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+    "text": "Published as a conference paper at ICLR 2026 FAME: FORMAL ABSTRACT MINIMAL EXPLANATION\nFOR NEURAL NETWORKS Ryma Boumazouza∗1,2, Raya Elsaleh∗3, Melanie Ducoffe1,2, Shahaf Bassan3 and Guy Katz3\n1Airbus SAS, France, 2IRT Saint-Exupery, France, 3The Hebrew University of Jerusalem, Israel We propose FAME (Formal Abstract Minimal Explanations), a new class of abductive explanations grounded in abstract interpretation. FAME is the first method\nto scale to large neural networks while reducing explanation size. Our main contri-2026 bution is the design of dedicated perturbation domains that eliminate the need for\ntraversal order. FAME progressively shrinks these domains and leverages LiRPAbased bounds to discard irrelevant features, ultimately converging to a formal\nabstract minimal explanation. To assess explanation quality, we introduce aMar procedure that measures the worst-case distance between an abstract minimal ex-\n11 planationattacks withandanaoptionaltrue minimalVERIX+explanation.refinementThisstep.procedureWe benchmarkcombinesFAMEadversarialagainst\nVERIX+ and demonstrate consistent gains in both explanation size and runtime\non medium- to large-scale neural networks. 1 INTRODUCTION[cs.AI] Figure 1: FAME Framework. The pipeline operates in two main phases (1) Abstract Pruning\n(Green) phase leverages abstract interpretation (LiRPA) to simultaneously free a large number of\nirrelevant (pixels that are certified to have no influence on the model's decision) features based\non a batch certificate (Section 4.2). This iterative process operates within a refined, cardinalityconstrained perturbation domain, Ωm(x, A) (Eq. 5) to progressively tighten the domain; To ensurearXiv:2603.10661v1 that the final explanation is as small as possible, the remaining features that could not be freed\nin batches are tested individually (Section 5). (2) Exact Refinement (Orange) phase identifies\nthe final necessary features using singleton addition attacks and, if needed, a final run of VERIX+\n(Section 6). The difference in size, |wAXpA⋆| −|AXp|, serves as an evaluation metric of phase 1. Neural network-based systems are being applied across a wide range of domains. Given AI tools'\nstrong capabilities in complex analytical tasks, a significant portion of these applications now involves tasks that require reasoning.",
+    "paper_id": "2603.10661",
+    "title": "FAME: Formal Abstract Minimal Explanation for Neural Networks",
+    "authors": [
+      "Ryma Boumazouza",
+      "Raya Elsaleh",
+      "Melanie Ducoffe",
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+    "text": "These tools often achieve impressive results in problems requiring intricate analysis to reach correct conclusions. Despite these successes, a critical challenge\nremains: understanding the reasoning behind neural network decisions. The internal logic of a neural\nnetwork is often opaque, with its conclusions presented without accompanying justifications. This\nlack of transparency undermines the trustworthiness and reliability of neural networks, especially in\nhigh-stakes or regulated environments. Consequently, the need for interpretable and explainable AI\n(XAI) has become a growing focus in recent research. Published as a conference paper at ICLR 2026 Two main approaches have emerged to address this challenge. The first employs statistical and\nheuristic techniques to infer explanations based on network's internal representations (Fel et al.,\n2022). While these methods estimate feature importance, that require empirical evaluation (such as\nthe µ-Fidelity metric (Bhatt et al., 2020)), the second approach leverages automated reasoners and\nformal verification to provide provably correct explanations grounded in logical reasoning. We ground our work in the formal definition of Abductive Explanations (AXp) (Ignatiev et al.,\n2019), a concept belonging to the broader family of \"formal XAI\" which includes minimal explanations, also known as local-minimal, minimal unsatisfiable subsets (Marques-Silva, 2010) and prime\nimplicants (Shih et al., 2018). An AXp is a subset of features guaranteed to maintain the model's\nprediction under any perturbation within a defined domain. In a machine learning context, these\nexplanations characterize feature sets where removing any single feature invalidates the guarantee,\neffectively representing subsets that preserve the decision's robustness. However, a major hurdle for\nformal XAI is its high computational cost due to the complexity of reasoning, preventing it from\nscaling to large neural networks (NNs) (Marques-Silva, 2023b). This limitation, combined with the\nscarcity of open-source libraries, significantly hinders its adoption. Initial hybrid approaches, such\nas the EVA method (Fel et al., 2023), have attempted to combine formal and statistical methods, but\nthese often fail to preserve the mathematical properties of the explanation.",
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+      "Ryma Boumazouza",
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+    "text": "However, robustnessbased approaches address the scalability challenges of formal XAI for NN by leveraging a fundamental connection between AXps and adversarial examples (Huang & Marques-Silva, 2023). In this work, we present FAME, a scalable framework for formal XAI that addresses the core limitations of existing methods. Our contributions are fourfold:",
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+    "text": "• Formal abstract explanations. We introduce the first class of abductive explanations\nderived from abstract interpretation, enabling explanation algorithms to handle highdimensional NNs. • Eliminating traversal order. We design perturbation domains and a recursive refinement\nprocedure that leverage Linear Relaxation based Perturbation Analysis (LiRPA)-based certificates to simultaneously discard multiple irrelevant features. This removes the sequential\nbottleneck inherent in prior work and yields an abstract minimal explanation. • Provable quality guarantees.",
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+    "title": "FAME: Formal Abstract Minimal Explanation for Neural Networks",
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+    "text": "We provide the first procedure to measure the worst-case\ngap between abstract minimal explanations and true minimal abductive explanations, combining adversarial search with optional VERIX+ refinement. • Scalable evaluation. We benchmark FAME on medium- and large-scale neural networks,\nshowing consistent improvements in both explanation size and runtime over VERIX+. Notably, we produce the first abstract formal abductive explanations for a ResNet architecture\non CIFAR-10, demonstrating scalability where exact methods become intractable. 2 ABDUCTIVE EXPLANATIONS & VERIFICATION",
+    "paper_id": "2603.10661",
+    "title": "FAME: Formal Abstract Minimal Explanation for Neural Networks",
+    "authors": [
+      "Ryma Boumazouza",
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+    "text": "Scalars are denoted by lower-case letters (e.g., x), and the set of real numbers by R. Vectors are\ndenoted by bold lower-case letters (e.g., x), and matrices by upper-case letters (e.g., W). The i-th\ncomponent of a vector x (resp. line of a matrix W) is written as xi (resp. The matrix W ≥0\n(resp. W ≤0) represents the same matrix with only nonnegative (resp. nonpositive) weights.",
+    "paper_id": "2603.10661",
+    "title": "FAME: Formal Abstract Minimal Explanation for Neural Networks",
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+    "text": "Sets are\nwritten in calligraphic font (e.g., S). We denote the perturbation domain by Ωand the property to\nbe verified by P. 2.2 THE VERIFICATION CONTEXT",
+    "paper_id": "2603.10661",
+    "title": "FAME: Formal Abstract Minimal Explanation for Neural Networks",
+    "authors": [
+      "Ryma Boumazouza",
+      "Raya Elsaleh",
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+    "text": "We consider a neural network as a function f : Rn →Rk. The core task of verification is to\ndetermine whether the network's output f(x′) satisfies a given property P for every possible input\nx′ within a specified domain Ω(x) ⊆Rn. When verification fails, it means there is at least one input\nx′ in the domain Ω(x) that violates the property P (a counterexample). The verification task can be\nwritten as: ∀x′ ∈Ω(x), does f(x′) satisfy P? This requires defining two components: Published as a conference paper at ICLR 2026 The Perturbation Domain (Ω): This domain defines the set of perturbations. It is often\nan lp-norm ball around a nominal input x, such as an l∞ball for modeling imperceptible\nnoise: Ω= {x′ ∈Rn | ∥x −x′∥∞≤ϵ}.\n2. The Property (P): This is the specification the network must satisfy. For a classification\ntask where the network correctly classifies an input x into class c, the standard robustness\nproperty P asserts that the logit for class c remains the highest for any perturbed input x′:\nP(x′) ≡min {fc(x′) −fi(x′)} > 0 (1)\ni̸=c For instance, given an MNIST image x of a '7' and a perturbation radius ϵ, the property P holds if the\nnetwork's logit for class '7' provably exceeds all other logits for every perturbed image x′ ∈Ω(x). A large body of work has investigated formal verification of NNs, with adversarial robustness being the most widely studied property (Urban & Min´e, 2021). Numerous verification tools are now\navailable off-the-shelf, and for piecewise-linear models f with corresponding input domains and\nproperties, exact verification is possible (Katz et al., 2017; Botoeva et al., 2020). In practice, however, exact methods quickly become intractable for realistic networks. To address this, we rely on\nAbstract Interpretation, a theory of sound approximation.",
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+    "text": "Specifically, we utilize Linear Relaxationbased Perturbation Analysis (LiRPA) (Zhang et al., 2018; Singh et al., 2019) which efficiently overapproximates the network's output by enclosing it between linear upper and lower bounds. Such\nabstractions enable sound but conservative verification: if the relaxed property holds, the original\none is guaranteed to hold as well. We provide a comprehensive background in Appendix A.",
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+    "text": "2.3 ABDUCTIVE EXPLANATIONS: PINPOINTING THE \"WHY\" Understanding Model Robustness with Formal Explanations: Neural networks often exhibit\nsensitivity to minor input perturbations, a phenomenon that certified training can mitigate but not\neliminate (De Palma et al., 2025). Even robustly trained models may only have provably safe regions\nspanning a few pixels for complex tasks like ImageNet classification (Serrurier et al., 2021). To build\nmore reliable systems, it is crucial to understand why a model's prediction is robust (or not) within\na given context. Formal explainability provides a rigorous framework for this analysis. We focus on abductive explanations (AXps, also called distance-restricted explanations (ϵ-AXp))\n(Ignatiev et al., 2019; Huang & Marques-Silva, 2023), which identify a subset of input features that\nare sufficient to guarantee that the property P holds. Formally, a local formal abductive explanation\nis defined as a subset of input features that, if collapsed to their nominal values (i.e., the sample x),\nensure that the local perturbation domain Ωsurrounding the sample contains no counterexamples. Definition 2.1 (Weak Abductive Explanation (wAXp) ). Formally, given a triple (x, Ω, P), an explanation is a subset of feature indices X ⊆F = {1, . . . , n} such that\nwAXp: ∀x′ ∈Ω(x), ^ (x′ i = xi) =⇒f(x′) |= P. (2)\ni∈X While many such explanations may exist (the set of all features F is a trivial one), the most useful\nexplanations are the most concise ones (Bassan & Katz, 2023). We distinguish three levels:",
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+    "text": "Minimal Explanation: An explanation X is minimal if removing any single feature from it would\nbreak the guarantee (i.e., X \\ {j} is no longer an explanation for any j ∈X). These are also known\nas minimal unsatisfiable subsets(Ignatiev et al., 2016; Bassan & Katz, 2023). Minimum Explanation: An explanation X is minimum if it has the smallest possible number of\nfeatures (cardinality) among all possible minimal explanations. The concept of an abductive explanation is illustrated using a classification task (details in Appendix\nD.1, Figure 4). The goal is to find a minimal subset of fixed features (X) that guarantees a sample's\nclassification within its perturbation domain. For the analyzed sample, fixing x2 alone is insufficient\ndue to the existence of a counterexample (Figure 5). However, fixing the set X = {x2, x3} creates a\n'safe' subdomain without counterexamples, confirming it is an abductive explanation. This explanation is minimal (neither x2 nor x3 work alone) but not minimum in cardinality, as X ′ = {x1} is also\na valid minimal explanation. In the rest of this paper, we will use the terms abductive explanation or\nformal explanation and the notation wAXp to refer to Definition 2.1. Published as a conference paper at ICLR 2026 Substantial progress has been made in the practical efficiency of computing formal explanations. While finding an abductive explanation (AXp) is tractable for some classifiers (Marques-Silva,\n2023a; Darwiche & Ji, 2022; Huang et al., 2022; 2021; Izza et al., 2020; Marques-Silva et al.,\n2020; 2021), it becomes computationally hard for complex models like random forests and neural\nnetworks (Ignatiev & Marques-Silva, 2021; Izza & Marques-Silva, 2021). To address this inherent\ncomplexity, these methods typically encode the problem as a logical formula, leveraging automated\nreasoners like SAT, SMT, and Mixed Integer Linear Programming (MILP) solvers (Audemard et al.,\n2022; Ignatiev, 2020; Ignatiev et al., 2022; Ignatiev & Marques-Silva, 2021; Izza & Marques-Silva,\n2021) . Early approaches, such as deletion-based (Chinneck & Dravnieks, 1991) and insertion-based\n(de Siqueira, 1988) algorithms, are inherently sequential, thus requiring an ordering of the input features traditionally denoted as traversal ordering. They require a number of costly verification calls\nlinear with the number of features, which prevents effective parallelization.",
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+    "text": "As an alternative, surrogate models have been used to compute formal explanations for complex models (Boumazouza\net al., 2021; 2023), but the guarantee does not necessary hold on the original model. Recent work aims to break the sequential bottleneck, by linking explainability to adversarial robustness and formal verification. DistanceAXp (Huang & Marques-Silva, 2023; La Malfa et al., 2021)\nis a key example, aligning with our definition of AXp and enabling the use of verification tools. The latest literature focuses on breaking the sequential bottleneck using several strategies that include parallelization. This is achieved either by looking for several counterexamples at once (Izza\net al., 2024; Bassan & Katz, 2023; La Malfa et al., 2021; Bassan et al., 2023) or by identifying a\nset of irrelevant features simultaneously, as seen in VERIX (Wu et al., 2023), VERIX+ (Wu et al.,\n2024b), and prior work (Bassan & Katz, 2023). For instance, VERIX+ introduced stronger traversal\nstrategies to alleviate the sequential bottleneck. Their binary search approach splits the remaining\nfeature set and searches for batches of consecutive irrelevant features, yielding the same result as sequential deletion but with fewer solver calls. They also adapted QuickXplain (Junker, 2004), which\ncan produce even smaller explanations at the cost of additional runtime by verifying both halves. Concurrently, (Bassan & Katz, 2023) proposed strategies like the singleton heuristic to reuse verification results and derived provable size bounds, but their approach remains significantly slower than\nVERIX+ and lacks publicly available code. The identified limitations are twofold. First, existing methods rely heavily on exact solvers such as\nMarabou (Katz et al., 2019; Wu et al., 2024a), which do not scale to large NNs and are restricted to\nCPU execution. Recent verification benchmarks (Brix et al., 2023; Ducoffe et al., 2024; Zhao et al.,\n2022) consistently demonstrate that GPU acceleration and distributed verification are indispensable\nfor achieving scalability. Second, these approaches critically depend on traversal order. As shown\nin VERIX, the chosen order of feature traversal strongly impacts both explanation size and runtime. Yet, determining an effective order requires prior knowledge of feature importance, precisely the\ninformation that explanations are meant to uncover, thus introducing a circular dependency.",
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+    "text": "Nevertheless, VERIX+ currently represents the SOTA for abductive explanations in NNs, achieving the\nbest trade-off between explanation size and computation time. Our work builds on this foundation by directly addressing the sequential bottleneck of formal explanation without requiring a traversal order, a first in formal XAI. We demonstrate that leveraging\nincomplete verification methods and GPU hardware is essential for practical scalability. Our approach offers a new solution to the core scalability issues, complementing other methods that aim to\nreduce explanation cost through different means (Bassan et al., 2025b;a). 4 FAME: FORMAL ABSTRACT MINIMAL EXPLANATION In this section, we introduce FAME, a framework that builds abstract abductive explanations (Definition 4.1). FAME proposes novel strategies to provide sound abstract abductive explanations\n(wAXpA) such as an Abstract Batch Certificate using Knapsack formulation, and a Recursive Refinement, relying on raw bounds provided by a formal framework (we use LiRPA in this paper). Definition 4.1 (Abstract Abductive Explanation (wAXpA)). Formally, given a triple (x, Ω, P), an\nabstract abductive explanation is a subset of feature indices X A ⊆F = {1, . . . , n} such that, under",
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+    "text": "Published as a conference paper at ICLR 2026 an abstract interpretation f of the model f, the following holds:\nwAXpA : ∀x′ ∈Ω(x), ^ (x′ i = xi) =⇒f(x′) |= P. (3)\ni∈X A Here, f = LiRPA(f, Ω) denotes the sound over-approximated bounds of the model outputs on the\ndomain Ω, as computed by the LiRPA method. If Eq. (3) holds, any feature outside X A can be\nconsidered irrelevant with respect to the abstract domain. This ensures that the concrete implication\nf(x′) |= P also holds for all x′ ∈Ω. In line with the concept of abductive explanations, we define\nan abstract minimal explanation as an abstract abductive explanation (wAXpA⋆) a set of features\nX A from which no feature can be removed without violating Eq. (3). Due to the over-approximation, as detailed in Section 2.2, any abstract abductive explanation is a\nweak abductive explanation for the model f. In the following we present the first steps described in\nFigure 1 to build such a wAXpA.",
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+    "text": "4.1 THE ASYMMETRY OF PARALLEL FEATURE SELECTION In the context of formal explanations, adding a feature means identifying it as essential to a model's\ndecision (causes the model to violate the desired property P), so its value must be fixed in the explanation.Conversely, freeing a feature means identifying it as irrelevant, allowing it to vary without\naffecting the prediction. A key insight is the asymmetry between these two actions: while adding\nnecessary features can be parallelized naively, freeing features cannot due to complex interactions. Proposition 4.1 (Simultaneous Freeing). it is unsound to free multiple features at once based only\non individual verification as two features may be individually irrelevant yet jointly critical. Parallelizing feature freeing based on individual verification queries is unsound due to hidden feature dependencies that stem from treating the verifier as a simple binary oracle (SAT/UNSAT; see\nAppendix A for formal definitions) (Proposition 4.1). To solve this, we introduce the Abstract Batch\nCertificate Φ(A) (Definition 4.2). Unlike naive binary checks, Φ(A) leverages abstract interpretation to compute a joint upper bound on the worst-case contribution of the entire set A simultaneously. If Φ(A) ≤0, it mathematically guarantees that simultaneously freeing A is sound, explicitly\naccounting for their combined interactions. The formal propositions detailing this asymmetry is\nprovided in the Appendix B. 4.2 ABSTRACT INTERPRETATION FOR SIMULTANEOUS FREEING Standard solvers act as a \"binary oracle\" and their outcomes (SAT/UNSAT) are insufficient to certify\nbatches of features for freeing without a traversal order. This is because of feature dependencies\nand the nature of the verification process. We address this by leveraging inexact verifiers based\non abstract interpretation (LiRPA) to extract proof objects (linear bounds) that conservatively track\nthe contribution of any feature set. Specifically, we use CROWN (Zhang et al., 2018) to define an\nabstract batch certificate Φ in Definition 4.2. If one succeeds in freeing a set of features A given Φ,\nwe denote such an explanation as a formal abstract explanation that satisfies Proposition 4.2.",
+    "paper_id": "2603.10661",
+    "title": "FAME: Formal Abstract Minimal Explanation for Neural Networks",
+    "authors": [
+      "Ryma Boumazouza",
+      "Raya Elsaleh",
+      "Melanie Ducoffe",
+      "Shahaf Bassan",
+      "Guy Katz"
+    ],
+    "published_date": "2026-03-11",
+    "primary_category": "",
+    "arxiv_url": "http://arxiv.org/abs/2603.10661v1",
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+    "text": "Let A be a set of features and Ωany perturbation domain. The abstract batch certificate is defined as:\nΦ(A; Ω) = max b i(x) + X ci,j ,\ni̸=c\nj∈A where the baseline bias (worst-case margin of the model's output) at x is b i(x) = W · x + wi, n i,≥0 i,≤0 oand the contribution of each feature j ∈A is ci,j = max W j (xj −xj), W j (xj −xj) ,\nwith xj = max{x′j : x′ ∈Ω(x)} and xj = min{x′j : x′ ∈Ω(x)}. The weights W and biases\nwi are obtained from LiRPA bounds, which guarantee for each target class i ̸= c, with c being the\ngroundtruth class:\n∀x′ ∈Ω(x), fi(x′) −fc(x′) ≤f i,c(x′) = W i · x′ + wi, Published as a conference paper at ICLR 2026 Proposition 4.2 (Batch-Certifiable Freeing). If Φ(A; Ω) ≤0, then F \\ A is a weak abductive\nexplanation (wAXp). If Φ(A) ≤0, freeing all features in A is sound; that is, the property P holds for every\nx′ ∈Ω(x) with {x′k = xk}k∈F\\A. The proof of Proposition 4.2 is given in Appendix B. The trivial case A = ∅always satisfies the\ncertificate, but our goal is to efficiently certify large feature sets. The abstract batch certificate also\nhighlights two extreme scenarii.",
+    "paper_id": "2603.10661",
+    "title": "FAME: Formal Abstract Minimal Explanation for Neural Networks",
+    "authors": [
+      "Ryma Boumazouza",
+      "Raya Elsaleh",
+      "Melanie Ducoffe",
+      "Shahaf Bassan",
+      "Guy Katz"
+    ],
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+    "text": "In the first, if Φ(F) ≤0, all features are irrelevant, meaning the\nproperty P holds across Ωwithout fixing any inputs. In the second, if b i(x) ≥0 for some i ̸= c,\nthen Φ(∅) > 0 and no feature can be safely freed; this situation arises when the abstract relaxation\nis too loose, producing vacuous bounds. Avoiding this degenerate case requires careful selection of\nthe perturbation domain, a consideration we highlight for the first time in the context of abductive\nexplanations. The choice of abstract domain is discussed in Section 5. 4.3 MINIMIZING THE SIZE OF AN ABSTRACT EXPLANATION VIA A KNAPSACK\nFORMULATION Between the trivial and degenerate cases lies the nontrivial setting: finding a maximal set of irrelevant features A to free given the abstract batch certificate Φ. Let F denote the index set of features. Maximizing |A| can be naturally formulated as a 0/1 Multidimensional Knapsack Problem (MKP). For each feature j ∈F, we introduce a binary decision variable yj indicating whether the feature is\nselected. The optimization problem then reads:\nmax X yj s.t. X cijyj ≤−b i(x), i ∈I, i ̸= c (4)\nj∈F j∈F\nwhere ci,j represents the contribution of feature j to constraint i, and −b i(x) is the corresponding\nknapsack capacity. The complexity of this MKP depends on the number of output classes. For binary\nclassification (k = 2), the problem is linear1. In the standard multiclass setting (k > 2), however,\nthe MKP is NP-hard. While moderately sized instances can be solved exactly using a MILP solver,\nthis approach does not scale to large feature spaces.",
+    "paper_id": "2603.10661",
+    "title": "FAME: Formal Abstract Minimal Explanation for Neural Networks",
+    "authors": [
+      "Ryma Boumazouza",
+      "Raya Elsaleh",
+      "Melanie Ducoffe",
+      "Shahaf Bassan",
+      "Guy Katz"
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+    "text": "To ensure scalability, we propose a simple\nand efficient greedy heuristic, formalized in Algorithm 1. Rather than solving the full MKP, the\nheuristic iteratively selects the feature j⋆that is least likely to violate any of the k −1 constraints, by\nminimizing the maximum normalized cost across all classes. An example is provided in Appendix\nD.2. This procedure is highly parallelizable, since all costs can be computed simultaneously.",
+    "paper_id": "2603.10661",
+    "title": "FAME: Formal Abstract Minimal Explanation for Neural Networks",
+    "authors": [
+      "Ryma Boumazouza",
+      "Raya Elsaleh",
+      "Melanie Ducoffe",
+      "Shahaf Bassan",
+      "Guy Katz"
+    ],
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+    "text": "While\nsuboptimal by design, it produces a set A such that Φ(A; Ω) ≤0. A key advantage of this greedy\nbatch approach is its computational efficiency. The cost is dominated by the computation of feature\ncontributions ci,j. This requires a single backward pass through the abstract network, which has\na complexity of O(L · N) (where L is depth and N is neurons) and is highly parallelizable on\nGPUs. In contrast, exact solvers require solving an NP-hard problem for each feature or batch. In\nSection 7, we compare the performance of this greedy approach against the optimal MILP solution,\ndemonstrating that it achieves competitive results with dramatically improved scalability. Algorithm 1 Greedy Abstract Batch Freeing (One Step)\n1: Input: model f, perturbation domain Ωm, candidate set F\n2: Initialize: A ←∅, linear bounds {W i, wi} = LiRPA(f, Ωm(x))\n3: Do: compute ci,j in parallel\n4: while Φ(A) ≤0 and |F| > 0 do\n5: pick j⋆= arg minj∈F \\A maxi̸=c ci,j/(−bi) ▷Parallel reduction\n6: if Φ(A ∪{j⋆}) ≤0 and |A| ≤m then\n7: A ←A ∪{j⋆}\n8: end if\n9: F ←F \\ {j⋆} ▷Remove candidate\n10: end while\n11: Return: A 1it can be solved optimally in O(n) time by sorting features by ascending contribution c1,j and greedily\nadding them until the capacity is exhausted. Published as a conference paper at ICLR 2026 5 REFINING THE PERTURBATION DOMAIN FOR ABDUCTIVE EXPLANATION Previous approaches for batch freeing reduce the perturbation domain using a traversal order π,\ndefining Ωπ,i(x) = {x′ ∈Rn : ∥x −x′∥∞≤ϵ, x′πi: = xπi:}. These methods only consider\nfreeing dimensions up to a certain order. However, as discussed previously, determining an effective\norder requires prior knowledge of feature importance, the very information that explanations aim to\nuncover, introducing a circular dependency. This reliance stems from the combinatorial explosion:\nthe number of possible subsets of input features grows exponentially, making naive enumeration of\nabstract domains intractable. To address this, we introduce a new perturbation domain, denoted the cardinality-constrained perturbation domain.",
+    "paper_id": "2603.10661",
+    "title": "FAME: Formal Abstract Minimal Explanation for Neural Networks",
+    "authors": [
+      "Ryma Boumazouza",
+      "Raya Elsaleh",
+      "Melanie Ducoffe",
+      "Shahaf Bassan",
+      "Guy Katz"
+    ],
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+    "chunk_id": "c9003cec-0f9d-41a8-9621-c4d38dd7bbf5",
+    "text": "For instance, one can restrict to ℓ0-bounded perturbations:\nΩm(x) = {x′ ∈Rn : ∥x −x′∥∞≤ϵ, ∥x −x′∥0 ≤m},\nwhich ensures that at most m features may vary simultaneously. This concept is closely related to\nthe ℓ0 norm and has been studied in verification (Xu et al., 2020), but, to the best of our knowledge,\nit is applied here for the first time in the context of abductive explanations. The greedy procedure\nin Algorithm 1 can then certify a batch of irrelevant features A under this domain. Once a set A\nis freed, the feasible perturbation domain becomes strictly smaller, enabling tighter bounds and the\nidentification of additional irrelevant features. We formalize this as the refined abstract domain that\nensures that at most m features can vary in addition to the set of previously seclected ones A:\nΩm(x; A) = {x′ ∈Rn : ∥x −x′∥∞≤ϵ, ∥xF\\A −x′ F\\A∥0 ≤m}. (5)\nBy construction, Ωm(x; A) ⊆Ωm+|A|(x), so any free set derived from Ωm(x; A) remains sound\nfor the original budget m + |A|. Recomputing linear bounds on this tighter domain often yields\nstrictly smaller abstract explanation. This refinement naturally suggests a recursive strategy: after\none round of greedy batch freeing, we restrict the domain to Ωm(x; A), recompute LiRPA bounds,\nand reapply Algorithm 1 for m = 1 . . . |F \\ A|. Unlike the static traversal of prior work (e.g.,\nVERIX+), FAME employs a dynamic, cost-based selection by re-evaluating abstract costs ci,j at\neach recursive step. This process functions as an adaptive abstraction mechanism: iteratively enforcing cardinality constraints tightens the domain, reducing LiRPA's over-approximation error and\nenabling the recovery of additional freeable features initially masked by loose bounds. As detailed\nin Algorithm 2, this process can be iterated, progressively shrinking the domain and expanding A. In practice, recursion terminates once no new features can be freed. Finally, any remaining candidate features can be tested individually using the binary search approach proposed by VeriX+ but\nreplacing Marabou by CROWN (see Algorithm 5).",
+    "paper_id": "2603.10661",
+    "title": "FAME: Formal Abstract Minimal Explanation for Neural Networks",
+    "authors": [
+      "Ryma Boumazouza",
+      "Raya Elsaleh",
+      "Melanie Ducoffe",
+      "Shahaf Bassan",
+      "Guy Katz"
+    ],
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+    "text": "This final step ensures that we obtain a formal\nabstract minimal explanation, as defined in Definition 4.1 Algorithm 2 Recursive Abstract Batch Freeing\n1: Input: model f, input x, candidate set F\n2: Initialize: A ←∅ ▷certified free set\n3: repeat\n4: Abest ←∅\n5: for m = 1 . . . |F \\ A| do\n6: Am ←GREEDYABSTRACTBATCHFREEING(f, Ωm(x; A), F \\ A)\n7: if |Am| > |Abest| then\n8: Abest ←Am\n9: end if\n10: end for\n11: A ←A ∪Abest\n12: until Abest = ∅\n13: A = ITERATIVE SINGLETON FREE(f, x, F, A) ▷refine by testing remaining features\n14: Return: A 6 DISTANCE FROM ABSTRACT EXPLANATION TO MINIMALITY Algorithm 2 returns a minimal abstract explanation: with respect to the chosen LiRPA relaxation,\nthe certified free set A cannot be further enlarged. This guarantee is strictly weaker than minimality Published as a conference paper at ICLR 2026 The remaining features may still include irrelevant coordinates that abstract\ninterpretation fails to certify, due to the coarseness of the relaxation. In other words, minimality is\nrelative to the verifier: stronger but more expensive verifiers (e.g., Verix+ with Marabou) are still\nrequired to converge to a true minimal explanation.",
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+    "title": "FAME: Formal Abstract Minimal Explanation for Neural Networks",
+    "authors": [
+      "Ryma Boumazouza",
+      "Raya Elsaleh",
+      "Melanie Ducoffe",
+      "Shahaf Bassan",
+      "Guy Katz"
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+    "text": "We achieve this via a two-phase pipeline (Figure 1). Phase 1 (Abstract Pruning) generates a\nsound abstract explanation wAXpA⋆. Phase 2 (Exact Refinement) minimizes this candidate using\nVERIX+, ensuring the final output is guaranteed minimal. The gap arises from the tradeoff between\nverifier accuracy and domain size. Abstract methods become more conservative as the perturbation\ndomain grows, while exact methods remain sound but scale poorly. This motivates hybrid strategies\nthat combine fast but incomplete relaxations with targeted calls to exact solvers. As an additional\nacceleration step, adversarial attacks can be used. By Lemma B.1, if attacks identify features that\nmust belong to the explanation, they can be added simultaneously (see Algorithm 4). Unlike abstract\ninterpretation, the effectiveness of adversarial search typically increases with the domain size: larger\nregions make it easier to find counterexamples. Towards minimal explanations. In formal XAI, fidelity is a hard constraint guaranteed by the\nverifier. Therefore, the explanation cardinality (minimality) becomes the only metric to compare\nformal abductive explanations. A smaller explanation is strictly better, provided it remains sufficient. Our strategy is to use the minimal abstract explanation (wAXpA⋆) as a starting point, and then\nsearch for the closest minimal explanation. Concretely, we aim to identify the largest candidate\nset of potentially irrelevant features that, if freed together, would allow all remaining features to be\nsafely added to the explanation at once.",
+    "paper_id": "2603.10661",
+    "title": "FAME: Formal Abstract Minimal Explanation for Neural Networks",
+    "authors": [
+      "Ryma Boumazouza",
+      "Raya Elsaleh",
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+    "text": "A good traversal order of the candidate space is crucial\nhere, as it determines how efficiently such irrelevant features can be pinpointed. Formally, if X A\ndenotes the minimal abstract explanation and X A⋆the closest minimal explanation, we define the\nabsolute distance to minimality as the number of irrelevant features not captured by the abstract\nmethod: d(X A, X A⋆) = X A \\ X A⋆ . To evaluate the benefits and reliability of our proposed explainability method, FAME, we performed\na series of experiments comparing its performance against the SoTA VERIX+ implementation. We assessed the quality of the explanations generated by FAME by comparing them to those of\nVERIX+ across four distinct models, including both fully connected and convolutional neural networks (CNNs). We considered two primary performance metrics: the runtime required to compute\na single explanation and the size (cardinality) of the resulting explanation. Our experiments, as in VERIX+ (Wu et al., 2024b), were conducted on two widely-used image\nclassification datasets: MNIST (Yann, 2010) and GTSRB (Stallkamp et al., 2012). Each score was\naveraged over non-robust samples from the 100 samples of each dataset.",
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+    "authors": [
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+      "Raya Elsaleh",
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+    "text": "For the comparison results,\nthe explanations were generated using the FAME framework only, and with a final run of VERIX+\nto ensure minimality (See Figure 1). VERIX+ (alone) FAME: Single-round FAME: Iterative refinement FAME-accelerated VERIX+\nTraversal order bounds / / / + bounds\nSearch procedure binary MILP Greedy MILP Greedy Greedy + binary\nMetrics ↓ |AXp| time |wAXpA| time |wAXpA| time |wAXpA| time |wAXpA| time ∥candidate-set∥ |AXp| time MNIST-FC 280.16 13.87 441.05 4.4 448.37 0.35 229.73 14.30 225.14 8.78 44.21 224.41 13.72\nMNIST-CNN 159.78 56.72 181.24 5.59 190.29 0.51 124.9 12.35 122.09 5.6 104.09 113.53 33.75\nGTSRB-FC 313.42 56.18 236.85 9.68 243.18 0.97 331.84 12.28 332.74 5.26 11.93 332.66 9.26\nGTSRB-CNN 338.28 185.03 372.66 12.45 379.34 1.35 322.42 17.63 322.42 7.42 219.57 322.42 138.12 Table 1: Average explanation size and generation time (in seconds) are compared for FAME (singleround and iterative MILP/Greedy) with FAME-accelerated VERIX+ to achieve minimality. Experimental Setup All experiments were carried out on a machine equipped with an Apple M2\nPro processor and 16 GB of memory. The analysis is conducted on fully connected (-FC) and\nconvolutional (-CNN) models from the MNIST and GTSRB datasets, with ϵ set to 0.05 and 0.01\nrespectively. The verified perturbation analysis was performed using the DECOMON library2, ap- 2https://github.com/airbus/decomon Published as a conference paper at ICLR 2026 Figure 2: FAME's iterative refinement approach against the VERIX+ baseline. The left plot\ncompares the size of the final explanations. The right plot compares the runtime (in seconds). The\ndata points for each model are distinguished by color, and the use of circles (card=True) and squares\n(card=False) indicates whether a cardinality constraint (||x −x′||0 ≤m) was applied. plying the CROWN method with an l∞-norm. The NN verifier Marabou (Katz et al., 2019) is used\nwithin VERIX+.",
+    "paper_id": "2603.10661",
+    "title": "FAME: Formal Abstract Minimal Explanation for Neural Networks",
+    "authors": [
+      "Ryma Boumazouza",
+      "Raya Elsaleh",
+      "Melanie Ducoffe",
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+    "text": "We included a sensitivity analysis covering: (1) Solver Choice, confirming the\nGreedy heuristic's near-optimality vs. MILP (Table 1); (2) Cardinality Constraints, showing that\ncard=True yields significantly smaller explanations (Figure 2); and (3) Perturbation Magnitude (ϵ),\nwhich we fixed to baseline used by VERIX+ for direct comparison. We include additional experimental results on the ResNet-2B architecture (CIFAR-10) from the VNN-COMP benchmark (Wang\net al., 2021) to demonstrate scalability on deeper models. The complete set of hyperparameters and\nthe detailed architectures of the models used are provided in Appendix E for full reproducibility.",
+    "paper_id": "2603.10661",
+    "title": "FAME: Formal Abstract Minimal Explanation for Neural Networks",
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+    "text": "MILP FOR ABSTRACT BATCH FREEING Performance in a Single Round This experiment, in the 'FAME: Single Round' column of Table\n1, compares the runtime and size of the largest free set obtained in a single round using the greedy\nmethod versus an exact MILP solver for the abstract batch freeing (Algorithm 1). Across all models, the greedy heuristic consistently provided a significant speedup (ranging from\n9× to 12×) while achieving an abstract explanation size very close (fewer than 9 features in average)\nto that of the optimal MILP solver. This demonstrates that, for single-round batch freeing, the greedy\nmethod offers a more practical and scalable solution. Performance with Iterative Refinement This experiment compares the two methods in an iterative\nsetting of the abstract batch freeing, where the perturbation domain is progressively refined (Section 5). For the iterative refinement process, the greedy approach maintained a substantial runtime\nadvantage over the MILP solver, with a speedup up to 2.4× on the GTSRB-CNN model, while\nproducing abstract explanations that were consistently close in size to the optimal solution. The\ndistinction between the circle and square markers is significant in Figure 2. The square markers\n(card=False) tend to lie closer to or even above the diagonal line. This suggests that the cardinalityconstrained domain, when successful, is highly effective at finding more compact explanations. Impact of Iterative Refinement: Comparing 'FAME: Single-round' vs. 'FAME: Iterative refinement' in Table 1 isolates the impact of Algorithm 2. For MNIST-CNN, iterative refinement reduces\nexplanation size by 36% (190.29 to 122.09). This highlights the trade-off: a modest increase in\nruntime yields significantly more compact explanations. 7.2 COMPARISON WITH STATE-OF-THE-ART (VERIX+) We compare in this section the results of VERIX+ (alone) vs. FAME-accelerated VERIX+.",
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+    "text": "Explanation Size and Runtime: FAME consistently produces smaller explanations than VERIX+\nwhile being significantly faster, mainly due to FAME's iterative refinement approach, as visually\nconfirmed by the plots in Figure 2 that show a majority of data points falling below the diagonal line Published as a conference paper at ICLR 2026 for both size and time comparisons. The runtime gains are particularly substantial for the GTSRB\nmodels (green and red markers), where FAME's runtime is often only a small fraction of VERIX+'s\nas shown in Table 1. In some cases, FAME delivers a non-minimal set that is smaller than VERIX+\n's minimal set, with up to a 25× speedup (322.42 features in 7.4s compared to 338.28 in 185.03s\nfor the GTSRB-CNN model) while producing wAXpA that were consistently close in size to the\noptimal solution. The Role of Abstract Freeing: The effectiveness of FAME's approach is further supported by the\n\"distance to minimality\" metric. The average distance to minimality was 44.21 for MNIST-FC and\n104.09 for MNIST-CNN. An important observation from our experiments is that when the abstract\ndomains in FAME are effective, they yield abstract abductive explanations wAXpA that are smaller\nthan the abductive explanations (AXp) from VERIX+. This is not immediately obvious from the\nsummary table, as the final explanations may differ.",
+    "paper_id": "2603.10661",
+    "title": "FAME: Formal Abstract Minimal Explanation for Neural Networks",
+    "authors": [
+      "Ryma Boumazouza",
+      "Raya Elsaleh",
+      "Melanie Ducoffe",
+      "Shahaf Bassan",
+      "Guy Katz"
+    ],
+    "published_date": "2026-03-11",
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+    "chunk_id": "e3af1c44-06f6-427e-b422-b45a1ee41f4e",
+    "text": "Conversely, when FAME's abstract domains\nfail to find a valid free set, our method defaults to a binary search approach similar to VERIX+. However, since we do not use the Marabou solver in this phase, the resulting wAXpA is larger than\nthe AXp provided by Marabou. This highlights the trade-off and the hybrid nature of our approach. Finally, to demonstrate the generality of our framework beyond standard benchmarks, in Appendix\nF we provide additional experiments on the ResNet-2B architecture Wang et al. (2021) trained on\nCIFAR-10. These results represent, to the best of our knowledge, the first formal explanations generated for such a complex architecture, highlighting FAME as an enabling technology for scalability. 8 CONCLUSION AND DISCUSSION In this work, we introduced FAME (Formal Abstract Minimal Explanations), a novel framework for\ncomputing abductive explanations that effectively scales to large neural networks. By leveraging a\nhybrid strategy grounded in abstract interpretation and dedicated perturbation domains, we successfully addressed the long-standing sequential bottleneck of traditional formal explanation methods. Our main contribution is a new approach that eliminates the need for traversal order by progressively\nshrinking dedicated perturbation domains and using LiRPA-based bounds to efficiently discard irrelevant features. The core of our method relies on a greedy heuristic for batch freeing that, as our\nanalysis shows, is significantly faster than an exact MILP solver while yielding comparable explanation sizes. Our experimental results demonstrate that the full hybrid FAME pipeline outperforms the current\nstate-of-the-art VERIX+ baseline, providing a superior trade-off between computation time and\nexplanation quality. We consistently observed significant reductions in runtime while producing\nexplanations that are close to true minimality. This success highlights the feasibility of computing\nformal explanations for larger models and validates the effectiveness of our hybrid strategy. Beyond its performance benefits, the FAME framework is highly generalizable. Although our evaluation focused on classification tasks, the framework can be extended to other machine learning\napplications, such as regression.",
+    "paper_id": "2603.10661",
+    "title": "FAME: Formal Abstract Minimal Explanation for Neural Networks",
+    "authors": [
+      "Ryma Boumazouza",
+      "Raya Elsaleh",
+      "Melanie Ducoffe",
+      "Shahaf Bassan",
+      "Guy Katz"
+    ],
+    "published_date": "2026-03-11",
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+    "text": "While we focused on robustness in continuous domains, FAME's\nhigh-level algorithms (batch certificate, greedy selection) support discrete features (see Appendix\nB). LiRPA natively handles discrete variables (e.g., one-hot encodings) via contiguous interval\nbounds. Furthermore, the framework can support other properties like local stability. Additionally,\nFAME can be configured to use exact solvers for the final refinement step, ensuring its adaptability\nand robustness for various use cases. Finally, we demonstrated FAME's scalability on the ResNet-2B (CIFAR-10) architecture. Although\nthe abstraction gap naturally widens with depth, FAME's ability to rapidly prune irrelevant features\nestablishes it as a critical enabling step for applying formal XAI to complex models where exact-only\nmethods are currently intractable. By designing a framework that natively leverages certificates from\nmodern, GPU-enabled verifiers, this work effectively bridges the gap between formal guarantees and\npractical scalability. Published as a conference paper at ICLR 2026 Our work has benefited from the AI Cluster ANITI and the research program DEEL.3 ANITI is\nfunded by the France 2030 program under the Grant agreement n°ANR-23-IACL-0002. DEEL is an\nintegrative program of the AI Cluster ANITI, designed and operated jointly with IRT Saint Exup´ery,\nwith the financial support from its industrial and academic partners and the France 2030 program under the Grant agreement n°ANR-10-AIRT-01. Within the DEEL program, we are especially grateful\nto Franck MAMALET for their constant encouragement, valuable discussions, and insightful feedback throughout the development of this work.",
+    "paper_id": "2603.10661",
+    "title": "FAME: Formal Abstract Minimal Explanation for Neural Networks",
+    "authors": [
+      "Ryma Boumazouza",
+      "Raya Elsaleh",
+      "Melanie Ducoffe",
+      "Shahaf Bassan",
+      "Guy Katz"
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+    "text": "The work of Elsaleh, Bassan, and Katz was partially\nfunded by the European Union (ERC, VeriDeL, 101112713). Views and opinions expressed are\nhowever those of the author(s) only and do not necessarily reflect those of the European Union or\nthe European Research Council Executive Agency. Neither the European Union nor the granting\nauthority can be held responsible for them. The work of Elsaleh, Bassan, and Katz was additionally\nsupported by a grant from the Israeli Science Foundation (grant number 558/24). Elsaleh is also\nsupported by the Ariane de Rothschild Women Doctoral Program. Gilles Audemard, Steve Bellart, Louenas Bounia, Fr´ed´eric Koriche, Jean-Marie Lagniez, and Pierre\nMarquis. Trading complexity for sparsity in random forest explanations. In Proceedings of the\nAAAI Conference on Artificial Intelligence, volume 36, pp. 5461–5469, 2022. Bassan, Yizhak Yisrael Elboher, Tobias Ladner, Matthias Althoff, and Guy Katz. Explaining,\nFast and Slow: Abstraction and Refinement of Provable Explanations. Conf.\non Machine Learning (ICML), 2025a. Shahaf Bassan and Guy Katz. Towards formal xai: formally approximate minimal explanations of\nneural networks. In International Conference on Tools and Algorithms for the Construction and\nAnalysis of Systems, pp. 187–207. Shahaf Bassan, Guy Amir, Davide Corsi, Idan Refaeli, and Guy Katz.",
+    "paper_id": "2603.10661",
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+    "text": "On guaranteed optimal robust explanations for nlp models. In Zhi-Hua Zhou (ed.),\nProceedings of the Thirtieth International Joint Conference on Artificial Intelligence, IJCAI-21,\npp. 2658–2665. International Joint Conferences on Artificial Intelligence Organization, 8 2021.\ndoi: 10.24963/ijcai.2021/366. URL https://doi.org/10.24963/ijcai.2021/366. Minimal unsatisfiability: Models, algorithms and applications (invited paper). In 40th IEEE International Symposium on Multiple-Valued Logic, ISMVL 2010, Barcelona, Spain,\n26-28 May 2010, pp. 9–14. IEEE Computer Society, 2010. doi: 10.1109/ISMVL.2010.11. URL\nhttps://doi.org/10.1109/ISMVL.2010.11. Disproving xai myths with formal methods–initial results. In 2023 27th International Conference on Engineering of Complex Computer Systems (ICECCS), pp. 12–21. Logic-based explainability in machine learning. Causality, Explanations and Declarative Knowledge: 18th International Summer School 2022, Berlin,\nGermany, September 27–30, 2022, Tutorial Lectures, pp. 24–104. Joao Marques-Silva, Thomas Gerspacher, Martin Cooper, Alexey Ignatiev, and Nina Narodytska.",
+    "paper_id": "2603.10661",
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+    "text": "Explaining naive bayes and other linear classifiers with polynomial time and delay. Advances in\nNeural Information Processing Systems, 33:20590–20600, 2020. Joao Marques-Silva, Thomas Gerspacher, Martin C Cooper, Alexey Ignatiev, and Nina Narodytska. Explanations for monotonic classifiers. In International Conference on Machine Learning, pp.\n7469–7479. Mathieu Serrurier, Franck Mamalet, Alberto Gonz´alez-Sanz, Thibaut Boissin, Jean-Michel Loubes,\nand Eustasio Del Barrio. Achieving robustness in classification using optimal transport with hinge\nregularization. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern\nRecognition, 2021. Andy Shih, Arthur Choi, and Adnan Darwiche.",
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+    "text": "A symbolic approach to explaining bayesian network classifiers. In Proceedings of the 27th International Joint Conference on Artificial Intelligence, IJCAI'18, pp. 5103–5111. Published as a conference paper at ICLR 2026 Gagandeep Singh, Timon Gehr, Markus P¨uschel, and Martin Vechev.",
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+    "text": "An abstract domain for certifying neural networks. Proceedings of the ACM on Programming Languages, 3(POPL):1–30,\n2019. Johannes Stallkamp, Marc Schlipsing, Jan Salmen, and Christian Igel. Man vs. computer: Benchmarking machine learning algorithms for traffic sign recognition. Neural networks, 32:323–332,\n2012. Caterina Urban and Antoine Min´e.",
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+    "text": "A review of formal methods applied to machine learning. ArXiv, abs/2104.02466, 2021. URL https://api.semanticscholar.org/\nCorpusID:233033440. Shiqi Wang, Huan Zhang, Kaidi Xu, Xue Lin, Suman Sekhar Jana, Cho-Jui Hsieh, and Zico Kolter.",
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+    "text": "Better verified explanations with applications\nto incorrectness and out-of-distribution detection. CoRR, abs/2409.03060, 2024b. doi: 10.48550/\nARXIV.2409.03060. URL https://doi.org/10.48550/arXiv.2409.03060. Kaidi Xu, Zhouxing Shi, Huan Zhang, Yihan Wang, Kai-Wei Chang, Minlie Huang, Bhavya\nKailkhura, Xue Lin, and Cho-Jui Hsieh.",
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+    "text": "Automatic perturbation analysis for scalable certified\nrobustness and beyond. Advances in Neural Information Processing Systems, 33:1129–1141,\n2020. Mnist handwritten digit database. Huan Zhang, Tsui-Wei Weng, Pin-Yu Chen, Cho-Jui Hsieh, and Luca Daniel. Efficient neural\nnetwork robustness certification with general activation functions. Advances in neural information\nprocessing systems, 31, 2018. Zhe Zhao, Yedi Zhang, Guangke Chen, Fu Song, Taolue Chen, and Jiaxiang Liu.",
+    "paper_id": "2603.10661",
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+    "authors": [
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+    "text": "Cleverest: accelerating cegar-based neural network verification via adversarial attacks. In International Static\nAnalysis Symposium, pp. 449–473. Published as a conference paper at ICLR 2026 The appendix collects proofs, model specifications, and supplementary experimental results that\nsupport the main paper. Appendix A contains additional background on formal verification terminology, Abstract Interpretation, and LiRPA. Appendix B contains the complete proofs of all propositions. Appendix C provides the pseudocode for the FAME algorithms and the associated baselines. Appendix D provides illustrative examples of abductive explanations and the greedy knapsack formulation. Appendix E provides specifications of the datasets and architectures used, along with supplementary experimental results. Appendix F details the scalability analysis on complex architectures (ResNet-2B on CIFAR-10). Appendix G provides the LLM usage disclosure.",
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+    "text": "A BACKGROUND ON FORMAL VERIFICATION A.1 ABSTRACT INTERPRETATION Abstract Interpretation is a theory of sound approximation of the semantics of computer programs. In the context of neural networks, it allows us to compute over-approximations of the network's\noutput range without executing the network on every single point in the input domain (which is\ninfinite). While exact verification methods (like MILP solvers) provide precise results, they are generally\nNP-hard and do not scale to large networks. Abstract interpretation trades precision for scalability\n(typically polynomial time) by operating on abstract domains (e.g., intervals, zonotopes, or polyhedra) rather than concrete values. A.2 LIRPA (LINEAR RELAXATION-BASED PERTURBATION ANALYSIS) LiRPA (Linear Relaxation-based Perturbation Analysis) is a specific, efficient instance of abstract\ninterpretation designed for neural networks. Instead of propagating simple intervals (which become\ntoo loose/imprecise in deep networks), LiRPA propagates linear constraints. For every neuron xj, it\ncomputes two linear bounds relative to the input x: wTj x + bj ≤fj(x) ≤wTj x + bj These linear bounds allow us to rigorously bound the \"worst-case\" behavior of the network much\nmore tightly than simple intervals. If the lower bound of the correct class minus the upper bound of\nthe target class is positive, we have a mathematically sound certificate of robustness. Illustrative Example: Consider a nominal input image ¯x from the MNIST dataset depicting the\ndigit '7'. In a standard local robustness verification task, we define the input domain Ω(¯x) as an\nl∞-norm ball with a radius of ϵ = 0.05.",
+    "paper_id": "2603.10661",
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+    "authors": [
+      "Ryma Boumazouza",
+      "Raya Elsaleh",
+      "Melanie Ducoffe",
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+    "text": "This implies that each pixel xi in the image is permitted to\nvary independently within the interval [¯xi −0.05, ¯xi + 0.05]. The verification objective is to prove that the property P holds: specifically, that for every possible\nperturbed image x ∈Ω(¯x), the network's output logit for the ground-truth class ('7') remains strictly\ngreater than the logit for any target class k (e.g., '1'). In the context of LiRPA, this is verified by\ncomputing a sound lower bound for the correct class (f 7) and a sound upper bound for the competing\nclass (f 1). If the verified margin f 7 −f 1 > 0, the network is guaranteed to be robust against all\nperturbations in Ω(¯x). Published as a conference paper at ICLR 2026 A.3 VERIFICATION TERMINOLOGY We formulate the check for explanation sufficiency as a constraint satisfaction problem. A query\nis SAT if a valid perturbation (counter-example) exists, and UNSAT if no such perturbation exists\n(meaning the explanation is valid). • Soundness (No False Positives): A verifier is sound if it guarantees that any certified property is truly holds. In Abstract Interpretation, soundness is achieved because the computed\nabstract bounds strictly enclose the true concrete values. If these conservative bounds satisfy the property (UNSAT), the actual network must also satisfy it.",
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+    "authors": [
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+    "text": "• Completeness (No False Negatives): A verifier is complete if it is capable of certifying any\nvalid explanation. Exact solvers (like MILP) are complete. In contrast, Abstract Interpretation is incomplete: due to over-approximation, the bounds may be too loose to prove a\ntrue property, leading to a \"don't know\" state where the explanation is valid, but the verifier\ncannot prove it. THE ASYMMETRY OF PARALLEL FEATURE SELECTION Proposition B.1 (Simultaneous Addition). Any number of essential features can be added to the\nexplanation simultaneously. This property allows us to leverage solvers capable of assessing multiple verification queries in parallel, leading to a substantial reduction in runtime. (a) Adding several features at once is sound. (b) Freeing several features at once is unsound. Figure 3: Toy example illustrating the asymmetry between adding and freeing features. Simultaneous Addition B.1. Let X be the current explanation candidate, and let R = {r1, . . . , rk}\nbe a set of features not in X. If, for every ri ∈R, removing the single feature ri from the set\nF \\ (X ∪{ri}) produces a counterexample, then all features in R are necessary and can be added\nto the explanation at once. Simultaneous freeing 4.1. If removing any feature from a set R ⊆F \\ X individually causes the\nexplanation to fail (i.e., produces a counterexample), then all features in R can be added to the\nexplanation X simultaneously. Batch-Certifiable Freeing 4.2.",
+    "paper_id": "2603.10661",
+    "title": "FAME: Formal Abstract Minimal Explanation for Neural Networks",
+    "authors": [
+      "Ryma Boumazouza",
+      "Raya Elsaleh",
+      "Melanie Ducoffe",
+      "Shahaf Bassan",
+      "Guy Katz"
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+    "text": "For any i ̸= c and x′ ∈Ω(x), lirpa bounds give fi(x′) −fc(x′) ≤\nb i(x) + Pj∈A ∆i,j(x′) with ∆i,j(x′) ≤ci,j. Taking the worst case over x′ and i yields fi(x′) −\nfc(x′) ≤Φ(A) ≤0, precluding a label flip. Published as a conference paper at ICLR 2026 PROPOSITION (CORRECTNESS OF THE RECURSIVE PROCEDURE) Let A be the set returned by Algorithm 2 augmented with the final singleton refinement step that\ntests each remaining feature individually with the LiRPA certificate Φ(·). (i) (No singleton extension) For every feature j ∈F \\ A we have i.e. no single feature can be added to A while preserving the certificate. Hence A is\nsingleton-maximal with respect to the LiRPA certificate. (ii) (Termination) Algorithm 2 terminates in at most |F| outer iterations (and finitely many\ninner steps). (iii) (Full abstract minimality — conditional) If the inner batch solver called by Algorithm 2\nreturns, for each tested budget p, a globally optimal certified free set (i.e., for the current\ndomain it finds a maximum-cardinality Ap satisfying Φ(Ap) ≤0), then the final A is a\nglobally maximal certified free set: there is no A′ ⊋A with Φ(A′) ≤0. In this case A is\na true minimal abstract explanation (with respect to the chosen LiRPA relaxation). Proof. (i) No singleton extension. By construction, the algorithm performs a final singleton refinement: it tests every feature j ∈F \\ A by evaluating the certificate on A ∪{j}. The algorithm\nonly adds j to A if Φ(A ∪{j}) ≤0. Since the refinement ends with no further additions, it follows\nthat for every remaining j we have Φ(A ∪{j}) > 0.",
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+    "authors": [
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+      "Melanie Ducoffe",
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+    "text": "Each time the algorithm adds at least one feature to A, the cardinality |A| strictly\nincreases and cannot exceed |F|. The outer loop therefore performs at most |F| successful additions. If an outer iteration yields no new features, the loop stops. Inner loops (scanning budgets\np or performing singleton checks) are finite since they iterate over finite sets. Hence the algorithm\nterminates in finite time. (iii) Full abstract minimality under optimal inner solver. Suppose that for every domain tested,\nthe inner routine (called for each p) returns a certified free set of maximum possible cardinality\namong all subsets that satisfy Φ(·) ≤0 on that domain. During each outer iteration the algorithm\nenumerates budgets p (or otherwise explores the space of allowed cardinalities) and selects the\nlargest Ap found; then A is augmented by that largest globally-feasible batch. If no nonempty\nglobally-feasible batch exists for any tested p, then no superset of the current A can be certified\n(because any superset would have some cardinality p′ tested and the solver would have returned it). After the final singleton checks (which also use the optimal verifier on singletons), there remains no\nsingle feature that can be added. Combining these facts yields that no superset of A is certifiable,\ni.e. A is a globally maximal certified free set, as claimed. Abstract Minimal Explanation Correctness of Iterative Singleton Freeing. Let F be the candidate feature set and let A0 ⊆F be\nan initial free set such that the LiRPA certificate verifies A0 (i.e. Φ(A0) ≤0).",
+    "paper_id": "2603.10661",
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+    "authors": [
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+    "text": "Run the Iterative\nSingleton Freeing procedure (Algorithm 5) with traversal order π. The algorithm returns a set A\nwith the following properties: 1. (Soundness) The final set A satisfies Φ(A) ≤0 (every added singleton was certified). 2. (Termination) The algorithm terminates after at most |F| −|A0| successful additions\n(hence in finite time). 3. (Singleton-maximality) For every j ∈F \\ A we have Φ(A ∪{j}) > 0, i.e. no remaining\nsingle feature can be certified as free. Published as a conference paper at ICLR 2026 Soundness (invariant). By assumption Φ(A0) ≤0. The algorithm only appends a feature\ni to the current free set after a LiRPA call returns success on A ∪{i}, i.e. Φ(A ∪{i}) ≤0. Since\nLiRPA certificates are sound, every update preserves the invariant \"current A is certified\". Therefore\nthe final A satisfies Φ(A) ≤0. Each successful iteration increases |A| by one and |A| ≤|F|. Thus there can be at\nmost |F| −|A0| successful additions.",
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+    "text": "The algorithm halts when a full scan yields no addition; since\nscans iterate over a finite set ordered by π, the procedure terminates in finite time. Singleton-maximality. Assume by contradiction that after termination there exists j ∈F \\ A with\nΦ(A ∪{j}) ≤0. The final scan that caused termination necessarily tested j (traversal order covers\nall remaining indices), so the algorithm would have added j, contradicting termination. Hence for\nevery j ∈F \\ A we must have Φ(A ∪{j}) > 0, proving singleton-maximality. Worked counterexample (illustrating joint freeing). Consider a toy binary classifier with two\ninput features x1, x2 and property P: the label remains class 0 iff f0(x′) −f1(x′) ≥0. Suppose the\nLiRPA relaxation yields conservative linear contributions such that b + c1 > 0, b + c2 > 0, but b + c1 + c2 ≤0, where ci is the worst-case contribution of feature i and b is the baseline margin.",
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+    "title": "FAME: Formal Abstract Minimal Explanation for Neural Networks",
+    "authors": [
+      "Ryma Boumazouza",
+      "Raya Elsaleh",
+      "Melanie Ducoffe",
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+    "text": "Then neither\nsingleton {1} nor {2} is certifiable (each violates the certificate), but the joint set {1, 2} is certifiable. The iterative singleton procedure terminates without adding either feature, while a batch routine (or\nan optimal MKP solver) would free both. This demonstrates the algorithm's limitation: it guarantees\nonly singleton-maximality, not global maximality over multi-feature batches. Complexity and practical cost. In the worst case the algorithm may attempt a\nLiRPA call for every remaining feature on each outer iteration. If r features are eventually added,\nthe total number of LiRPA calls is bounded by −1) + 1) (n) + (n −1) + · · · + (n −r + 1) = r · n −r(r ≤n(n = O(n2).\n2 2\nThus worst-case LiRPA call complexity is quadratic in n. In practice, however, each successful addition reduces the candidate set and often many iterations terminate early; empirical behavior tends\nto be much closer to linear in n for structured data because (i) many features are certified in early\npasses and (ii) LiRPA calls are highly parallelizable across features and can exploit GPU acceleration. Finally, the dominant runtime factor is the per-call cost of LiRPA (forward/backward bound\npropagation); therefore hybrid strategies (batch pre-filtering, prioritized traversal orders, occasional\nexact-solver checks on promising subsets) are useful to reduce the number of expensive LiRPA\nevaluations. FAME FOR DISCRETE DATA FAME, as presented, uses LiRPA, which is designed for continuous ( ) domains. A discrete feature j\nwith admissible values in a finite set Sj can be incorporated by specifying an interval domain, which\nis the standard abstraction used in LiRPA-based verification. Consequently, FAME allows a discrete feature to vary over its admissible values. LiRPA supports\nthis by assigning\nx′j ∈[min Sj, max Sj],\nor, if only a subset S′j ⊆Sj is permitted,\nx′j ∈[min S′j, max S′j], provided that the values form a contiguous range.",
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+    "text": "If a feature belongs to the explanation, it is fixed to its nominal value, which corresponds to assigning\nthe zero-width interval [xj, xj]. Note that freeing a feature to a non-contiguous set (e.g., allowing {1, 4} but excluding {2, 3}) cannot\nbe represented exactly, since LiRPA abstractions are convex intervals. Extending LiRPA to arbitrary\nfinite non-convex domains is left for future work. In practice, such cases are rare: when categorical Published as a conference paper at ICLR 2026 values have no meaningful numeric ordering, one-hot encodings are standard, and each coordinate\nbecomes a binary {0, 1} feature naturally supported by interval domains.",
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+    "text": "Since FAME only requires sound per-feature lower and upper bounds, all its components, including the batch certificate Φ(A) and the refinement steps, apply directly to discrete and categorical\nfeatures. This appendix details the algorithmic procedures supporting the FAME framework and its baselines. We present four key algorithms: • Algorithm 3 (BINARYSEARCH): An enhanced version of the binary search traversal\nstrategy used in Verix+. It employs a divide-and-conquer approach to identify irrelevant\nfeatures, accepting a generic verification oracle (e.g., Marabou or LiRPA) as an input parameter. • Algorithm 4 (Simultaneous Add): An acceleration heuristic that uses adversarial attacks\nto quickly identify necessary features. By checking if relaxing a specific feature immediately leads to a counterexample via attacks (e.g., PGD), we can efficiently add necessary\nfeatures to the explanation without expensive verification calls. • Algorithm 5 (Iterative Singleton Freeing): A refinement procedure that iterates sequentially through the remaining candidate features. It utilizes LiRPA certificates to check if\nindividual features can be safely freed, serving as a final cleanup step for features that\ncould not be certified in batches. • Algorithm 5 (Recursive Abstract Batch Freeing): The core recursive loop of our framework. It iteratively tightens the perturbation domain using cardinality constraints (varying\nm) and invokes the greedy batch-freeing heuristic to maximize the size of the abstract explanation, concluding with a singleton refinement step. In this enhanced BINARYSEARCH algorithm, the solver (e.g., Marabou or Lirpa) is passed as an\nexplicit parameter to enable the CHECK function, which performs the core verification queries. Published as a conference paper at ICLR 2026",
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+    "text": "Algorithm 3 BINARYSEARCH(f, xΘ, solver)\n1: function BINARYSEARCH(f, xΘ, solver)\n2: if |xΘ| = 1 then\n3: if CHECK(f, xB ∪xΘ, solver) then\n4: xB ←xB ∪xΘ\n5: return\n6: else\n7: xA ←xA ∪xΘ\n8: return\n9: end if\n10: end if\n11: xΦ, xΨ = split(xΘ, 2)\n12: if CHECK(f, xB ∪xΦ, solver) then\n13: xB ←xB ∪xΦ\n14: if CHECK(f, xB ∪xΨ, solver) then\n15: xB ←xB ∪xΨ\n16: else\n17: if |xΨ| = 1 then\n18: xA ←xA ∪xΨ\n19: else\n20: BINARYSEARCH(f, xΨ, solver)\n21: end if\n22: end if\n23: else\n24: if |xΦ| = 1 then\n25: xA ←xA ∪xΦ\n26: else\n27: BINARYSEARCH(f, xΦ, solver)\n28: end if\n29: end if\n30: end function Algorithm 4 Simultaneous Add\n1: Input: model f, input x, candidate set F, current free set A, adversarial procedure ATTACK(,)\nproperty P\n2: Initialize: E ←∅ ▷set of necessary features\n3: for i ∈F \\ A do\n4: F′ ←F \\ {i}\n5: if ATTACK(f, Ω(x, F′), P) succeeds then\n6: E ←E ∪{i} ▷i must remain fixed\n7: end if\n8: end for\n9: Return: E Published as a conference paper at ICLR 2026 C.3 ITERATIVE SINGLETON FREEING Algorithm 5 Iterative Singleton Free\n1: Input: model f, input x, candidate set F, free set A, certificate method LIRPA(,) traversal\norder π, property P\n2: repeat\n3: found ←false\n4: for i ∈π with i ∈F \\ A do\n5: if LIRPA(f, Ω(x, A ∪{i}), P) succeeds then\n6: A ←A ∪{i}\n7: found ←true\n8: break ▷restart scan from beginning of π\n9: end if\n10: end for\n11: until found = false\n12: Return: A C.4 RECURSIVE SIMULTANEOUS FREE Algorithm 6 Recursive Abstract Batch Freeing\n1: Input: model f, input x, candidate set F\n2: Initialize: A ←∅ ▷certified free set\n3: repeat\n4: Abest ←∅\n5: for m = 1 . . . |F \\ A| do\n6: Am ←GREEDYABSTRACTBATCHFREEING(f, Ωm(x; A), F \\ A)\n7: if |Am| > |Abest| then\n8: Abest ←Am\n9: end if\n10: end for\n11: A ←A ∪Abest\n12: until Abest = ∅\n13: A = ITERATIVE SINGLETON FREE(f, x, F, A) ▷refine by testing remaining features\n14: Return: A D.1 ILLUSTRATION OF ABDUCTIVE EXPLANATION Figure 4 illustrates a 3D classification task. For the starred sample, we seek an explanation for\nits classification within a local cube-shaped domain. As shown in Figure 5, fixing only feature x2\n(i.e. freeing {x1, x3}, restricting perturbations to the orange plane) is not enough to guarantee the\nproperty, since a counterexample exists. However, fixing both x2 and x3 (orange line on free x1)\ndefines a 'safe' subdomain where the desired property holds true, since no counterexample exists in\nthat subdomain. Therefore, X = {x2, x3} is an abductive explanation. Since neither {x2} nor {x3}\nare explanations on their own, {x2, x3} is minimal. But it is not minimum since X = {x1} is also\na minimal abductive explanation with a smaller cardinality.",
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+      "Ryma Boumazouza",
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+    "text": "Two special cases are worth noting: an\nempty explanation (all features are irrelevant) and a full explanation (the entire input is necessary). If all features are irrelevant, the explanation is the empty set, and no valid explanation exists. Conversely, if perturbing any feature in the input x changes the prediction, the entire input must be fixed,\nmaking the full feature set the explanation.",
+    "paper_id": "2603.10661",
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+      "Ryma Boumazouza",
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+    "text": "Published as a conference paper at ICLR 2026 Figure 4: A 3D classification task. Figure 5: AXps with different properties. D.2 ILLUSTRATION OF THE KNAPSACK FORMULATION This is an example demonstrating how the greedy heuristic described in Algorithm 1 works. Given\na multi-class classification problem with three classes: 0, 1, and 2. The model correctly predicts\nclass 0 for a given input. We want to free features from the irrelevant set A based on the abstract\nbatch certificate. We have three candidate features to free: j1, j2, and j3.",
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+    "text": "The baseline budgets for\nthe non-ground-truth classes are: • Class 1: −b = 10\n• Class 2: −b = 20 The normalized costs for each feature are calculated as ci,j/(−b i): Table 2: Example of Greedy Heuristic Decision Making\nFeature Normalized Cost for Class 1 Normalized Cost for Class 2 Maximum Normalized Cost\n(j) (c1,j/(−b 1)) (c2,j/(−b 2)) (maxi)\nj1 2/10 = 0.2 8/20 = 0.4 0.4\nj2 7/10 = 0.7 4/20 = 0.2 0.7\nj3 3/10 = 0.3 3/20 = 0.15 0.3 The algorithm's objective is to minimize the maximum normalized cost across all non-ground-truth\nclasses. As shown in the table, the minimum value in the \"Maximum Normalized Cost\" column is\n0.3, which corresponds to feature j3. Therefore, the greedy heuristic selects feature j3 to be added\nto the free set in this step, as it represents the safest choice.",
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+    "text": "E.1 MODEL SPECIFICATION We evaluated our framework on standard image benchmarks including the MNISTYann (2010) and\nGTSRBStallkamp et al. (2012) datasets. We used both fully connected and convolutional models\ntrained in a prior state-of-the-art VERIX+Wu et al. (2024b) to perform our analysis. The MNIST dataset consists of 28 × 28 × 1 grayscale handwritten images. The architectures of\nthe fully connected and convolutional neural networks trained on this dataset are detailed in Table\n3 and Table 4, respectively. These models achieved prediction accuracies of 93.76% for the fully\nconnected model and 96.29% for the convolutional model. The GTSRB dataset contains colored images of traffic signs with a shape of 32×32×3 and includes\n43 distinct categories. In the models used for our experiments, which were trained by the authors\nof VERIX+, only the 10 most frequent categories were used to mitigate potential distribution shift\nand obtain higher prediction accuracies.",
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+    "text": "The architectures of the fully connected and convolutional Published as a conference paper at ICLR 2026 Table 3: Architecture of the MNIST-FC model. Layer Type Input Shape Output Shape Activation\nFlatten 28 × 28 × 1 784 -\nFully Connected 784 10 ReLU\nFully Connected 10 10 ReLU\nOutput 10 10 - Table 4: Architecture of the MNIST-CNN model. Layer Type Input Shape Output Shape Activation\nConvolution 2D 28 × 28 × 1 13 × 13 × 4 -\nConvolution 2D 13 × 13 × 4 6 × 6 × 4 -\nFlatten 6 × 6 × 4 144 -\nFully Connected 144 20 ReLU\nOutput 20 10 - models trained on GTSRB are presented in Table 5 and Table 6, respectively. These networks\nachieved prediction accuracies of 85.93% and 90.32%, respectively. Table 5: Architecture of the GTSRB-FC model. Layer Type Input Shape Output Shape Activation\nFlatten 32 × 32 × 3 3072 -\nFully Connected 3072 10 ReLU\nFully Connected 10 10 ReLU\nOutput 10 10 - Table 6: Architecture of the GTSRB-CNN model. Layer Type Input Shape Output Shape Activation\nConvolution 2D 32 × 32 × 3 15 × 15 × 4 -\nConvolution 2D 15 × 15 × 4 7 × 7 × 4 -\nFlatten 7 × 7 × 4 196 -\nFully Connected 196 20 ReLU\nOutput 20 10 - The CIFAR-10 dataset contains colored images of common objects with a shape of 32 × 32 × 3 and\nincludes 10 distinct categories. The architecture of the ResNet-2B model used is detailed in Table\n7. This model (sourced from the Neural Network Verification Competition (VNN-COMP) Wang\net al. (2021)) is a compact residual network benchmark designed for neural network verification\non CIFAR-10.",
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+    "text": "Intended to help verification tools evolve beyond simple feedforward networks, this\nmodel was adversarially trained with an L∞perturbation epsilon of 2/255. Table 7: Architecture of the ResNet-2B model (CIFAR-10). Layer Type Input Shape Output Shape Activation\nReshape 3072 32 × 32 × 3 -\nConvolution 2D 32 × 32 × 3 16 × 16 × 8 ReLU\nResidual Block (Downsample) 16 × 16 × 8 8 × 8 × 16 ReLU\nResidual Block 8 × 8 × 16 8 × 8 × 16 ReLU\nFlatten 8 × 8 × 16 1024 -\nFully Connected 1024 100 ReLU\nOutput 100 10 - Published as a conference paper at ICLR 2026 E.2 DETAILED EXPERIMENTAL SETUP We configured the VERIX+ implementation with the following settings: binary search=true,\nlogit ranking=true, and traversal order=bounds. To identify necessary features, we used the Fast\nGradient Sign (FGS) technique for singleton attack addition, though the Projected Gradient Descent\n(PGD) is also available for this purpose. We performed a comprehensive sensitivity analysis covering: (1) Solver Choice: Table 1 shows the\nGreedy heuristic finds explanations nearly identical in size to the optimal MILP solver (gap < 9\nfeatures), validating its near-optimality. (2) Cardinality Constraints: Figure 4 confirms that using\nthe constraint (card=True) yields significantly smaller explanations. (3) Perturbation Magnitude (ϵ):\nWhile we adhered to standard baseline values used by the baseline VERIX+ (e.g., 0.05 for MNIST,\n0.01 for GTSRB) to ensure a direct and fair comparison, we acknowledge that explanation size is\ninversely related to ϵ, as larger radii result in looser bounds. E.3 SUPPLEMENTARY EXPERIMENTAL RESULTS PERFORMANCE WITH ITERATIVE REFINEMENT The three plots compare the performance of a greedy heuristic with an exact MILP solver for an iterative refinement task. The central\nfinding across all three visualizations is that the greedy heuristic provides a strong trade-off between\nspeed and solution quality, making it a more practical approach for large-scale problems. Figure 6: Performance Comparison of FAME's Abstract Batch Freeing Methods. These three\nplots compare the greedy heuristic against the exact MILP solver for the iterative refinement task\nfor all the models. The first plot shows the runtime comparison of the two methods on a log-log\nscale. The second plot compares the size of the freed feature set for both methods.",
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+    "text": "The third plot\nillustrates the distribution of the optimality gap (MILP size - Greedy size). Analysis of FAME's Abstract Batch Freeing The visualizations demonstrate that the greedy\nheuristic provides a strong trade-off between speed and solution quality for the iterative refinement\ntask. • Runtime Performance: As shown in the first plot, the greedy algorithm is consistently\nfaster than the MILP solver. This is evidenced by the data points for all models lying\nsignificantly below the diagonal line, confirming a substantial gain in runtime. • Solution Quality: The second plot shows that the greedy algorithm produces solutions of\ncomparable quality to the optimal MILP solver. The tight clustering of data points along\nthe diagonal line for all models indicates a strong correlation between the sizes of the freed\nfeature sets. • Optimality Gap: The histogram of the final plot reinforces these findings by showing that\nthe greedy heuristic frequently achieves the optimal solution, with the highest frequency\nof samples occurring at a gap of zero. The distribution further confirms that any suboptimality is typically minimal.",
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+    "text": "Published as a conference paper at ICLR 2026 F SCALABILITY ANALYSIS ON COMPLEX ARCHITECTURES (RESNET-2B ON\nCIFAR-10) To validate the scalability of FAME on architectures significantly deeper and more complex than\nstandard benchmarks, we conducted an evaluation on the ResNet-2B model (2 residual blocks, 5\nconvolutional layers, 2 linear layers) trained on the CIFAR-10 dataset Wang et al. (2021). We\nutilized an L∞perturbation budget of ϵ = 2/255. These additional experiments were conducted on\na server equipped with an NVIDIA A100 80GB GPU. For these experiments, we define the feature set F at the pixel level. Consequently, the total number of features is N = 32 × 32 = 1024. Freeing a feature in this context\ncorresponds to simultaneously relaxing the constraints on all three color channels (RGB) for that\nspecific pixel location. Feasibility and Comparison. Running exact formal explanation methods (such as the complete\nVERIX+ pipeline with Marabou) on this architecture resulted in consistent timeouts or memory\nexhaustion, confirming that exact minimality is currently out of reach for this complexity class. In\ncontrast, FAME successfully terminated for all processed samples.",
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+    "text": "Detailed Quantitative Results by Configuration. To rigorously assess the contribution of each\ncomponent in the FAME framework, we evaluated three configurations (N = 100 samples). The\nresults are summarized below: • Single-Round Abstract Freeing (Algorithm 1 only). This baseline represents a static\napproach without domain refinement.",
+    "paper_id": "2603.10661",
+    "title": "FAME: Formal Abstract Minimal Explanation for Neural Networks",
+    "authors": [
+      "Ryma Boumazouza",
+      "Raya Elsaleh",
+      "Melanie Ducoffe",
+      "Shahaf Bassan",
+      "Guy Katz"
+    ],
+    "published_date": "2026-03-11",
+    "primary_category": "",
+    "arxiv_url": "http://arxiv.org/abs/2603.10661v1",
+    "chunk_index": 74,
+    "total_chunks": 75,
+    "char_count": 335,
+    "word_count": 46,
+    "chunking_strategy": "semantic"
+  },
+  {
+    "chunk_id": "1ac691d2-ee81-4fda-9310-d87d6d4e0caa",
+    "text": "– Performance: It freed an average of only 5.53 features (pixels).\n– Insight: This confirms that on deep networks, the initial abstract bounds are too loose\nto certify meaningful batches in a single pass. A static traversal strategy would fail\nhere.\n– Solver Comparison: The Greedy heuristic (5.53 features, 50.8s) performed identically\nto the optimal MILP solver (5.37 features, 50.8s), validating the heuristic's quality. • Recursive Abstract Refinement (Algorithm 5). This configuration enables the iterative\ntightening of the domain Ωm(x; A). – Performance: The average number of freed features jumped to 476.38 pixels (approx\n46% of the image).\n– Insight: This dramatic increase (from ∼5 to ∼477) proves that the adaptive abstraction mechanism is critical. By iteratively constraining the cardinality, FAME recovers\nfeatures that were previously masked by over-approximation.\n– Solver Comparison: Remarkably, even in this complex iterative setting, the Greedy\napproach (size 476.38) remained extremely close to the optimal MILP solution (size\n477.76), with a negligible gap of < 0.4%. This strongly justifies using the faster\nGreedy heuristic for scalability.\n– Runtime: The average runtime for this intensive recursive search was approximately\n1934.94 seconds (∼32 minutes). • Full Pipeline (Iteration + Singleton Refinement). This represents the final output of the\ncomplete FAME pipeline, including final safety checks and singleton refinement. – Explanation Compactness: The pipeline successfully certified a robust explanation\nwith an average of 240.84 freed features (pixels) across the full dataset.\n– Efficiency: The breakdown confirms that FAME can navigate the search space of\ndeep networks where exact enumerations fail, producing sound abstract explanations\n(WAXpA) significantly faster than the timeout threshold of exact solvers. Discussion and Future Directions.",
+    "paper_id": "2603.10661",
+    "title": "FAME: Formal Abstract Minimal Explanation for Neural Networks",
+    "authors": [
+      "Ryma Boumazouza",
+      "Raya Elsaleh",
+      "Melanie Ducoffe",
+      "Shahaf Bassan",
+      "Guy Katz"
+    ],
+    "published_date": "2026-03-11",
+    "primary_category": "",
+    "arxiv_url": "http://arxiv.org/abs/2603.10661v1",
+    "chunk_index": 75,
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+    "chunking_strategy": "semantic"
+  },
+  {
+    "chunk_id": "305d1375-7be7-4484-a57c-a03ac364e6ed",
+    "text": "While the computational cost (∼32 mins) is higher than for\nsmaller models, these results establish that the Abstract Batch Certificate (Φ) and recursive refinement scale mathematically to residual connections without theoretical blockers. Published as a conference paper at ICLR 2026 the abstract explanation size and the true minimal explanation is driven primarily by the looseness\nof the abstract bounds (LiRPA CROWN) on deep networks. Future work integrating tighter abstract\ninterpretation methods (e.g., α-CROWN) into the FAME engine will directly improve these results. G DISCLOSURE: USAGE OF LLMS An LLM was used solely as a writing assistant to correct grammar, fix typos, and enhance clarity. It played no role in generating research ideas, designing the study, analyzing data, or interpreting\nresults; all of these tasks were carried out exclusively by the authors.",
+    "paper_id": "2603.10661",
+    "title": "FAME: Formal Abstract Minimal Explanation for Neural Networks",
+    "authors": [
+      "Ryma Boumazouza",
+      "Raya Elsaleh",
+      "Melanie Ducoffe",
+      "Shahaf Bassan",
+      "Guy Katz"
+    ],
+    "published_date": "2026-03-11",
+    "primary_category": "",
+    "arxiv_url": "http://arxiv.org/abs/2603.10661v1",
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+  }
+]
\ No newline at end of file