Rule representations significantly influence the search capabilities and decision boundaries within the search space of Learning Classifier Systems (LCSs). However, it is very difficult to choose an appropriate rule representation for each problem. Additionally, some problems benefit from using different representations for different subspaces within the input space. Thus, an adaptive mechanism is needed to choose an appropriate rule representation for each rule in LCSs. This article introduces a flexible rule representation using a four-parameter beta distribution and integrates it into a fuzzy-style LCS. The four-parameter beta distribution can form various function shapes, and this flexibility enables our LCS to automatically select appropriate representations for different subspaces. Our rule representation can represent crisp/fuzzy decision boundaries in various boundary shapes, such as rectangles and bells, by controlling four parameters, compared to the standard representations such as trapezoidal ones. Leveraging this flexibility, our LCS is designed to adapt the appropriate rule representation for each subspace. Moreover, our LCS has a generalization bias to produce as many crisp rules as possible. Experimental results on real-world classification tasks show that our LCS significantly outperformed LCSs with popular rule representations in test classification accuracy on up to 17 of the 25 datasets tested.

The decision-making process significantly influences the predictions of machine learning models. This is especially important in rule-based systems such as Learning Fuzzy-Classifier Systems (LFCSs) where the selection and application of rules directly determine prediction accuracy and reliability. LFCSs combine evolutionary algorithms with supervised learning to optimize fuzzy classification rules, offering enhanced interpretability and robustness. Despite these advantages, research on improving decision-making mechanisms (i.e., class inference schemes) in LFCSs remains limited. Most LFCSs use voting-based or single-winner-based inference schemes. These schemes rely on classification performance on training data and may not perform well on unseen data, risking overfitting. To address these limitations, this article introduces a novel class inference scheme for LFCSs based on the Dempster–Shafer Theory of Evidence (DS theory). The proposed scheme handles uncertainty well. By using the DS theory, the scheme calculates belief masses (i.e., measures of belief) for each specific class and the “I don’t know” state from each fuzzy rule and infers a class from these belief masses. Unlike the conventional schemes, the proposed scheme also considers the “I don’t know” state that reflects uncertainty, thereby improving the transparency and reliability of LFCSs. Applied to a variant of LFCS (i.e., Fuzzy-UCS), the proposed scheme demonstrates statistically significant improvements in terms of test macro F1 scores across 30 real-world datasets compared to conventional voting-based and single-winner-based fuzzy inference schemes. It forms smoother decision boundaries, provides reliable confidence measures, and enhances the robustness and generalizability of LFCSs in real-world applications.

Function approximation is a critical task in various fields. However, existing neural network approaches struggle with locally complex or discontinuous functions due to their reliance on a single global model covering the entire problem space. We propose X-KAN, a novel method that optimizes multiple local Kolmogorov-Arnold Networks (KANs) through an evolutionary rule-based machine learning framework called XCSF. X-KAN combines KAN's high expressiveness with XCSF's adaptive partitioning capability by implementing local KAN models as rule consequents and defining local regions via rule antecedents. Our experimental results on artificial test functions and real-world datasets demonstrate that X-KAN significantly outperforms conventional methods, including XCSF, Multi-Layer Perceptron, and KAN, in terms of approximation accuracy. Notably, X-KAN effectively handles functions with locally complex or discontinuous structures that are challenging for conventional KAN, using a compact set of rules (average 7.2 rules). These results validate the effectiveness of using KAN as a local model in XCSF, which evaluates the rule fitness based on both accuracy and generality. Our X-KAN implementation and an extended version of this paper, including appendices, are available at https://doi.org/10.48550/arXiv.2505.14273.

This paper focuses on “prototypes” as representative samples that can serve as a summary of the original dataset and proposes a novel Learning Classifier System (LCS) to generate a small number of prototypes that can achieve high classification accuracy in class classification. Concretely kNNUCS-PG is proposed to improve the sUpervised Classifier System (UCS) as one of LCSs by changing from the interval (rule) representation to the point representation to represent a prototype by employing k-nearest neighbors (kNN) to match the neighborhood prototypes. For a high classification accuracy with data reduction, selecting and generating prototypes that can maintain decision boundaries is necessary. Since UCS has generalization mechanisms, including rule merging, unnecessary rule deletion, and new rule discovery using niche Genetic Algorithm (GA), such techniques are highly useful for evolving indispensable prototypes as generalized prototypes. This paper compared conventional prototype-based methods with kNNUCS-PG in experiments using three benchmark problems and eight table datasets. kNNU CS-PG can maintain high classification accuracy even for complex problems with much overlap in the datasets. The results show that kNNUCS-PG is a robust PG that can maintain high classification accuracy while reducing data for various complex problems.

This paper focuses on the impact of rule representation in Michigan-style Learning Fuzzy-Classifier Systems (LFCSs) on its classification performance. A well-representation of the rules in an LFCS is crucial for improving its performance. However, conventional rule representations frequently need help addressing problems with unknown data characteristics. To address this issue, this paper proposes a supervised LFCS (i.e., Fuzzy-UCS) with a self-adaptive rule representation mechanism, entitled Adaptive-UCS. Adaptive-UCS incorporates a fuzzy indicator as a new rule parameter that sets the membership function of a rule as either rectangular (i.e., crisp) or triangular (i.e., fuzzy) shapes. The fuzzy indicator is optimized with evolutionary operators, allowing the system to search for an optimal rule representation. Results from extensive experiments conducted on continuous space problems demonstrate that Adaptive-UCS outperforms other UCSs with conventional crisp-hyperrectangular and fuzzy-hypertrapezoidal rule representations in classification accuracy. Additionally, Adaptive-UCS exhibits robustness in the case of noisy inputs and real-world problems with inherent uncertainty, such as missing values, leading to stable classification performance.

To mine high-dimensional rules in Learning Classifier Systems (LCSs) through a reduction of the dimensionality of input data, this paper proposes a novel approach that indirectly learns the rules in the "latent space" based on the rewards of the reconstructed rules in the "observation space". We call this approach Learning Strategy by exploring rules in Observation space via Latent space (LS-OvL), which is based on two rule representations in the observation and latent space. Concretely, LS-OvL explores the rules by searching the latent space as the reduced dimensional input space by an autoencoder and evaluates them in the observation space by reconstructing them from the latent space. Such a design is significant because it prevents the generation of inaccurate rules during the reconstitution process from the latent space to the observation space. Through a comparison LS-OvL with the conventional learning strategy, which explores and evaluates the rules in the only latent space and reconstructs them in the observation space, the experimental results show that (1) LS-OvL outperforms the conventional learning strategy in terms of the acquired reward and the population size, and (2) LS-OvL can generate explainable and classifiable high-dimensional rules.

This paper proposes the Beta Distribution based XCS Classifier System (called j3-XCS) as the novel XCS having the new representation (1) that can handle curved surface hyperpolyhedral conditions, including hyperellipsoids, (2) that can “quickly” and “stably” evolve classifiers that appropri-ately classify the area, and (3) that is robust to the initial hyperparameters of the representation. Concretely, j3-XCS is composed of classifiers that condition part in each dimension is represented by the beta distribution that can change a flexible distribution shape according to its parameters. Through the intensive experiments of the different types of continuous space problems, the following implications have been revealed: (1) j3-X CS can show higher classification performance and function approximation performance with fewer classifiers than other XCSs with the conventional representations such as XCS with the hyperrectangular representation (i.e., XCSR) and XCS with the hyperellipsoidal representations (i.e., hyperellipsoid-based XCS); (2) fJ-XCS can quickly and stably evolve the classifiers that can appropriately match the line and curved shapes in comparison with XCSR and the ellipsoidal-based XCS; and (3) while the performance of the conventional XCSs is highly sensitive to the hyperparameter that defines the generality of the covering classifier, the performance of fJ-XCS is the most robust to its values.

This paper focuses on the matching mechanism of Learning Classifier System (LCS) in a continuous space and proposes a novel matching mechanism based on Gaussian distribution. This mechanism can match the “area” instead of the “point (one value)” in the continuous space unlike the conventional LCS such as XCSR (XCS with Continuous-Valued Inputs). Such an area matching contributes to generating the rules (called classifiers) with less-overlapped with other rules. Concretely, the proposed area matching mechanism employed in XCSR using VAE can generate appropriate classifiers for latent variables with high-dimensional inputs by VAE and create a human-interpretable observation space of human-interpretable classifiers. Since the latent variable in VAE is followed by Gaus-sian distribution, the following three matching mechanisms are compared: (i) the (single) point matching that selects the classifier which condition covers the mean of Gaussian distribution M; (ii) the multiple points matching that selects the classifier which condition covers the data sampled from Gaussian distribution (M, u); and (iii) the area matching that selects the classifier which condition roughly covers a certain area of Gaussian distribution (M, o). Through the intensive experiments on the high dimension maze problem, the following implications have been revealed: (1) the point matching in XCSR with VAE generates the ambiguous classifiers which conditions are overlapped with the other classifiers with the different action; (2) the sampling multiple points matching in XCSR with VAE has a potential of generating the less-overlapped classifiers by improving the data set through sampling. (3) the proposed area matching can generate the less-overlapped classifiers with the same learning steps, which corresponds to the time of the point matching.

This paper focuses on the concept of "absumption" which restrains over-general rules by decomposing them into several concrete rules, proposes the novel "absumption" for continuous spaces by improving the conventional absumption to achieve high performance (e.g., the acquired rewards) in a noisy environment, and integrates it into the XCS for real-valued inputs (XCSR) to evaluate it through a comparison with the conventional absumption. Concretely, the proposed absumption mechanism based on Overgenerality and Condition-clustering based specialization (called Absumption-OC) manipulates the balance between the Overgenerality of the rules and the specialization of Condition of the rules. Through the intensive experiments of three different types of continuous space problems, the following implications have been revealed: (1) XCSR with Absumption-OC shows the statistically significant performance in the acquired reward, the system error, and the population size against XCSR with the conventional absumption; and (2) this effectiveness of Absumption-OC becomes to be clear in noisy reward environments in comparison within noiseless reward environments.

This paper focuses on the rule representation in Learning Classifier System (LCS) and proposes a flexible representation mechanism that can generate a variety of shapes of its matching area with one rule condition of a classifier. Concretely, the proposed representation mechanism changes the shape of the matching area according to the logical product or multiplication of the values of the probability distribution in the multiple dimension. As one of its implementation, this paper introduces the beta distribution in XCS for continuous space. Through intensive experiments of different types of continuous space problems, the following implications have been revealed: XCS based on the beta distribution (1) can match line and curved shapes using the same classifier; (2) can obtain higher reward values with the same or fewer classifiers than the conventional representations (i.e., the hyperrectangular and hyperellipsoidal representations); and (3) is robust to variations in the shape of the class boundary, contributing to stable classification performance.

This paper proposes ELPSDeCS (Encoding, Learning, "Plausible" Sampling, and Decoding Classifier System) by extending ELSDeCS (Encoding, Learning, Sampling, and Decoding Classifier System) to increase both the accuracy and interpretability of the generated classifiers which matches the high-dimensional input such as images. The experimental results on the complex multi-class classification problem of the handwritten numerals show that both the accuracy and interpretability of ELPSDeCS are higher than that of ELSDeCS.