MC3G: Causally Constrained Counterfactuals

• The paper introduces MC3G, which leverages a rule-based surrogate to generate counterfactuals that are both causally consistent and model-agnostic.
• It refines recourse cost by charging only for user-initiated feature changes, thereby ensuring realistic intervention burden.
• Empirical results show lower L1 and L2 costs compared to C3G, validating MC3G’s effectiveness in high-stakes decision settings.

Searching arXiv for MC3G and closely related causally constrained counterfactual generation papers. Model-Agnostic Causally Constrained Counterfactual Generation (MC3G) is a counterfactual explanation framework for black-box classifiers on tabular data that combines three commitments: model-agnostic approximation of the predictor through an explainable rule-based surrogate, causal compliance of the generated counterfactual states, and a refined notion of recourse cost that charges only for direct user-initiated feature changes rather than for downstream changes that occur automatically because of causal dependencies. In the formulation introduced in 2025, MC3G is intended to provide actionable recourse in high-stakes decision settings without exposing proprietary model internals, while ensuring that recommended feature changes remain causally realistic and that effort is computed in a way the paper describes as more realistic and fair (Dasgupta et al., 24 Aug 2025).

1. Definition and conceptual scope

MC3G starts from the standard counterfactual explanation premise: for a binary classifier $f : X \to \{0,1\}$, counterfactuals are alternative inputs that change the predicted outcome. The framework restates the counterfactual set for an input $x$ as

$$CF_f(x) = \{ \hat{x} \in X \mid f(x) \neq f(\hat{x}) \}.$$

Its claim is that this standard definition is insufficient unless three additional conditions are imposed: the method must be usable for black-box models, the resulting state must satisfy causal dependencies among features, and the cost of recourse must not be inflated by automatic downstream changes (Dasgupta et al., 24 Aug 2025).

The framework is explicitly motivated by a tension common in explainable AI for finance, hiring, and law. Individuals need recourse that explains why a decision was made and what can be changed to obtain a favorable outcome, but organizations may not wish to disclose the internal logic of a proprietary classifier. MC3G addresses this by learning an interpretable surrogate model from black-box predictions and then performing rule-based causal reasoning on that surrogate rather than on the original model internals (Dasgupta et al., 24 Aug 2025).

The paper positions itself against three shortcomings in earlier counterfactual work. First, many methods are not model-agnostic because they require access to model parameters or internal structure. Second, many methods ignore causal dependencies and therefore treat features as independently mutable. Third, earlier causal methods can still overestimate effort because they penalize every changed feature, including those that change only as a consequence of a direct intervention. MC3G is defined precisely to correct that third point while preserving the first two desiderata (Dasgupta et al., 24 Aug 2025).

2. Formal problem formulation

MC3G uses a state-space formalization. If the feature set is $F_1,\dots,F_n$ with domains $D_1,\dots,D_n$, then the state space is

$$S = \{ (V_1,V_2,\dots,V_n) \mid V_i \in D_i,\ \text{for each } i \in 1,\dots,n \}.$$

A state is therefore a complete feature assignment. Causal knowledge is represented by a rule set $C$, and the paper defines a filtering operator

$\theta_C : P(S) \to P(S),$

with

$$\theta_C(S') = \{ s \in S' \mid s \text{ satisfies all causal rules in } C \}.$$

The causally consistent state space is then

$S_C = \theta_C(S).$

Decision logic is represented by a rule set $x$0 for the undesired outcome. The corresponding operator is

$x$1

with

$x$2

and therefore

$x$3

A counterfactual generation problem is the triple

$x$4

where $x$5 is the initial state, and the goal set is

$x$6

A valid counterfactual is any $x$7: it is causally consistent and it no longer satisfies the undesired decision rules (Dasgupta et al., 24 Aug 2025).

This formulation matters because it makes causal admissibility a hard feasibility condition rather than a soft preference. A state is not merely “good” if it flips the prediction; it must first remain in $x$8. This sharply distinguishes MC3G from unconstrained recourse formulations that optimize only proximity to the original instance.

3. Surrogate extraction and the generation pipeline

The framework is a three-stage pipeline. First, it approximates the original classifier

$x$9

with an interpretable rule-based surrogate

$$CF_f(x) = \{ \hat{x} \in X \mid f(x) \neq f(\hat{x}) \}.$$0

learned using the rule-based machine learning algorithm FOLD-SE. If the original model is already rule-based, its rules are used directly; otherwise the black-box model is queried on the training data and the resulting labels are used to train the surrogate (Dasgupta et al., 24 Aug 2025).

The paper’s extract_logic procedure is summarized as follows: if the original model $$CF_f(x) = \{ \hat{x} \in X \mid f(x) \neq f(\hat{x}) \}.$$1 is rule-based, set $$CF_f(x) = \{ \hat{x} \in X \mid f(x) \neq f(\hat{x}) \}.$$2; otherwise compute black-box predictions $$CF_f(x) = \{ \hat{x} \in X \mid f(x) \neq f(\hat{x}) \}.$$3, train the RBML learner on $$CF_f(x) = \{ \hat{x} \in X \mid f(x) \neq f(\hat{x}) \}.$$4, and return the learned rule set $$CF_f(x) = \{ \hat{x} \in X \mid f(x) \neq f(\hat{x}) \}.$$5. The paper does not define an explicit surrogate-fidelity objective or report agreement-to-black-box scores, but the framework conceptually depends on the surrogate mimicking the original model sufficiently well for downstream counterfactual search (Dasgupta et al., 24 Aug 2025).

Second, MC3G searches over a set of candidate states $$CF_f(x) = \{ \hat{x} \in X \mid f(x) \neq f(\hat{x}) \}.$$6. The search is state-based rather than a continuous per-instance optimization routine. For each candidate state $$CF_f(x) = \{ \hat{x} \in X \mid f(x) \neq f(\hat{x}) \}.$$7, the framework checks whether it is causally consistent with $$CF_f(x) = \{ \hat{x} \in X \mid f(x) \neq f(\hat{x}) \}.$$8 and no longer compliant with the undesired decision rules $$CF_f(x) = \{ \hat{x} \in X \mid f(x) \neq f(\hat{x}) \}.$$9. This is implemented through is_counterfactual, which returns a Boolean validity flag together with an adjusted feature-weight vector. Third, for all valid candidates, MC3G computes a weighted $F_1,\dots,F_n$0 cost and returns the minimum-cost solution (Dasgupta et al., 24 Aug 2025).

The top-level algorithm is therefore enumerative. It initializes the best solution, loops over candidates, tests validity via is_counterfactual, evaluates the adjusted weighted distance, and retains the candidate with the smallest cost. The paper does not present an asymptotic complexity analysis, and it explicitly notes that the current framework incurs higher computational cost and that future work should optimize the search space (Dasgupta et al., 24 Aug 2025).

This makes MC3G operationally distinct from amortized one-shot generators. In contrast, the reinforcement-learning framework in "Model-agnostic and Scalable Counterfactual Explanations via Reinforcement Learning" produces target-conditional counterfactuals in a single forward pass and is model-agnostic in the sense of using only prediction feedback, but it does not impose explicit causal or structural constraints (Samoilescu et al., 2021).

4. Causal rules, intervention semantics, and the cost model

The central causal idea in MC3G is that a counterfactual should distinguish direct interventions from downstream consequences. Causal dependencies are encoded as symbolic rules $F_1,\dots,F_n$1, and a candidate is valid only if it belongs to the causally consistent state space $F_1,\dots,F_n$2. The paper’s motivating loan example makes the distinction concrete. A naive suggestion may directly raise both bank balance and credit score, but if credit score causally depends on debt, then the causally compliant recommendation is to clear the debt and increase the bank balance, after which credit score changes automatically (Dasgupta et al., 24 Aug 2025).

The refined cost model is the paper’s main novelty relative to earlier causal work. Let $F_1,\dots,F_n$3 be the original feature-weight vector. MC3G constructs an adjusted weight vector $F_1,\dots,F_n$4 such that

$F_1,\dots,F_n$5

This means that only user-initiated changes incur effort; features altered automatically through causal propagation are assigned zero cost (Dasgupta et al., 24 Aug 2025).

The framework then evaluates candidate states with weighted $F_1,\dots,F_n$6, $F_1,\dots,F_n$7, or $F_1,\dots,F_n$8 costs. For states $F_1,\dots,F_n$9,

$D_1,\dots,D_n$0

$D_1,\dots,D_n$1

$D_1,\dots,D_n$2

MC3G replaces $D_1,\dots,D_n$3 with $D_1,\dots,D_n$4, yielding the adjusted distances $D_1,\dots,D_n$5. The implicit optimization problem is therefore

$D_1,\dots,D_n$6

subject to causal consistency and decision reversal (Dasgupta et al., 24 Aug 2025).

This cost definition changes the semantics of recourse. The framework argues that earlier methods can overestimate effort by charging for all changed features even when some changes are merely causal side effects. MC3G instead charges only for what the user must actually do. A plausible implication is that the framework is better interpreted as estimating intervention burden than as merely minimizing geometric proximity in feature space.

The symbolic treatment of causal rules also places MC3G near logic-based systems such as CoGS, which use goal-directed ASP and s(CASP) to search for paths from undesired to desired states while respecting causal dependencies. The difference is that CoGS is naturally defined for rule-based models, whereas MC3G inserts a surrogate-learning stage so that the same symbolic machinery can be applied to an arbitrary black-box model after approximation (Dasgupta et al., 2024).

5. Empirical evaluation

The empirical study uses three tabular datasets: Adult, Statlog (German Credit), and Car Evaluation. Adult and German are used because the authors state that they have well-defined causal dependencies known to them; Car is used as a comparison case where no causal dependencies are used. The reported baselines are Borderline-CF, DiCE, MACE, and C3G (Dasgupta et al., 24 Aug 2025).

The first evaluation reports causal compliance and causal consistency. On Adult, Borderline-CF is marked causally compliant = FALSE with causal consistency $D_1,\dots,D_n$7, DiCE is marked Indirectly with $D_1,\dots,D_n$8, MACE is FALSE with $D_1,\dots,D_n$9, and both C3G and MC3G are TRUE with $$S = \{ (V_1,V_2,\dots,V_n) \mid V_i \in D_i,\ \text{for each } i \in 1,\dots,n \}.$$0. On German, Borderline-CF is FALSE with $$S = \{ (V_1,V_2,\dots,V_n) \mid V_i \in D_i,\ \text{for each } i \in 1,\dots,n \}.$$1, DiCE is Indirectly with $$S = \{ (V_1,V_2,\dots,V_n) \mid V_i \in D_i,\ \text{for each } i \in 1,\dots,n \}.$$2, MACE is FALSE with $$S = \{ (V_1,V_2,\dots,V_n) \mid V_i \in D_i,\ \text{for each } i \in 1,\dots,n \}.$$3, and again both C3G and MC3G achieve TRUE and $$S = \{ (V_1,V_2,\dots,V_n) \mid V_i \in D_i,\ \text{for each } i \in 1,\dots,n \}.$$4. On Car, causal consistency is listed as N/A because no causal dependencies are used (Dasgupta et al., 24 Aug 2025).

The second evaluation compares MC3G directly with C3G using $$S = \{ (V_1,V_2,\dots,V_n) \mid V_i \in D_i,\ \text{for each } i \in 1,\dots,n \}.$$5, $$S = \{ (V_1,V_2,\dots,V_n) \mid V_i \in D_i,\ \text{for each } i \in 1,\dots,n \}.$$6, and $$S = \{ (V_1,V_2,\dots,V_n) \mid V_i \in D_i,\ \text{for each } i \in 1,\dots,n \}.$$7 for the nearest counterfactual ($$S = \{ (V_1,V_2,\dots,V_n) \mid V_i \in D_i,\ \text{for each } i \in 1,\dots,n \}.$$8), the furthest among top-$$S = \{ (V_1,V_2,\dots,V_n) \mid V_i \in D_i,\ \text{for each } i \in 1,\dots,n \}.$$9 ($C$0), and the average over top-$C$1. On Adult, C3G reports $C$2 values $C$3, while MC3G reports $C$4. For $C$5, C3G reports $C$6, while MC3G reports $C$7. Both methods report $C$8. On German, C3G reports $C$9, $\theta_C : P(S) \to P(S),$0, and $\theta_C : P(S) \to P(S),$1, whereas MC3G reports $\theta_C : P(S) \to P(S),$2, $\theta_C : P(S) \to P(S),$3, and $\theta_C : P(S) \to P(S),$4. On Car, MC3G and C3G are identical on all metrics: $\theta_C : P(S) \to P(S),$5, $\theta_C : P(S) \to P(S),$6, and $\theta_C : P(S) \to P(S),$7 (Dasgupta et al., 24 Aug 2025).

These results support the paper’s intended interpretation. When causal dependencies matter, MC3G matches earlier causal methods on validity while producing lower-cost recommendations because automatic causal changes are excluded from the effort calculation. When no causal structure is used, MC3G collapses to the same behavior as C3G. This suggests that its distinctive contribution is not a new notion of validity, but a new accounting of intervention cost.

6. Position in the literature, misconceptions, and limitations

MC3G belongs to a broader family of causality-aware counterfactual methods, but it occupies a specific point in that design space. It is not an amortized generator like the reinforcement-learning method of 2021, which is model-agnostic and scalable but explicitly lacks causal or structural constraints (Samoilescu et al., 2021). It is also not a purely symbolic planner like CoGS, whose native setting is a rule-based predictor and an ASP search program over state transitions (Dasgupta et al., 2024). MC3G combines those two design motives: black-box approximation through a rule-based surrogate and symbolic causal filtering through explicit causal rules (Dasgupta et al., 24 Aug 2025).

It should also be distinguished from dependency-preserving realism methods that do not impose explicit causal rules. "RealAC: A Domain-Agnostic Framework for Realistic and Actionable Counterfactual Explanations" preserves pairwise feature dependencies and supports user-frozen features, but it explicitly requires a differentiable neural classifier and does not provide SCM-style or rule-based causal constraints (Arefeen et al., 14 Aug 2025). MC3G is narrower in modality and relies on supplied causal dependencies, but its counterfactual admissibility criterion is explicitly causal rather than associational.

A common misconception is that “model-agnostic” in MC3G means no surrogate or no symbolic representation. In the paper’s actual formulation, model-agnosticism means that any black-box classifier can be approximated by an explainable rule-based surrogate and then handled through rule-based causal reasoning. Another misconception is that causal consistency alone determines quality. MC3G argues that two causally valid counterfactuals can differ materially in effort once downstream automatic changes are excluded from the cost calculation (Dasgupta et al., 24 Aug 2025).

The framework also has clear limitations. It is currently limited to tabular data; it depends on the quality and completeness of the causal rule set; it depends on surrogate fidelity to the original black-box; and the main pseudocode does not include a separate final validation step against the original black-box, even though the narrative claims that the generated counterfactuals produce a favorable outcome for the underlying model (Dasgupta et al., 24 Aug 2025). The experimental section also does not specify which black-box models were used. More broadly, the causal counterfactual literature has emphasized that exact counterfactual distributions may be non-identifiable even with a causal graph in high-dimensional settings, which suggests caution in transporting strong causal claims across modalities (Pan et al., 2024).

Taken together, MC3G is best characterized as a surrogate-based, rule-driven, causally filtered recourse framework whose main technical distinction is the replacement of ordinary weighted distance $\theta_C : P(S) \to P(S),$8 with a causally adjusted distance $\theta_C : P(S) \to P(S),$9. In that sense, its contribution is less a new definition of counterfactual validity than a more intervention-faithful definition of counterfactual cost (Dasgupta et al., 24 Aug 2025).