Fuzzy Logic

This study presents the Neuro-Memory Fuzzy Inference System (NeMeFIS), a hierarchical machine learning architecture that asymmetrically models acceleration and deceleration in car following behavior by integrating five human memory types procedural, working, episodic, semantic, and declarative. By linking external variables to memory functions via metaheuristics and validating them through factor and p-value analyses, NeMeFIS uncovers latent cognitive influences across Arterial, Collector, and Rural Highway corridors for different types of vehicles. Results from 54 different trained models emphasize cognitive thresholds shaped by driver perception limits and cognitive load. The trained NeMeFIS models outperform traditional statistical and conventional machine learning models in replicating realistic driving behavior, including comparisons with Linear Regression, ANFIS, and LSTM architectures. Fuzzy rule analysis reveals that declarative memory demands the highest rule, especially during deceleration, indicating complex braking decisions. Procedural memory drives acceleration, while semantic and declarative memory guide deceleration. Risk perception also emerges as a key factor, particularly on urban roads. Validated on both heterogeneous and homogeneous datasets, NeMeFIS offers a robust framework for modeling driver cognition. The findings support psychotherapeutic applications and the development of adaptive, human-like decision systems in Connected and Autonomous Vehicles (CAVs) to enhance traffic safety.

Foundation models pretrained on large-scale datasets demonstrate strong transferability to medical imaging tasks. However, understanding how their latent representations encode clinically relevant information remains an open challenge in safety-critical domains. This study proposes a prototype-based fuzzy-rule framework that interprets the patch-level features produced by the inner layers of pretrained foundation models, without any fine-tuning. Class-specific prototypes are learned by clustering in the feature space, yielding compact visual patterns. Patch features are then expressed as prototype similarities and classified by fuzzy rules with linguistic IF-THEN conditions that are human readable. The framework is applied across the final two blocks of ViT-S/16 backbones pretrained on ImageNet-1K and GastroNet-5M, and benchmarked against k-nearest neighbours, kernel SVM, and linear probing under identical frozen features, on wireless capsule endoscopy classification, gastrointestinal endoscopy classification, and colonic polyp segmentation. The experimental analysis shows that the proposed method, without backbone fine-tuning, reaches accuracy comparable to these black-box classifiers, and that domain-specific pretraining yields features that are both discriminative and symbolically compressible. Because the resulting rules are extracted from real data and expressed in interpretable terms, they are further used as an instrument to investigate synthetic medical images, providing a human-readable account of which real prototypes and rules a generator reproduces or fails to reproduce, localising where a synthetic image departs from real tissue rather than summarising it with a single score. The framework thus offers a transparent, depth-resolved view of how foundation models organise clinically relevant structure, together with a practical downstream use of the extracted rules.

Synthetic tabular data generation provides an effective alternative for the training of machine learning models when real-world data is limited or inaccessible. However, the heterogeneous, non-smooth, and incomplete nature of tabular data poses fundamental challenges to conventional probabilistic and deep generative models, where their interpretability remains limited. This paper proposes a novel fuzzy distribution modeling methodology for synthetic tabular data generation based on fuzzy sets theory. Feature distributions are represented using fuzzy sets and feature dependencies are modeled through Fuzzy Cognitive Maps, resulting in a low-parameter, and an interpretable data representation. Synthetic samples are generated by sampling fuzzy concepts rather than raw values, enabling native support for mixed data types, missing values, and domain constraints. The methodology further supports linguistic queries and IF-THEN reasoning, facilitating transparent simulation of decision-making processes. Experimental results on benchmark datasets demonstrate competitive performance with respect to utility, fidelity and privacy compared to state-of-the-art methods, while offering substantially improved interpretability. These results establish fuzzy distribution modeling as a principled and effective approach for synthetic tabular data generation in fuzzy systems and decision support applications.

The shipping industry has a significant impact on the global economy, emphasizing the need for operational availability and safety through the use of effective maintenance techniques. During the last decades, predictive maintenance (PdM) has emerged as a promising solution compared to the existing conventional maintenance systems. This is because it offers several advantageous functions, such as damage predictions for vessel components, reduced downtime, improved and extended life of machinery, as well as higher safety during voyages. However, existing methodologies developed for performing PdM do not provide explanations of their results to users, so that they can understand the failures that may occur. To address this limitation, this paper proposes a novel framework based on a fuzzy decision tree and a deep residual neural network, aiming to perform explainable PdM on naval vessels. The proposed framework is able to generate fuzzy local rules based on the dataset used, and can provide explanations of its outcomes, using cause-and-effect relationships, in a way that are understandable to users, thereby gaining their trust. Experiments using a publicly available dataset demonstrate the effectiveness of the proposed framework, as it achieves an accuracy of 99.24%.

In-space servicing has been receiving great attention to extend the operation of spacecraft with defective components. This requires rendezvous and proximity operations for a chaser to provide service to a target. This work constructs a fuzzy inference system-based controller for the chaser to reach the cooperative target on a circular orbit in the final approach phase while minimizing the energy consumption of the chaser. The offline training process performed by a genetic algorithm deals with multiple initial relative positions of the chaser, and the trained controller is validated using a testing environment with disturbances, which differs from the training scenarios.

This paper proposes a decentralized approach for a multi-robot system (MRS) using a genetic fuzzy system to perform a collaborative object transportation task that minimizes the total path length of the MRS in unstructured environment while avoiding obstacles. For an environment given by an elevation map, terrain traversability analysis with respect to the slope is performed to reduce the dimension and identify non-traversable areas that can be considered as obstacles, and the given map is converted into a traversability map in two dimensional space. In the training process, proposed fuzzy inference systems (FISs) to generate the MRS's velocity for transporting an object to a target position are optimized by a genetic algorithm with several scenarios, such as a local minima, a target that is close to an obstacle, and a cluttered environment. The trained FIS models are applied to the testing environment, which is the converted traversability map, and validated using multiple scenarios.

The adaptive neuro-fuzzy inference system (ANFIS) is an interpretable reasoning framework capable of generating explicit IF-THEN fuzzy rules, making it suitable for tasks requiring transparent reasoning. However, existing ANFIS models generally construct rule antecedents and perform inference in Euclidean space, limiting their representational capacity and predictive performance. To address this issue, we propose Hyperbolic ANFIS (HyperANFIS), a hyperbolic extension of ANFIS. HyperANFIS preserves the fuzzy semantics and core architecture of conventional ANFIS while performing rule-prototype learning, rule activation, and consequent aggregation in hyperbolic space. It also retains the ability to generate interpretable IF-THEN rules. By exploiting the representational properties of hyperbolic geometry, HyperANFIS strengthens the fuzzy inference process, thereby improving predictive accuracy, inter-rule collaboration, and the credibility of its interpretable rules. Experimental results show that HyperANFIS consistently outperforms the standard ANFIS baseline and various ANFIS variants across all datasets, while also generating higher-quality fuzzy rules.

The current state-of-the-art (SOTA) deep randomized neural networks, such as deep Random Vector Functional Link (dRVFL) and ensemble deep RVFL (edRVFL), treat all training samples uniformly, which limits their robustness and effectiveness when applied to real-world datasets containing noise and outliers. Furthermore, the propagation of contaminated features across hidden layers negatively influences the decision-making capability of these models. To overcome these limitations, we propose intuitionistic fuzzy dRVFL (IF-dRVFL) and intuitionistic fuzzy edRVFL (IF-edRVFL) frameworks that enhance model robustness. The proposed models unify intuitionistic fuzzy theory to exploit sample neighborhood information in the kernel space by jointly considering membership and non-membership degrees for each sample. Membership degrees are computed based on the distance of samples from their respective class centroids, while non-membership degrees quantify sample heterogeneity within local neighborhoods. These measures are employed to assign adaptive weights to training samples, enabling effective discrimination among clean, noisy, and outlier data points. Extensive experiments conducted on UCI and KEEL benchmark datasets, with and without the presence of Gaussian noise, demonstrate the superiority of the proposed IF-dRVFL and IF-edRVFL models over existing SOTA fuzzy and non-fuzzy approaches. The source code is available at https://github.com/mtanveer1/IF-edRVFL.

Interpretable classification often requires more than accurate predictions for real-life deployment: models should be transparent about the evidence behind their decisions and abstain when they cannot decide reliably. We introduce Fast Evidential Rule Learning (FERL), a method that learns interpretable, accurate fuzzy rule models whose outputs are evidential. Unlike post-hoc calibration, FERL's belief, plausibility, and abstention capabilities arise directly from the fuzzy memberships in a single deterministic pass, with no auxiliary head, held-out set, or repeated inference. Our theoretical analysis further shows that FERL is Lipschitz stable, which means that its evidential outputs vary smoothly with the input. Against state-of-the-art rule learners, FERL is statistically significantly more accurate across a 30 tabular-dataset benchmark ($+2.6\%$ average accuracy over the second best). Its native set predictions attain the best utility-discounted accuracy among credal classifiers ($u_{65}/u_{80}=0.80/0.83$ vs.\ $0.79/0.80$ for the naive credal classifier), at higher set coverage ($0.92$ vs.\ $\le0.82$). FERL also matches dedicated out-of-distribution detectors on tabular near-OOD detection ($77.7$ vs.\ $77.4$ AUROC for the strongest baseline). Under detector-class-disjoint concept-bottleneck evaluation, its it is within $2.3$ AUROC points of the strongest dedicated detector on both CUB and AwA2, while attaining the best AwA2 AUPR-Out ($68.3$) and novel-class rejection ($57.2$), while being able to name which attributes are anomalous.

Vision-based guidance of unmanned aerial vehicles (UAVs) toward unmanned ground vehicles (UGVs) supports cooperative aerial--ground robotics, but reliable continuous yaw estimation from onboard vision remains challenging because of sensing uncertainty, limited computation, and the need for interpretable control. Existing deep-learning and geometric-reconstruction approaches often require large datasets, external localization, or complex modeling assumptions, reducing transparency and deployment suitability on resource-constrained platforms. We present an interpretable fuzzy-inference framework that generates continuous yaw commands from low-dimensional features extracted from YOLO boxes: target centroid location, area, and aspect ratio. No explicit geometric modeling is required. A Mamdani fuzzy system serves as an interpretable baseline using a shoulder--triangle--shoulder input partition. It is followed by a first-order Takagi--Sugeno model with three antecedent membership terms per input, whose parameters are derived from training-set quantiles, yielding a compact 27-rule structure. Evaluation uses 6{,}169 labeled samples from a VICON motion-capture environment. Across five randomized train--test splits, the Takagi--Sugeno model achieves a test-set mean absolute error of $0.140^\circ \pm 0.003^\circ$, a root mean squared error of $0.200^\circ \pm 0.008^\circ$, and a maximum absolute error of $1.254^\circ \pm 0.121^\circ$. Within-threshold accuracies are $99.676$ for $\pm1^\circ$ and $100.000$ for both $\pm3^\circ$ and $\pm5^\circ$. Directional consistency between image-plane horizontal displacement and predicted yaw sign reaches $90.254$. These results show that the framework is transparent, data-efficient, computationally lightweight, and suitable for real-time vision-based UAV guidance toward mobile ground targets.

Standard automated sewer pipe severity assessment relies on direct image classification, creating a "black box" where the link between visual defects and final severity scores remains implicit. This study introduces a modular, fuzzy rule-based neuro-symbolic framework that bridges this gap by decoupling neural perception from symbolic reasoning. The perception module utilizes a Swin Transformer to predict 14 multilabel inspection CODE degrees directly from images. For reasoning, a DT, specifically Weka's J48, algorithm is trained on ground-truth CODEs and severity labels, and its paths are converted into 19 fixed IF--THEN rules. Inference operates via fuzzy logic: t-norm activations from CODE conditions are weighted by rule confidence and combined with corresponding s-norms to produce interpretable class evidence. We assessed Product, Łukasiewicz, and Hamacher operator pairs using a dataset of 3,244 images spanning five highly imbalanced severity classes. Ground-truth labels were robustly generated via consensus from five independent large language models analyzing original inspector notes. Our results show an improvement of accuracy, balanced accuracy, Macro F1 and MCC by 17.9%, 12.2%, 23.0%, and 17.3%, respectively, over image-only based classification. Overall, the framework combines competitive class-balanced performance with traceable reasoning from predicted CODE degrees to rule supports and severity evidence.

Dissolved gas analysis (DGA) is crucial for diagnosing early power transformer failures. Traditional DGA interpretation methods like Duval Triangle, IEC ratio, Roger ratio, Doernenburg ratio and Key Gas are inconsistent and vary in accuracy, especially for multiple fault conditions. We propose an Adaptive Multi-Fuzzy Logic (AMFL) model integrating multiple DGA methods with fuzzy logic and a dynamic weight adjustment mechanism. Unlike existing approaches with fixed weights, this system iteratively evaluates each method's diagnostic performance, identifies multiple fault types, and adjusts weights based on fault prediction accuracy. A feedback-based optimization recalibrates weights after each cycle to ensure optimal solution convergence. The model, implemented in MATLAB/Simulink, is validated against DGA datasets with known error conditions. Results show the AMFL model significantly improves diagnostic accuracy, especially in complex error scenarios, and enhances adaptability to new datasets. Comparative analysis demonstrates the proposed method outperforms traditional fixed weight multi-fuzzy systems in accuracy, consistency, and reliability of error detection. This work provides a robust, flexible diagnostic tool for transformer condition monitoring and supports more accurate asset management decisions.

Fuzzy Integral (FI) based aggregation provides a powerful mechanism for nuanced aggregation, for example, in ensemble approaches or decision-level fusion more generally. The main challenge of this approach is the appropriate parametrization of the Fuzzy Measure (FM), which captures the worths of the individual components--and their combinations--which are being fused. Here, widely used approaches including the Sugeno-$λ$ and Decomposable FMs, parametrize the FM by extrapolating from the densities, i.e. the weights associated with individual sources, while respecting the FM's monotonicity constraint. This paper articulates that this information is, in general, insufficient to uniquely identify a discrete FM; but shows how an interval-valued FM can indeed be determined uniquely. We proceed to show how the incorporation of additional information beyond the above, such as the choice of a specific FI and a dataset, then allows for obtaining even more specific interval-valued FMs. In practice, establishing the quality of an empirically determined FM is not trivial. To help address this, we show how the likelihood with which a resulting interval FM encompasses the ideal', i.e. the commonly intangible, best, or ground-truth numeric FM, can be determined, producing a confidence interval at a given confidence level. Finally, based on a series of experiments, we demonstrate empirically that the Choquet FI output based on this FM can also be regarded as the confidence interval for the ideal' information fusion result, providing a novel means to characterize FI fusion outcomes a priori and charting a pathway for future research.

Machine learning models achieve high predictive accuracy in regression tasks, but their deployment in safety-critical and regulated domains requires interpretability. While fuzzy rule-based systems offer transparent, linguistically explicit interpretable models, Mamdani-style fuzzy regression remains underrepresented in modern machine learning software libraries. This paper presents an interpretable regression extension for the Ex-Fuzzy library, enabling Mamdani fuzzy inference with scalar consequents learned directly from data. For this, a target-aware partition initialisation strategy based on Fuzzy C-Means clustering is introduced, in which linguistic variables are derived from an augmented input-output space to emphasise output-relevant regions of the feature space. The proposed extension is evaluated on ten regression datasets from the KEEL repository, comparing Gaussian and trapezoidal partition strategies against standard baselines including linear regression, multilayer perceptron, and random forests. Experimental results show that Gaussian partitions consistently outperform uniform trapezoidal partitions, achieving a mean coefficient of determination of approximately 0.86 while producing compact rule bases of 10-15 human-readable rules. The proposed implementation provides a transparent and competitive alternative to black-box regression models, supporting practical interpretability with competitive predictive performance.

Private blockchain networks run with fixed node configurations that cannot adapt to changing workload conditions. Too many nodes serving a light workload waste resources; too few nodes facing heavy demand slow block production and degrade finalisation. The right validator count is hard to determine, as it depends on overlapping factors that shift over time. This paper presents a Takagi-Sugeno (TS) fuzzy inference system that reads live blockchain parameters (block production time, block size, and active node count) and outputs a continuous efficiency score alongside a scaling recommendation: Scale Up, Maintain, or Scale Down. The controller uses triangular membership functions across three linguistic variables, evaluated through a complete 27-rule base with product t-norm aggregation. A key contribution is an empirical recalibration of the membership functions, anchoring linguistic terms to the observed operating range of the testbed rather than to theoretical extremes. The system is evaluated on a 10-node Substrate blockchain network storing real smart water meter data hashes from the Queensland Government open data portal. Statistical analysis across configurations of 4, 7, and 10 active nodes confirms that the controller produces distinct operational profiles reflecting each configuration's provisioning state. In closed-loop experiments, the controller autonomously adjusts validator participation in both directions, activating validators under rising load and removing them under over-provisioning, converging to the same stable equilibrium from both directions. Compared against three threshold-based baselines, it shows fewer scaling oscillations while maintaining comparable block production times. Results show that TS fuzzy inference can support autonomous validator management in private blockchain deployments, with stable scaling behaviour threshold approaches cannot match.

Human reasoning often operates through qualitative concepts expressed by linguistic labels such as high, low, expensive, or cheap, whose interpretation depends on context and is usually vague, despite being rooted in numerical data. This paper explores a novel fuzzy-logic-based qualitative extension of Answer Set Programming (ASP) to bridge numerical information and qualitative reasoning. The underlying language, formally introduced in a separate work, provides a principled framework that avoids rigid thresholds and supports robust reasoning under vagueness. Focusing on a representative use case, we illustrate how the framework integrates numerically grounded inputs (such as outputs of machine learning models) with symbolic reasoning over qualitative labels. Key features, including learning-based membership functions and semantically enriched predicates, enable the combination of expert knowledge, contextual factors, and subjective interpretations within a unified declarative setting.

A transformer's feed-forward (FFN) sublayer materializes the distinctions attention gathers, yet gives no account of what it computes. In a parameter-neutral replacement, each hidden unit is an explicit fuzzy set operation on sigmoid-bounded [0,1] memberships: intersection AB and set-difference A(1-B), the latter a bounded positive negation ("A but not B") that gated/bilinear units lack -- a negation-capable FFN (NC-FFN). On N-bit parity they are the most parameter-efficient reasoning basis at shallow depth; at scale (125M, OpenWebText) NC-FFN ties the GELU baseline's perplexity, every unit carrying explicit logical form. Two limits share one cause: two-operand logic localizes to layer 0 and erodes under training, and the one robust grammatical deficit concentrates in licensing and quantifiers, beyond within-token operators. We resolve both with a small block of sequence quantifiers: a soft existential and a soft proportion, each with a per-unit learned forgetting rate from a sticky init. This recovers the deficit at epoch one (halving the wider epoch-two gap), modestly leads on LAMBADA, and makes the FFN legible: the structure now holds and migrates into depth; the decay un-learns its stickiness (median half-life ~1.5 tokens; zero latch units); and at the semantic layers the units read, without dictionary learning, as grammatical licensing detectors: each fires on a licensor (a comparative, a passive participle, a negative-polarity item) and carries its memory forward to predict the licensed word (than, by, nor). This legibility is localized and free only up to a partition (a fully Boolean FFN diverges in training), but the result is a parameter-neutral, language-model-quality transformer with a readable, interpretable-by-construction grammatical mechanism -- an account not just of what a feed-forward layer represents but how it licenses.

Increasing demand for precise and reliable control in complex scenarios has led to the development of increasingly sophisticated controllers, including data-driven approaches employing closed box models and mathematically rigorous yet complex designs. This complexity highlights the needs for explainable control that can provide human-understandable insights into controller behavior. In this paper, an explainable control framework (XCF) along with supporting algorithms and user interface are proposed to explain how controllers determine their control actions and their underlying working mechanism. The novel contributions of this work are threefold: First, the XCF is designed to provide model-agnostic explanations for controllers in closed-loop systems and can optionally refine local explanations by system response dynamics. Second, a novel explanation method, hierarchical fuzzy model-agnostic explanation for control systems (HFMAE-C), is proposed based on the designed framework. The HFMAE-C employs a fuzzy logic system to approximate the controller's behavior and system dynamics, providing sample, local, domain and universe level explanations via IF-THEN rules revealing the controller's decision logic and salience values quantifying the contribution of system states to control actions. Third, a large language model agent-supported user interface is developed to automatically analyze user requirements, select appropriate algorithms, interpret the generated explanations to a natural language report, and provide interactive consultation. Case studies on inverted pendulum system and Turtlebot obstacle avoidance demonstrate the effectiveness of the proposed method through simulated user experiments and quantitative comparisons with mainstream explainable control approaches.

This paper presents a versatile framework for evaluating fuzzy quantification queries over both standard and fuzzy ontologies as well as knowledge graphs. The primary objective is the retrieval of individuals that satisfy queries articulated via Type I or Type II fuzzy quantified expressions. A key advantage of the proposed approach is its inherent adaptability: it remains entirely agnostic to the quantifier type, the underlying evaluation method, and the specific data source of the ontology (i.e., OWL ontologies or RDFS knowledge graphs). Furthermore, we present Q2S2, a publicly accessible implementation of this system developed to support future research.

Modern intrusion detection systems generate thousands of alerts daily, but alert fatigue severely limits security operations effectiveness due to too many false positives or low-impact events. We address this by proposing a principled framework for alert prioritization based on subnormal Gaussian fuzzy numbers, explicitly modeling three sources of uncertainty: threat severity, detection confidence, and organizational risk attitude. Each alert is represented as a fuzzy number with the core indicating severity, spread indicating uncertainty, and height reflecting detection reliability. We apply ranking indices to prioritize alerts, allowing organizations to tune security posture through a risk-attitude parameter. Experimental validation on CIC-IDS2017 and NSL-KDD demonstrates greater robustness than baselines under detector degradation (0.9963 vs 0.8215 NDCGrel@100), with distinct differentiation in mid-confidence alerts and near-parity with baselines under robust detectors. The framework is theoretically grounded, computationally efficient, provides interpretable reasoning, and remains robust across detector families and miscalibration scenarios.

Mediative Fuzzy Logic was conceived as a practical scheme for reconciling hesitant or conflicting assessments in fuzzy control and decision-making. However, its logical and semantic foundations remain underdeveloped, especially beyond operational type-1 settings. This article develops a unified account of the type-1 core together with interval type-2, granular type-3, and quantum extensions. We characterize the mediative operator as a convex aggregation controlled by hesitation and contradiction, model mediative truth values as independent truth-falsity pairs in a continuous bilattice-like structure, and introduce a propositional system extending a standard t-norm-based fuzzy logic with a mediative connective. We establish soundness, paraconsistency, and conservativity over the underlying fuzzy base for formulas without mediation, and formulate coherent semantic extensions to interval type-2 truth values, granule-indexed local evaluations, and effects and density operators on Hilbert spaces. An autonomous-braking sensor-fusion example illustrates how the framework supports transparent, conservative, and safety-first decisions under incomplete, heterogeneous, and mildly contradictory evidence. Under suitable assumptions, the higher-level formulations reduce to the type-1 case, clarifying coherence across levels and reliably supporting future work in intelligent decision systems.

Reinforcement learning (RL) often struggles in real-world tasks with high-dimensional state spaces and long horizons, where sparse or fixed rewards severely slow down exploration and cause agents to get trapped in local optima. This paper presents a fuzzy logic based reward shaping method that integrates human intuition into RL reward design. By encoding expert knowledge into adaptive and interpreable terms, fuzzy rules promote stable learning and reduce sensitivity to hyperparameters. The proposed method leverages these properties to adapt reward contributions based on the agent state, enabling smoother transitions between fast motion and precise control in challenging navigation tasks. Extensive simulation results on autonomous drone racing benchmarks show stable learning behavior and consistent task performance across scenarios of increasing difficulty. The proposed method achieves faster convergence and reduced performance variability across training seeds in more challenging environments, with success rates improving by up to approximately 5 percent compared to non fuzzy reward formulations.

Recent advances in Deep Learning (DL) have strengthened data-driven System Identification (SysID), with Neural and Fuzzy Ordinary Differential Equation (NODE/FODE) models achieving high accuracy in nonlinear dynamic modeling. Yet, system states in these frameworks are often reconstructed without clear physical meaning, and input contributions to the state derivatives remain difficult to interpret. To address these limitations, we propose Explainable FODE (xFODE), an interpretable SysID framework with integrated DL-based training. In xFODE, we define states in an incremental form to provide them with physical meanings. We employ fuzzy additive models to approximate the state derivative, thereby enhancing interpretability per input. To provide further interpretability, Partitioning Strategies (PSs) are developed, enabling the training of fuzzy additive models with explainability. By structuring the antecedent space during training so that only two consecutive rules are activated for any given input, PSs not only yield lower complexity for local inference but also enhance the interpretability of the antecedent space. To train xFODE, we present a DL framework with parameterized membership function learning that supports end-to-end optimization. Across benchmark SysID datasets, xFODE matches the accuracy of NODE, FODE, and NLARX models while providing interpretable insights.

Recent advances in Deep Learning (DL) have boosted data-driven System Identification (SysID), but reliable use requires Uncertainty Quantification (UQ) alongside accurate predictions. Although UQ-capable models such as Fuzzy ODE (FODE) can produce Prediction Intervals (PIs), they offer limited interpretability. We introduce Explainable Type-2 Fuzzy Additive ODEs for UQ (xFODE+), an interpretable SysID model which produces PIs alongside point predictions while retaining physically meaningful incremental states. xFODE+ implements each fuzzy additive model with Interval Type-2 Fuzzy Logic Systems (IT2-FLSs) and constraints membership functions to the activation of two neighboring rules, limiting overlap and keeping inference locally transparent. The type-reduced sets produced by the IT2-FLSs are aggregated to construct the state update together with the PIs. The model is trained in a DL framework via a composite loss that jointly optimizes prediction accuracy and PI quality. Results on benchmark SysID datasets show that xFODE+ matches FODE in PI quality and achieves comparable accuracy, while providing interpretability.

Adaptive Neuro-Fuzzy Inference System (ANFIS) was designed to combine the learning capabilities of neural network with the reasoning transparency of fuzzy logic. However, conventional ANFIS architectures suffer from structural complexity, where the product-based inference mechanism causes an exponential explosion of rules in high-dimensional spaces. We herein propose the Kolmogorov-Arnold Neuro-Fuzzy Inference System (KANFIS), a compact neuro-symbolic architecture that unifies fuzzy reasoning with additive function decomposition. KANFIS employs an additive aggregation mechanism, under which both model parameters and rule complexity scale linearly with input dimensionality rather than exponentially. Furthermore, KANFIS is compatible with both Type-1 (T1) and Interval Type-2 (IT2) fuzzy logic systems, enabling explicit modeling of uncertainty and ambiguity in fuzzy representations. By using sparse masking mechanisms, KANFIS generates compact and structured rule sets, resulting in an intrinsically interpretable model with clear rule semantics and transparent inference processes. Empirical results demonstrate that KANFIS achieves competitive performance against representative neural and neuro-fuzzy baselines.

The probabilistic linguistic term has been proposed to deal with probability distributions in provided linguistic evaluations. However, because it has some fundamental defects, it is often difficult for decision-makers to get reasonable information of linguistic evaluations for group decision making. In addition, weight information plays a significant role in dynamic information fusion and decision making process. However, there are few research methods to determine the dynamic attribute weight with time. In this paper, I propose the concept of double fuzzy probability interval linguistic term set (DFPILTS). Firstly, fuzzy semantic integration, DFPILTS definition, its preference relationship, some basic algorithms and aggregation operators are defined. Then, a fuzzy linguistic Markov matrix with its network is developed. Then, a weight determination method based on distance measure and information entropy to reducing the inconsistency of DFPILPR and obtain collective priority vector based on group consensus is developed. Finally, an aggregation-based approach is developed, and an optimal investment case from a financial risk is used to illustrate the application of DFPILTS and decision method in multi-criteria decision making.