Concept Rule Sets in Interpretable ML

• CRS are hierarchical rule-based models that structure classification as multi-layer compositions of logical conjunctions and disjunctions with binary weights.
• They integrate transparent symbolic reasoning with differentiable training via Multilayer Logical Perceptron methods to extract interpretable rule sets.
• CRS facilitate model simplification through techniques like random binarization and L0 sparsification while exploring near-optimal, diverse rule set explanations.

Concept Rule Sets (CRS) are hierarchical rule-based models for classification in which the model is represented as a multi-level composition of conjunctions and disjunctions with binary weights, yielding a transparent inner structure and a discrete logical form that is directly readable as rules and rule sets. In the formulation developed with the Multilayer Logical Perceptron (MLLP), CRS is the binarized, interpretable version of a continuous logical network, so training is performed in a differentiable space and inference is ultimately expressed as propositional structure. Within interpretable machine learning, CRS is also naturally connected to sparse rule-set learning more broadly, including recent work on exploring the Rashomon set of near-optimal rule sets rather than selecting a single explanation (Wang et al., 2019, Bergamin et al., 26 Sep 2025, Ciaperoni et al., 2024).

1. Formal definition and logical architecture

CRS is defined as a multi-level hierarchical rule-based model with binary weights. One formulation states that a CRS is “an instance of a multi-layer logical perceptron network” where each weight satisfies

$w_{i,j}^{(l)} \in \{0,1\}.$

A binary weight indicates whether a logical connection exists. The core layer definitions are

$$\mathbf{r}_i^{(l)}=\bigwedge_{w_{i, j}^{(l)}=1} \mathbf{s}_j^{(l-1)}, \qquad \mathbf{s}_i^{(l+1)}=\bigvee_{w_{i, j}^{(l+1)}=1} \mathbf{r}_j^{(l)} .$$

Here, $\mathbf{r}_i^{(l)}$ is a conjunction node, i.e. a rule, and $\mathbf{s}_i^{(l)}$ is a disjunction node, i.e. a rule set in DNF form. A CRS with $L$ levels has $2L+1$ layers: an input layer, then alternating conjunction and disjunction layers, and the model outputs the last disjunction layer $\mathcal{S}^{(2L)}$, where $\mathbf{s}_i^{(2L)}=1$ means the classifier predicts class $i$ (Bergamin et al., 26 Sep 2025).

An equivalent presentation defines a rule as a conjunction of Boolean variables,

$$\mathbf{r}_i=\mathbf{b}_{i_1} \wedge \mathbf{b}_{i_2} \wedge \dots \wedge \mathbf{b}_{i_p},$$

and a rule set as a disjunction of rules,

$$\mathbf{r}_i^{(l)}=\bigwedge_{w_{i, j}^{(l)}=1} \mathbf{s}_j^{(l-1)}, \qquad \mathbf{s}_i^{(l+1)}=\bigvee_{w_{i, j}^{(l+1)}=1} \mathbf{r}_j^{(l)} .$$0

A CRS with $$\mathbf{r}_i^{(l)}=\bigwedge_{w_{i, j}^{(l)}=1} \mathbf{s}_j^{(l-1)}, \qquad \mathbf{s}_i^{(l+1)}=\bigvee_{w_{i, j}^{(l+1)}=1} \mathbf{r}_j^{(l)} .$$1 levels alternates conjunction layers $$\mathbf{r}_i^{(l)}=\bigwedge_{w_{i, j}^{(l)}=1} \mathbf{s}_j^{(l-1)}, \qquad \mathbf{s}_i^{(l+1)}=\bigvee_{w_{i, j}^{(l+1)}=1} \mathbf{r}_j^{(l)} .$$2 and disjunction layers $$\mathbf{r}_i^{(l)}=\bigwedge_{w_{i, j}^{(l)}=1} \mathbf{s}_j^{(l-1)}, \qquad \mathbf{s}_i^{(l+1)}=\bigvee_{w_{i, j}^{(l+1)}=1} \mathbf{r}_j^{(l)} .$$3, with binary adjacency matrices determining which lower-layer nodes participate in each rule or rule set (Wang et al., 2019).

This architecture makes transparency structural rather than post hoc. Each internal node is itself interpretable: conjunction nodes correspond to rules, disjunction nodes correspond to rule sets, and the hidden representation at every layer can be read as a logical composition. A plausible implication is that CRS occupies an intermediate position between classical shallow rule learning and layered representation learning: it preserves explicit symbolic semantics while allowing multi-level compositions of rules.

2. Differentiable realization through the Multilayer Logical Perceptron

The MLLP is the continuous differentiable counterpart of CRS. It is designed so that each neuron corresponds to a CRS node, but its weights are real-valued in $$\mathbf{r}_i^{(l)}=\bigwedge_{w_{i, j}^{(l)}=1} \mathbf{s}_j^{(l-1)}, \qquad \mathbf{s}_i^{(l+1)}=\bigvee_{w_{i, j}^{(l+1)}=1} \mathbf{r}_j^{(l)} .$$4. Logical operations are approximated by differentiable conjunction and disjunction functions: $$\mathbf{r}_i^{(l)}=\bigwedge_{w_{i, j}^{(l)}=1} \mathbf{s}_j^{(l-1)}, \qquad \mathbf{s}_i^{(l+1)}=\bigvee_{w_{i, j}^{(l+1)}=1} \mathbf{r}_j^{(l)} .$$5 with

$$\mathbf{r}_i^{(l)}=\bigwedge_{w_{i, j}^{(l)}=1} \mathbf{s}_j^{(l-1)}, \qquad \mathbf{s}_i^{(l+1)}=\bigvee_{w_{i, j}^{(l+1)}=1} \mathbf{r}_j^{(l)} .$$6

The weights are constrained to $$\mathbf{r}_i^{(l)}=\bigwedge_{w_{i, j}^{(l)}=1} \mathbf{s}_j^{(l-1)}, \qquad \mathbf{s}_i^{(l+1)}=\bigvee_{w_{i, j}^{(l+1)}=1} \mathbf{r}_j^{(l)} .$$7 via

$$\mathbf{r}_i^{(l)}=\bigwedge_{w_{i, j}^{(l)}=1} \mathbf{s}_j^{(l-1)}, \qquad \mathbf{s}_i^{(l+1)}=\bigvee_{w_{i, j}^{(l+1)}=1} \mathbf{r}_j^{(l)} .$$8

When the weights become binary, the MLLP behaves identically to the corresponding CRS (Bergamin et al., 26 Sep 2025).

The earlier MLLP formulation presents the same selectors as exact logical surrogates in the binary regime: $$\mathbf{r}_i^{(l)}=\bigwedge_{w_{i, j}^{(l)}=1} \mathbf{s}_j^{(l-1)}, \qquad \mathbf{s}_i^{(l+1)}=\bigvee_{w_{i, j}^{(l+1)}=1} \mathbf{r}_j^{(l)} .$$9 and defines

$\mathbf{r}_i^{(l)}$0

If $\mathbf{r}_i^{(l)}$1 and $\mathbf{r}_i^{(l)}$2 are binary, these reduce to exact Boolean conjunction and disjunction (Wang et al., 2019).

Training in the continuous space uses gradient descent. One formulation uses

$\mathbf{r}_i^{(l)}$3

with $\mathbf{r}_i^{(l)}$4 as $\mathbf{r}_i^{(l)}$5 regularization (Wang et al., 2019). The model design also confronts vanishing-gradient effects arising from product structure. The derived gradients contain multiplicative terms such as

$\mathbf{r}_i^{(l)}$6

and

$\mathbf{r}_i^{(l)}$7

which can become very small as layer width increases. The reported practical remedy is to initialize weights with $\mathbf{r}_i^{(l)}$8, so that $\mathbf{r}_i^{(l)}$9 and $\mathbf{s}_i^{(l)}$0 remain near $\mathbf{s}_i^{(l)}$1 and gradients remain healthier (Wang et al., 2019).

3. Discrete extraction, random binarization, and symbolic simplification

CRS is extracted from a trained MLLP by binarizing the learned weights with a threshold: $\mathbf{s}_i^{(l)}$2 The corresponding discrete model is then a symbolic logical network whose nodes correspond directly to rules and rule sets (Bergamin et al., 26 Sep 2025).

A central technical issue is the mismatch between continuous MLLP training and discrete CRS inference. To reduce this mismatch, Random Binarization (RB) binarizes a random subset of weights during training. Under RB, each weight is randomly selected with probability $\mathbf{s}_i^{(l)}$3: $\mathbf{s}_i^{(l)}$4 and then replaced by

$\mathbf{s}_i^{(l)}$5

RB is intended to make the continuous network more faithful to its future binarized CRS, but it is random and does not explicitly optimize sparsity (Bergamin et al., 26 Sep 2025). The earlier formulation further specifies that gradients are blocked for binarized weights and reports that RB makes the extracted CRS behave “almost the same” as the MLLP, whereas post-hoc thresholding alone often fails, especially for deeper networks (Wang et al., 2019).

After extraction, the transparent logical structure allows explicit simplification. Two procedures are described: dead node detection, which removes nodes with no path from top to bottom or nodes that are never activated on the training set, and redundant rules elimination, based on the subset test

$\mathbf{s}_i^{(l)}$6

A redundant-weight set is then identified by

$\mathbf{s}_i^{(l)}$7

These operations reduce complexity without changing behavior (Wang et al., 2019).

Empirically, RB is decisive for faithful discretization. On 12 public data sets, the extracted CRS without RB, denoted CRS($\mathbf{s}_i^{(l)}$8), has average Macro F1 $\mathbf{s}_i^{(l)}$9, even though the corresponding continuous MLLP($L$0) averages $L$1. With RB, the discrete CRS tracks the continuous model closely, and deep CRS models benefit especially when the binarization rate is roughly $L$2 to $L$3 on connect-4 (Wang et al., 2019).

4. Complexity control and differentiable $L$4 sparsification

CRS complexity is measured quantitatively as the total length of all rules, i.e. the total number of literals across all conjunction and disjunction nodes: $L$5 The terms $L$6 and $L$7 are interpreted as the number of nodes contained in a rule and a rule set, respectively; one example in the reported experiments has complexity $L$8 (Bergamin et al., 26 Sep 2025).

A later development adapts differentiable $L$9 regularization to the MLLP framework so that sparsity is induced directly by the loss function rather than indirectly via RB. The method uses hard concrete stochastic gates,

$2L+1$0

$2L+1$1

$2L+1$2

with active-gate probability

$2L+1$3

In the grouped form used for MLLP,

$2L+1$4

At test time, a deterministic estimator is used: $2L+1$5 The regularized objective becomes

$2L+1$6

with

$2L+1$7

This directly pushes many gates to zero, removing whole input-neuron groups from the computational graph and simplifying the extracted CRS (Bergamin et al., 26 Sep 2025).

The reported results emphasize a performance–interpretability trade-off that is not purely negative. On connect-4, for $2L+1$8, the F1 scores of MLLP and CRS are “similar or considerably improved” with $2L+1$9 regularization. For $\mathcal{S}^{(2L)}$0, performance tends to deteriorate, especially for shallow models. At $\mathcal{S}^{(2L)}$1, the total rule length is reduced by approximately eight times for the largest model, but with a slight performance decrease. The bank dataset shows regularized models surpassing baseline performance while being smaller, whereas letRecog becomes too small and performance drops. The same study notes that there are no guarantees of stability, that the method is limited to binary input features, and that difficult problems such as connect-4 can still produce large rule sets (Bergamin et al., 26 Sep 2025).

5. Rashomon-set exploration and the multiplicity of CRS-style explanations

Rule-set learning is inherently multi-solution: many distinct rule sets can achieve essentially the same predictive performance. Recent work on the Rashomon set of rule set models studies not only how to find one good rule set, but how to enumerate, count, or sample all rule sets whose objective is near-optimal. For binary prediction, the model class is a decision set or rule set over a pre-mined rule library with binary indicator vector

$\mathcal{S}^{(2L)}$2

where $\mathcal{S}^{(2L)}$3 means rule $\mathcal{S}^{(2L)}$4 is included, and the selected rules form an OR-of-rules classifier. The Rashomon set is defined operationally by

$\mathcal{S}^{(2L)}$5

with objective

$\mathcal{S}^{(2L)}$6

In the SAT-style encoding this is

$\mathcal{S}^{(2L)}$7

This framework is described as closely aligned with CRS-style interpretability because it treats sparse, human-readable rule sets as the relevant explanatory objects while explicitly acknowledging explanation multiplicity (Ciaperoni et al., 2024).

The exact exploration method is a branch-and-bound search over rule subsets with three pruning bounds: a hierarchical objective lower bound, a look-ahead bound, and a rule-set size bound. The appendix gives the incremental lower-bound update

$\mathcal{S}^{(2L)}$8

and objective update

$\mathcal{S}^{(2L)}$9

Approximate exploration uses random XOR or parity constraints for counting and sampling, estimating the number of constraints $\mathbf{s}_i^{(2L)}=1$0, counting solutions in a cell, and scaling by $\mathbf{s}_i^{(2L)}=1$1 (Ciaperoni et al., 2024).

The experiments show that exact pruning matters materially. On COMPAS with $\mathbf{s}_i^{(2L)}=1$2 and $\mathbf{s}_i^{(2L)}=1$3, removing the length bound increases runtime by about $\mathbf{s}_i^{(2L)}=1$4, removing the look-ahead bound increases runtime by about $\mathbf{s}_i^{(2L)}=1$5, and removing the hierarchical bound increases runtime by about $\mathbf{s}_i^{(2L)}=1$6. When the Rashomon threshold is tight, branch-and-bound scales well as the number of rules grows; when the threshold is larger and the Rashomon set explodes, approximate counting and sampling scale much better than exact enumeration. The sampled rule sets also preserve feature-frequency and model-reliance statistics closely enough that fair-model selection from samples yields test performance close to the optimal fair model found by exhaustive Rashomon-set search (Ciaperoni et al., 2024).

A plausible implication for CRS is that explanation should often be treated as a family of near-optimal sparse logical models rather than a single canonical rule set. In that reading, Rashomon-set exploration provides machinery for explanation diversity, stability analysis, and fairness-aware model selection within the same sparse rule-set regime.

6. Related rule-set frameworks and acronym ambiguity

The acronym “CRS” is used for several distinct constructs in the literature. This suggests a persistent bibliographic ambiguity that is especially relevant in arXiv-scale searches.

In causal inference, “Causal Rule Sets” denotes a Bayesian model for interpretable subgroup discovery. A Causal Rule Sets model uses a small set of short decision rules to capture a subgroup in which the average treatment effect is elevated, with subgroup membership defined by satisfaction of at least one rule in a disjunctive set (Wang et al., 2017). This is rule-based and interpretable, but its target is enhanced treatment effect rather than transparent classification.

In federated learning, “CRS-FL” refers to “Conditional Random Sampling for Communication-Efficient and Privacy-Preserving Federated Learning.” That CRS is a stochastic coordinate-sampling and sparsification scheme inserted after local training and before upload; it is explicitly not a concept rule set (Wang et al., 2023). In conversational recommendation, “CRS” stands for conversational recommender systems, as in the evaluation protocol “Concept,” which formalizes system-centric and user-centric evaluation factors for CRS models such as KBRD, BARCOR, UNICRS, and CHATCRS (Huang et al., 2024). In mathematical physics, the generalized CRS model refers to the one-dimensional Cariñena–Ranada–Santander model related to the generalized Higgs model (Song et al., 2013).

For work specifically on Concept Rule Sets, the defining lineage is the transparent classification program built around CRS and MLLP, followed by subsequent simplification through differentiable $\mathbf{s}_i^{(2L)}=1$7 regularization and by rule-set multiplicity analyses that are directly applicable to sparse interpretable models (Wang et al., 2019, Bergamin et al., 26 Sep 2025, Ciaperoni et al., 2024). The common misconception is therefore terminological rather than substantive: identical initials conceal substantially different model classes, objectives, and mathematical settings.