v g suff is supermodular (Proposition 3), we have that v g suff ([ n ]) − v g suff (([ n ] \ { i } ) . It follo ws that 1 ≤ v g suff ([ n ]) − v g suff ([ n ] \ { i } ) ≤ 1 | D | 2 . This implies: 1 ≤ min i ∈ [ n ] v g suff ([ n ]) − v g suff ([ n ] \{ i } ) v g suff ( { i } ) − v g suff ( ∅ ) ≤ | D | 2 , and therefore, 1 − | D | 2 ≤ k f ≤ 0 . This yields 1 1 − k f ≤ | D | 2 , demonstrating that the overall approximation bound based on the one established by Shi et al. is constant. While this bound may be signiﬁcantly smaller in practice, our goal here is simply to show that it remains constant — unlike in the local setting, where no bounded approximation is achiev able. Lemma 13. Unless PTIME = NP , ther e is no bounded approximation for computing a local cardinally minimal δ -sufﬁcient r eason for any ⟨ f , x ⟩ , e ven when | D | = 1 . Pr oof. The proof follows a similar approach to Lemma 11. Suppose, for contradiction, that there exists a polynomial-time algorithm that provides a bounded approximation to a cardinally minimal local δ -suf ﬁcient reason for some ⟨ f , x ⟩ , where f is either a neural network or a tree ensemble—ev en when | D | = 1 . Y et, Lemma 6 shows that even in the extreme case where the dataset consists of a single baseline z = 0 n , deciding whether a suf ﬁcient reason exists is NP-hard unless PTIME = NP . Hence, the existence of such an approximation algorithm would contradict this hardness result, as it would entail the ability to decide the existence of a suf ﬁcient reason. M H A R D N E S S O F A P P R OX I M A T I N G G L O BA L C O N T R A S T I V E R E A S O N S F O R N E U R A L N E T W O R K S U S I N G E M P I R I C A L D I S T R I B U T I O N S Proposition 1. Assume a neural network f : F → [ c ] with ReLU activations, some dataset D , such that D is the empirical distribution deﬁned over D , and some thr eshold δ ∈ Q . Then unless NP ⊆ DTIME ( n O ( l og ( log ( n )) ) , ther e does not e xist a polynomial-time O (ln( | D | ) − ϵ ) appr oximation algorithm for computing a cardinally minimal global contr astive reason for f concerning D , for any ﬁxed ϵ > 0 . Pr oof. The proof proceeds via a polynomial-time approximation-preserving reduction from a speciﬁc variant of Set Cov er that is, similarly to regular Set Cov er , prov ably hard to approximate within the factor gi ven in the proposition (V azirani, 2001). Partial Set Cov er : Input : A uni verse U := { u 1 , . . . , u n } , a collection of sets S := { S 1 , . . . S m } such that for all j S j ⊆ U , and a coverage threshold δ ∈ { 0 , 1 , . . . , n } . Output : A minimum-size subset C ⊆ S such that |∪ S j ∈ C S j | ≥ δ . W e reduce this problem to that of computing a global , cardinally-minimal contrastiv e reason S for a neural network f with respect to a distribution D deﬁned ov er a dataset D , and some threshold δ ′ . W e begin by recalling the deﬁnition of a global contrasti ve reason and then extend it to the setting where the explanation is e valuated o ver an empirical dataset D . The deﬁnition is as follows: E x ∼D [ Pr z ∼D ( f ( z )  = f ( x ) | z ¯ S = x ¯ S )] ≥ δ. (73) More speciﬁcally , when we assume an empirical distribution o ver D , this implies that: E x ∼D [ Pr z ∼D ( f ( z )  = f ( x ) | z ¯ S = x ¯ S )] = 1 | D | X x ∈ D [ Pr z ∼D ( f ( z )  = f ( x ) | z ¯ S = x ¯ S )] = 1 | D | X x ∈ D  P z ∈ D ¯ S ( z , x ) 1 { f ( z )  = f ( x ) } D ¯ S ( z , x )  (74) where we denote D ¯ S ( z , x ) := { z ∈ D : z ¯ S = x ¯ S } . W e now construct a ReLU neural network f . Before doing so, we specify the functional form that we intend t he constructed neural network f to 40 Published as a conference paper at ICLR 2026 implement. For each set S j , we introduce two vectors x j and y j , both belonging to Z n and drawn from the dataset D , such that they agree on all coordinates outside S j ; that is, they share identical feature values on the complement ¯ S j . Formally: x j ¯ S j = y j ¯ S j (75) and we assign strictly dif ferent values to the features in S j by selecting them from tw o distinct vectors z and z ′ such that: x j = ( x j ¯ S j ; z j S j )  = ( x j ¯ S j ; z ′ j S j ) = ( y j ¯ S j ; z ′ j S j ) = y j (76) W e aim to construct f such that it satisﬁes the following: f ( x j ) = f ( x j ¯ S j ; z j S j )  = f ( x j ¯ S j ; z ′ j S j ) = f ( y j ¯ S j ; z ′ j S j ) = f ( y j ) (77) Let us no w assume that both x j and y j occur uniquely in the dataset D across all [ n ] coordinates — that is, no other v ector in D matches either of them on an y coordinate. Under this assumption, we obtain the following relation: E x ∼D [ Pr z ∼D ( f ( z )  = f ( x ) | z ¯ S j = x ¯ S j )] = 1 | D | X x ∈ D  P z ∈ D ¯ S j ( z , x ) 1 { f ( z )  = f ( x ) } D ¯ S j ( z , x )  = 2 2 | D | = 1 | D | (78) Here, the factor of 2 appearing in both the numerator and denominator arises from the fact that only two vector pairs — ( x j , z j ) and ( z j , x j ) — contribute to the respecti ve sums. W e repeat this construction for each S i in the set-cover instance, each time choosing two distinct vectors x i and y i that dif fer on every coordinate in [ n ] . In total, this yields 2 m vectors that form the dataset D from which we construct f . W e then construct a neural network f that simulates this function. This is achiev ed using a kno wn result on neural network memorization (V ardi et al., 2022), which guarantees that a ReLU network capable of representing an y ﬁnite set of input–output pairs can be b uilt in polynomial time. Using this result, we construct f accordingly , and set δ ′ := δ | D | . W e now ar gue that a subset S j is a global contrasti ve reason for f under tolerance δ ′ if and only if C is a partial set cover for U under tolerance δ . By Equation 78, each constructed pair of vectors ( x j , y j ) contributes e xactly 1 | D | to the v alue term. Moreov er , because each S j uses its o wn distinct pair of vectors in Z n , there is no ov erlap between these pairs. Consequently , the coverage achiev ed by any set cov er C corresponds exactly to the accumulated contribution of the selected subsets in the input domain of f . This completes the reduction. N T R AC TA B I L I T Y R E S U LT S F O R O RT H O G O N A L D N F S Lemma 14. If f is an orthogonal DNFs and the pr obability term v g suff can be computed in polynomial time given the distribution D (which holds for independent distributions, among others), then obtaining a subset-minimal global δ -sufﬁcient r eason can be obtained in polynomial time . Pr oof. W e begin by deﬁning orthogonal DNFs (Crama & Hammer, 2011) before presenting the full proof. Deﬁnition 2 (Orthogonal DNFs) . A Boolean function φ is in orthogonal disjunctiv e normal form (orthogonal DNF) (Crama & Hammer, 2011) if it is a disjunction of terms T 1 ∨ T 2 ∨ · · · ∨ T m , such that for every pair of distinct terms T i and T j ( i  = j ): T i ∧ T j | = ⊥ This means that no single variable assignment can satisfy mor e than one term simultaneously . 41 Published as a conference paper at ICLR 2026 The proof mirrors the argument used in Theorem 1, relying on the f act that v g suff can be computed in polynomial time for any set S . For a given set S , let C S T i be the constraints that the term T i imposes on the variables in S . W e enumerate all pairs of terms ( T i , T j ) . Any pair whose constraints on S are inconsistent (i.e., C S T i ∧ C S T j | = ⊥ ) is ignored, since it contrib utes zero to v g suff. For each consistent pair of terms ( T i , T j ) , we e valuate Pr ( x S ∈ C S T i ,T j ) , where C S T i ,T j = C S T i ∩ C S T j . The contribution of this pair to v g suff is then Pr ( x S ∈ C S T i ,T j ) · Pr ( x ¯ S ∈ C ¯ S T i ) · Pr ( x ¯ S ∈ C ¯ S T j ) . Under feature independence, these probabilities factor as Pr( x S ) = Q i ∈ S p ( x i ) . Since there are at most m 2 such term pairs, the v alue of v g suff can be computed in polynomial time under an independent distribution D . O D I S C L O S U R E : U S AG E O F L L M S An LLM was used exclusi vely as a writing assistant to reﬁne grammar and typos and improv e clarity . It did not contribute to the generation of research ideas, study design, data analysis, or interpretation of results, all of which were carried out solely by the authors. 42

Unifying Formal Explanations: A Complexity-Theoretic Perspective

Original Paper

Comments & Academic Discussion

Leave a Comment

Original Paper

Related Papers

Comments & Academic Discussion

Leave a Comment