Weak Abductive Explanation (WAXp)

• WAXp is a formal notion where fixing a set of features is sufficient to preserve a classifier's output without enforcing subset minimality.
• It is defined via a monotonic property in logic-based XAI and modal logic frameworks, linking sufficient conditions to minimal abductive explanations.
• WAXp extends to probabilistic settings and adversarial examples, offering computational methods and establishing duality with contrastive explanations.

Weak Abductive Explanation (WAXp) denotes a weak, non-minimal notion of abductive explanation whose precise formal meaning depends on the framework in which it is used. In logic-based XAI, the common core is sufficiency: a set of hypotheses or features is a WAXp when fixing them is enough to preserve the observed decision or class, without yet requiring subset-minimality. In the modal-logic account of classifier explanation, the “weak” aspect is not a separate explanatory primitive but the use of the weaker proof system $WBCL$ for countably infinite feature vocabularies; in later formal-XAI work, “WeakAXp” is the explicit non-minimal base notion from which minimal abductive explanations (AXps) are obtained by subset minimization (Liu et al., 2021, Izza et al., 2023, Huang et al., 16 Aug 2025).

1. Core notion and terminological scope

A direct formalization of weak abductive explanation appears in the feature-based setting of classifier explanation. Given an instance $(\mathbf{v}, c)$ with $\kappa(\mathbf{v}) = c$, a set of features $X \subseteq F$ is a weak abductive explanation precisely when fixing those features to their values in $\mathbf{v}$ suffices to force the prediction $c$ for all completions: $$WeakAXp(X) := \forall \mathbf{x}\in \mathbf{F}.\; \left[\bigwedge_{i\in X}(x_i = v_i)\right] \rightarrow (\kappa(\mathbf{x}) = c).$$ An abductive explanation, or $AXp$, is then a subset-minimal weak AXp: $$AXp(X) := WeakAXp(X) \land \forall X' \subsetneq X.\; \neg WeakAXp(X'),$$ and, by monotonicity of $WeakAXp$, equivalently

$(\mathbf{v}, c)$0

This makes the distinction exact: WAXp is sufficiency without minimality; AXp is minimal sufficiency (Izza et al., 2023).

The same pattern is used in a more general explanation problem $(\mathbf{v}, c)$1, where $(\mathbf{v}, c)$2 is either a classification or regression model, $(\mathbf{v}, c)$3 is the feature set, and $(\mathbf{v}, c)$4 is the target input. The similarity predicate $(\mathbf{v}, c)$5 is defined by

$(\mathbf{v}, c)$6

for classification, and by

$(\mathbf{v}, c)$7

for regression. A weak abductive explanation is then any $(\mathbf{v}, c)$8 such that

$(\mathbf{v}, c)$9

while

$\kappa(\mathbf{v}) = c$0

In this formulation, WAXp is the sufficient set of fixed features and AXp is the subset-minimal sufficient set (Huang et al., 16 Aug 2025).

The modal-logic framework for explanations in classifier systems uses a different presentation. There, weak abductive explanation is not introduced as a separate primitive symbol “WAXp”; rather, abductive explanation is defined in the object language, and the “weak” qualifier is tied to the weaker logic $\kappa(\mathbf{v}) = c$1, the infinite-variable counterpart of $\kappa(\mathbf{v}) = c$2 (Liu et al., 2021). This suggests that “WAXp” is best understood as a family resemblance term across formal frameworks rather than a universally fixed syntactic operator.

2. Ceteris paribus modal formalization

The unified logical framework for classifier explanations is based on the modal language $\kappa(\mathbf{v}) = c$3, where $\kappa(\mathbf{v}) = c$4 is a set of atomic propositions for input features and $\kappa(\mathbf{v}) = c$5 are decision atoms. Formulas are generated by

$\kappa(\mathbf{v}) = c$6

with $\kappa(\mathbf{v}) = c$7 and $\kappa(\mathbf{v}) = c$8 finite. The intended reading of $\kappa(\mathbf{v}) = c$9 is that $X \subseteq F$0 holds after arbitrary perturbations of all features not in $X \subseteq F$1, while the features in $X \subseteq F$2 are kept fixed. Semantically, for a classifier model $X \subseteq F$3 with states $X \subseteq F$4,

$X \subseteq F$5

This is the paper’s ceteris paribus modality: the value of the features in $X \subseteq F$6 is held constant, while everything else may vary (Liu et al., 2021).

Within this language, abductive explanation is defined via prime implicants. For a term $X \subseteq F$7 and a decision value $X \subseteq F$8, “$X \subseteq F$9 is a prime implicant” is introduced by the abbreviation

$\mathbf{v}$0

The actualized abductive explanation is

$\mathbf{v}$1

Equivalently,

$\mathbf{v}$2

This equivalence makes the explanatory structure explicit. The conjunct $\mathbf{v}$3 states sufficiency: if the features occurring in $\mathbf{v}$4 are fixed, the decision $\mathbf{v}$5 is invariant under arbitrary perturbations of all other features. The remaining conjuncts state minimality: each literal in $\mathbf{v}$6 is needed, because if one of them is allowed to vary while the others are fixed, a state where $\mathbf{v}$7 fails becomes reachable. In this framework, abductive explanation is therefore a minimal sufficient reason expressed by modal invariance (Liu et al., 2021).

3. Weakness, minimality, existence, and computational behavior

The distinction between strong and weak abductive settings in the modal framework is determined by the cardinality of the feature vocabulary. When $\mathbf{v}$8 is finite, the logic is $\mathbf{v}$9; when $c$0 is countably infinite, the logic is $c$1. The latter is literally $c$2 minus the finitary axiom

$c$3

which cannot be written finitarily when the feature set is infinite. The paper proves soundness and completeness of $c$4 for classifier models in the finite-variable case, and soundness and completeness of $c$5 for classifier models in the infinite-variable case (Liu et al., 2021).

This weakening has a substantive effect on existence. In finite settings, every Boolean classifier has a prime implicant, hence an abductive explanation exists. In the infinite-variable framework, this fails: if $c$6 is countably infinite and $c$7, there exist models with no abductive explanation at a state. On the positive side, if the classifier is $c$8-definite for some finite $c$9, then every state admits an abductive explanation: $$WeakAXp(X) := \forall \mathbf{x}\in \mathbf{F}.\; \left[\bigwedge_{i\in X}(x_i = v_i)\right] \rightarrow (\kappa(\mathbf{x}) = c).$$0 Accordingly, in the $$WeakAXp(X) := \forall \mathbf{x}\in \mathbf{F}.\; \left[\bigwedge_{i\in X}(x_i = v_i)\right] \rightarrow (\kappa(\mathbf{x}) = c).$$1 setting WAXp-style explanation is present, but its existence is conditional rather than automatic (Liu et al., 2021).

In the feature-based formal-XAI setting, weak abductive explanation is monotone: if a set is weak, then all supersets are weak. That monotonicity is exactly why minimality can be checked by one-feature deletion: $$WeakAXp(X) := \forall \mathbf{x}\in \mathbf{F}.\; \left[\bigwedge_{i\in X}(x_i = v_i)\right] \rightarrow (\kappa(\mathbf{x}) = c).$$2 A common misconception is therefore to identify WAXp with AXp. The formal definitions rule this out: WAXp is the monotone sufficient base notion, whereas AXp is the subset-minimal specialization (Izza et al., 2023).

The computational profile also changes with the underlying logic. For the full language $$WeakAXp(X) := \forall \mathbf{x}\in \mathbf{F}.\; \left[\bigwedge_{i\in X}(x_i = v_i)\right] \rightarrow (\kappa(\mathbf{x}) = c).$$3, satisfiability over classifier models is polynomial-time when $$WeakAXp(X) := \forall \mathbf{x}\in \mathbf{F}.\; \left[\bigwedge_{i\in X}(x_i = v_i)\right] \rightarrow (\kappa(\mathbf{x}) = c).$$4 is finite and fixed; when $$WeakAXp(X) := \forall \mathbf{x}\in \mathbf{F}.\; \left[\bigwedge_{i\in X}(x_i = v_i)\right] \rightarrow (\kappa(\mathbf{x}) = c).$$5 is countably infinite, satisfiability becomes NEXPTIME-complete. The fragment using only the universal modality $$WeakAXp(X) := \forall \mathbf{x}\in \mathbf{F}.\; \left[\bigwedge_{i\in X}(x_i = v_i)\right] \rightarrow (\kappa(\mathbf{x}) = c).$$6 is NP-complete in the infinite-variable case. These results matter indirectly for WAXp because abductive explanation formulas are expressed in the same language (Liu et al., 2021).

4. Probabilistic generalizations and local minimality

Probabilistic abductive explanation extends the weak abductive pattern by relaxing universal preservation to thresholded conditional preservation. A weak probabilistic abductive explanation is a feature set $$WeakAXp(X) := \forall \mathbf{x}\in \mathbf{F}.\; \left[\bigwedge_{i\in X}(x_i = v_i)\right] \rightarrow (\kappa(\mathbf{x}) = c).$$7 such that

$$WeakAXp(X) := \forall \mathbf{x}\in \mathbf{F}.\; \left[\bigwedge_{i\in X}(x_i = v_i)\right] \rightarrow (\kappa(\mathbf{x}) = c).$$8

with the conditional probability written explicitly as

$$WeakAXp(X) := \forall \mathbf{x}\in \mathbf{F}.\; \left[\bigwedge_{i\in X}(x_i = v_i)\right] \rightarrow (\kappa(\mathbf{x}) = c).$$9

A $AXp$0 is then the subset-minimal weak $AXp$1: $AXp$2 The paper emphasizes that, unlike $AXp$3, $AXp$4 is not monotone; fixing more features can increase or decrease the relevant conditional probability (Izza et al., 2023).

Because of this non-monotonicity, local one-feature deletion is no longer equivalent to subset minimality. The proposed relaxation is the locally-minimal probabilistic abductive explanation: $AXp$5 For classical AXps, local minimality and subset minimality coincide due to monotonicity; for probabilistic explanations they do not necessarily coincide. There may therefore exist locally minimal $AXp$6s that are not subset-minimal $AXp$7s. The paper treats $AXp$8 as a practical surrogate for exact $AXp$9 computation (Izza et al., 2023).

The computational method is deletion-based. DelLmPAXp starts with either all features $$AXp(X) := WeakAXp(X) \land \forall X' \subsetneq X.\; \neg WeakAXp(X'),$$0 or a known AXp $$AXp(X) := WeakAXp(X) \land \forall X' \subsetneq X.\; \neg WeakAXp(X'),$$1, iteratively attempts feature removal, and invokes an approximation subroutine approxCount to test the weak probabilistic condition. Two implementations are described: approximate model counting using ApproxMC / ApproxMCPB, and Monte Carlo sampling with Hoeffding/Chernoff-style guarantees. The approximate counting approach yields

$$AXp(X) := WeakAXp(X) \land \forall X' \subsetneq X.\; \neg WeakAXp(X'),$$2

while the sampling approach uses

$$AXp(X) := WeakAXp(X) \land \forall X' \subsetneq X.\; \neg WeakAXp(X'),$$3

This line of work preserves the WAXp idea of sufficiency while replacing universal certainty by a probability threshold and exact minimality by a locally checkable approximation (Izza et al., 2023).

5. Contrastive duality, adversarial examples, and feature attribution

Abductive explanation is sharply distinguished from contrastive explanation in the modal classifier framework. Contrastive explanation is defined by

$$AXp(X) := WeakAXp(X) \land \forall X' \subsetneq X.\; \neg WeakAXp(X'),$$4

The paper proves that contrastive explanation implies a counterfactual conditional statement,

$$AXp(X) := WeakAXp(X) \land \forall X' \subsetneq X.\; \neg WeakAXp(X'),$$5

and, for literals under completeness, also proves the converse characterization

$$AXp(X) := WeakAXp(X) \land \forall X' \subsetneq X.\; \neg WeakAXp(X'),$$6

In the paper’s wording, AXp answers “why?” and CXp answers “why not?”. WAXp belongs to the sufficiency side of this divide, whereas contrastive explanation is about minimal decision-changing perturbation (Liu et al., 2021).

A later formalization makes the duality operational through adversarial examples. Around a target input $$AXp(X) := WeakAXp(X) \land \forall X' \subsetneq X.\; \neg WeakAXp(X'),$$7, an adversarial example is

$$AXp(X) := WeakAXp(X) \land \forall X' \subsetneq X.\; \neg WeakAXp(X'),$$8

A weak contrastive explanation is a set $$AXp(X) := WeakAXp(X) \land \forall X' \subsetneq X.\; \neg WeakAXp(X'),$$9 such that

$WeakAXp$0

and a $WeakAXp$1 is the subset-minimal $WeakAXp$2: $WeakAXp$3 The paper states the duality in three equivalent forms: a WAXp is a hitting set of the $WeakAXp$4s; equivalently, it hits every minimal contrastive explanation; equivalently again, it excludes all adversarial examples in the local neighborhood (Huang et al., 16 Aug 2025).

This duality underlies a recent use of WAXp in rigorous feature attribution. Such methods typically exploit WAXp as the characteristic function, but the paper argues that this neglects the information carried by non-WAXp sets. To incorporate graded exclusion of adversarial examples, it defines the $WeakAXp$5-Forest

$WeakAXp$6

and then derives Shapley-like and Banzhaf-like scores: $WeakAXp$7 A central conceptual move is thus to start from WAXp as the sufficient-set notion, then replace the binary WAXp indicator by a characteristic function quantifying how many adversarial examples a coalition excludes (Huang et al., 16 Aug 2025).

6. Related refinements and broader logical settings

Several neighboring lines of work refine or weaken abductive explanation along dimensions that are orthogonal to WAXp itself. One such dimension is arbitrariness. In abductive logic programming with integrity constraints, the degree of arbitrariness $WeakAXp$8 is defined as the maximum number of pairwise independent replacement functions preserving explanatory status; an explanation is constrained exactly when $WeakAXp$9. The paper explicitly states that minimum arbitrariness is orthogonal to subset minimality and cardinality minimality, so it supplies a preference criterion for weak explanations rather than a replacement for them (Caroprese et al., 2014).

Another direction studies abduction under inconsistency. In ABox abduction under repair semantics, a brave-hypothesis makes the observation true in some repair, while an AR-hypothesis makes it true in all repairs. The paper introduces the notion of conflict-confining hypotheses, defined by preservation of the conflict sets of the original inconsistent knowledge base. This gives a weak-explanation-like refinement in which the hypothesis need not restore global consistency, but should not create new conflicts (Haak et al., 29 Jul 2025).

Paraconsistent abduction in the Belnap–Dunn setting generalizes explanation to inconsistent theories without repairs. In $(\mathbf{v}, c)$00, formulas $(\mathbf{v}, c)$01 express that the information about $(\mathbf{v}, c)$02 is reliable; in $(\mathbf{v}, c)$03, formulas $(\mathbf{v}, c)$04 express that there is information that $(\mathbf{v}, c)$05 is true. Explanations are terms satisfying entailment and non-triviality, and the framework studies both $(\mathbf{v}, c)$06-minimal and theory-minimal solutions. This is not WAXp terminology, but it provides a formally weaker and non-classical explanatory environment (Bienvenu et al., 2024).

Contextual abductive reasoning under weak completion semantics extends explanation with side-effects and inspection points. It introduces $(\mathbf{v}, c)$07 and $(\mathbf{v}, c)$08 as consumer-like abducibles and defines necessary and possible contextual side-effects, contestable contextual side-effects, contextual relevant consequences, and jointly supported contextual relevant consequences. This line of work does not define WAXp, but it shows how weak explanation can be enriched by contextual dependence between explanations (Pereira et al., 2014).

A different neighboring development appears in ontology-mediated query answering. For negative query answers in DL-Lite, explanations are ABox additions $(\mathbf{v}, c)$09 such that $(\mathbf{v}, c)$10 is consistent and the previously missing tuple becomes a certain answer. The framework studies arbitrary, subset-minimal, and cardinality-minimal explanations, showing that weak explanation can also be understood as non-minimal abductive completion of an ontology-query pair (Calvanese et al., 2014).

Across these frameworks, the recurring invariant is that WAXp names sufficiency prior to minimality, but the source of the “weakness” differs: weaker axiomatics in $(\mathbf{v}, c)$11, non-minimal feature fixing in formal XAI, thresholded sufficiency in probabilistic abduction, or tolerance of inconsistency in repair-based and paraconsistent reasoning. This suggests that WAXp is less a single formal object than a stable explanatory pattern: preserve the target outcome under an admissible space of variations, and postpone stronger optimality criteria to a later stage.