The combination of higher-order theories and fuzzy logic can be useful in decision-making tasks that involve reasoning across abstract functions and predicates, where exact matches are often rare or unnecessary. Developing efficient reasoning and computational techniques for such a combined formalism presents a significant challenge. In this paper, we adopt a more straightforward approach aiming at integrating two well-established and computationally well-behaved components: higher-order patterns on one side and fuzzy equivalences expressed through similarity relations based on minimum T-norm on the other. We propose a unification algorithm for higher-order patterns modulo these similarity relations and prove its termination, soundness, and completeness. This unification problem, like its crisp counterpart, is unitary. The algorithm computes a most general unifier with the highest degree of approximation when the given terms are unifiable.

Informative representations enhance model performance and generalisability in downstream tasks. However, learning self-supervised representations for spatially characterised time series, like traffic interactions, poses challenges as it requires maintaining fine-grained similarity relations in the latent space. In this study, we incorporate two structure-preserving regularisers for the contrastive learning of spatial time series: one regulariser preserves the topology of similarities between instances, and the other preserves the graph geometry of similarities across spatial and temporal dimensions. To balance contrastive learning and structure preservation, we propose a dynamic mechanism that adaptively weighs the trade-off and stabilises training. We conduct experiments on multivariate time series classification, as well as macroscopic and microscopic traffic prediction. For all three tasks, our approach preserves the structures of similarity relations more effectively and improves state-of-the-art task performances. The proposed approach can be applied to an arbitrary encoder and is particularly beneficial for time series with spatial or geographical features. Furthermore, this study suggests that higher similarity structure preservation indicates more informative and useful representations. This may help to understand the contribution of representation learning in pattern recognition with neural networks. Our code is made openly accessible with all resulting data at https://github.com/yiru-jiao/spclt.

In the Taiwanese judicial system, Cause of Actions (COAs) are essential for identifying relevant legal judgments. However, the lack of standardized COA labeling creates challenges in filtering cases using basic methods. This research addresses this issue by leveraging embedding and clustering techniques to analyze the similarity between COAs based on cited legal articles. The study implements various similarity measures, including Dice coefficient and Pearson's correlation coefficient. An ensemble model combines rankings, and social network analysis identifies clusters of related COAs. This approach enhances legal analysis by revealing inconspicuous connections between COAs, offering potential applications in legal research beyond civil law.

Understanding and predicting the behavior of complex physical systems is a cornerstone of scientific and engineering endeavors. In fluid mechanics, for instance, accurately simulating real operational conditions is essential for the design and optimization of pipelines, aerospace components, and various industrial processes. However, full-scale simulations of such systems are often prohibitively expensive and time-consuming due to the intricate dynamics and vast parameter spaces involved. This poses a significant challenge for researchers and engineers who seek to explore and optimize these systems efficiently. One promising approach to mitigate these challenges is the identification of scaling similarities and symmetry groups within physical systems. By uncovering the correct scaling relations, we can develop smaller, more manageable models that accurately capture the essential behavior of real-world scenarios. These scaled models not only reduce computational costs but also accelerate the design and testing processes by allowing for efficient exploration of the parameter space. Moreover, understanding these scaling laws deepens our insight into the fundamental principles governing these systems, enabling us to generalize findings from simplified models to full-scale applications with greater confidence. In recent years, the application of machine learning in fluid mechanics has been on the rise, offering innovative tools to address complex problems that are difficult to solve analytically.

This paper introduces a novel Choquet distance using fuzzy rough set based measures. The proposed distance measure combines the attribute information received from fuzzy rough set theory with the flexibility of the Choquet integral. This approach is designed to adeptly capture non-linear relationships within the data, acknowledging the interplay of the conditional attributes towards the decision attribute and resulting in a more flexible and accurate distance. We explore its application in the context of machine learning, with a specific emphasis on distance-based classification approaches (e.g. k-nearest neighbours). The paper examines two fuzzy rough set based measures that are based on the positive region. Moreover, we explore two procedures for monotonizing the measures derived from fuzzy rough set theory, making them suitable for use with the Choquet integral, and investigate their differences.

Detecting and exploiting similarities between seemingly distant objects is without doubt an important human ability. This paper develops \textit{from the ground up} an abstract algebraic and qualitative justification-based notion of similarity based on the observation that sets of generalizations encode important properties of elements. We show that similarity defined in this way has appealing mathematical properties. As we construct our notion of similarity from first principles using only elementary concepts of universal algebra, to convince the reader of its plausibility, we show that it can be naturally embedded into first-order logic via model-theoretic types.

Evaluating counterfactual statements is a fundamental problem for many approaches to causal reasoning [40]. Such reasoning can for instance be used to explain erroneous system behavior with a counterfactual statement such as'If the input i at the first position of the observed computation π had not been enabled then the system would not have reached an error e.' which can be formalized using the counterfactual operator and the temporal operator F: π ( i) ( Fe). Since the foundational work by Lewis[38] on the formal semantics of counterfactual conditionals, many applications for counterfactuals [28, 5, 34, 46, 3, 15] and some theoretical results on the decidability of the original theory [37] and related notions [20, 2] have been discovered. Still, certain domains have proven elusive for a long time, for instance, theories involving higher-order reasoning and an infinite number of variables. In this paper, we consider a domain that combines both of these aspects: temporal reasoning over infinite sequences. In particular, we consider counterfactual conditionals that relate two properties expressed in temporal logics, such as the temporal property F e from the introductory example. Temporal logics are used ubiquitously as high-level specifications for verification [21, 4] and synthesis [22, 41], and recently have also found use in specifying reinforcement learning tasks [32, 39]. Our work lifts the language of counterfactual reasoning to similar high-level expressions.

This paper develops a {\em qualitative} and logic-based notion of similarity from the ground up using only elementary concepts of first-order logic centered around the fundamental model-theoretic notion of type.

Graph Neural Networks share with Logic Programming several key relational inference mechanisms. The datasets on which they are trained and evaluated can be seen as database facts containing ground terms. This makes possible modeling their inference mechanisms with equivalent logic programs, to better understand not just how they propagate information between the entities involved in the machine learning process but also to infer limits on what can be learned from a given dataset and how well that might generalize to unseen test data. This leads us to the key idea of this paper: modeling with the help of a logic program the information flows involved in learning to infer from the link structure of a graph and the information content of its nodes properties of new nodes, given their known connections to nodes with possibly similar properties. The problem is known as graph node property prediction and our approach will consist in emulating with help of a Prolog program the key information propagation steps of a Graph Neural Network's training and inference stages. We test our a approach on the ogbn-arxiv node property inference benchmark. To infer class labels for nodes representing papers in a citation network, we distill the dependency trees of the text associated to each node into directed acyclic graphs that we encode as ground Prolog terms. Together with the set of their references to other papers, they become facts in a database on which we reason with help of a Prolog program that mimics the information propagation in graph neural networks predicting node properties. In the process, we invent ground term similarity relations that help infer labels in the test set by propagating node properties from similar nodes in the training set and we evaluate their effectiveness in comparison with that of the graph's link structure. Finally, we implement explanation generators that unveil performance upper bounds inherent to the dataset. As a practical outcome, we obtain a logic program, that, when seen as machine learning algorithm, performs close to the state of the art on the node property prediction benchmark.

Contrastive learning aims to extract distinctive features from data by finding an embedding representation where similar samples are close to each other, and different ones are far apart. We study generalization in contrastive learning, focusing on its simplest representative: Siamese Neural Networks (SNNs). We show that Double Descent also appears in SNNs and is exacerbated by noise. We point out that SNNs can be affected by two distinct sources of noise: Pair Label Noise (PLN) and Single Label Noise (SLN). The effect of SLN is asymmetric, but it preserves similarity relations, while PLN is symmetric but breaks transitivity. We show that the dataset topology crucially affects generalization. While sparse datasets show the same performances under SLN and PLN for an equal amount of noise, SLN outperforms PLN in the overparametrized region in dense datasets. Indeed, in this regime, PLN similarity violation becomes macroscopical, corrupting the dataset to the point where complete overfitting cannot be achieved. We call this phenomenon Density-Induced Break of Similarity (DIBS). We also probe the equivalence between online optimization and offline generalization for similarity tasks. We observe that an online/offline correspondence in similarity learning can be affected by both the network architecture and label noise.