How many key-value associations can a $d\times d$ linear memory store? We show that the answer depends not only on the $d^2$ degrees of freedom in the memory matrix, but also on the retrieval criterion. In an isotropic Gaussian model for the stored pairs, we show that top-1 retrieval, where every signal must beat its largest distractor, requires the logarithmic model-size scale $d^2\asymp n\log n$. We prove that the correlation matrix memory construction, which stores associations by superposing key-target outer products, achieves this scale through a sharp phase transition, and that the same scaling is necessary for any linear memory. Thus the logarithm is the intrinsic extreme-value price of winner-take-all decoding. We next consider listwise retrieval, where the correct target need not be the unique top-scoring item but should remain among the strongest candidates. To formalize this regime, we propose the Tail-Average Margin (TAM), a convex upper-tail criterion that certifies inclusion of the correct target in a controlled candidate list. Under this listwise retrieval criterion, the capacity follows the quadratic scale $d^2\asymp n$. At load $n/d^2\toα$, we develop an exact asymptotic theory for the TAM empirical-risk minimizer through a two-parameter scalar variational principle. The theory has a rich phenomenology: in the ridgeless limit it yields a closed-form critical load separating satisfiable and unsatisfiable phases, and it predicts the limiting laws of true scores, competitor scores, margins, and percentile profiles. Finally, a small-tail extrapolation further leads to the conjectural sharp top-1 threshold $d^2\sim 2n\log n$.

In machine teaching, a concept is represented by (and inferred from) a small number of labeled examples.

Artificial Neural Networks (ANNs) are powerful machine-learning models capable of capturing intricate non-linear relationships. They are widely used nowadays across numerous scientific and engineering domains, driving advancements in both research and real-world applications. In our recent work, we focused on the statics and dynamics of a particular subclass of ANNs, which we refer to as binary ANNs. A binary ANN is a feed-forward network in which both inputs and outputs are restricted to binary values, making it particularly suitable for a variety of practical use cases. Our previous study approached binary ANNs through the lens of belief-change theory, specifically the Alchourron, Gardenfors and Makinson (AGM) framework, yielding several key insights. Most notably, we demonstrated that the knowledge embodied in a binary ANN (expressed through its input-output behaviour) can be symbolically represented using a propositional logic language. Moreover, the process of modifying a belief set (through revision or contraction) was mapped onto a gradual transition through a series of intermediate belief sets. Analogously, the training of binary ANNs was conceptualized as a sequence of such belief-set transitions, which we showed can be formalized using full-meet AGM-style belief change. In the present article, we extend this line of investigation by addressing some critical limitations of our previous study. Specifically, we show that Dalal's method for belief change naturally induces a structured, gradual evolution of states of belief. More importantly, given the known shortcomings of full-meet belief change, we demonstrate that the training dynamics of binary ANNs can be more effectively modelled using robust AGM-style change operations -- namely, lexicographic revision and moderate contraction -- that align with the Darwiche-Pearl framework for iterated belief change.

We study how linear orders can be employed to realise choice functions for which the set of potential choices is restricted, i.e., the possible choice is not possible among the full powerset of all alternatives. In such restricted settings, constructing a choice function via a relation on the alternatives is not always possible. However, we show that one can always construct a choice function via a linear order on sets of alternatives, even when a fallback value is encoded as the minimal element in the linear order. The axiomatics of such choice functions are presented for the general case and the case of union-closed input restrictions. Restricted choice structures have applications in knowledge representation and reasoning, and here we discuss their applications for theory change and abstract argumentation.

Forgetting as a knowledge management operation deliberately ignores parts of the knowledge and beliefs of an agent, for various reasons. Forgetting has many facets, one may want to forget parts of the syntax, a proposition, or a conditional. In the literature, two main operators suitable for performing forgetting have been proposed and investigated in depth: First, variable elimination is a syntactical method that blends out certain atomic variables to focus on the rest of the language. It has been mainly used in the area of logic programming and answer set programming. Second, contraction in AGM belief revision theory effectively removes propositions from belief sets under logical deduction. Both operations rely mainly on classical logics. In this article, we take an epistemic perspective and study forgetting operations in epistemic states with richer semantic structures, but with clear links to propositional logic. This allows us to investigate what forgetting in the epistemic background means, thereby lifting well-known and novel forgetting operations to the epistemic level. We present five general types of epistemic forgetting and instantiate them with seven concrete forgetting operations for Spohn's ranking functions. We take inspiration from postulates of forgetting both from logic programming and AGM theory to propose a rich landscape of axioms for evaluating forgetting operations. Finally, we evaluate all concrete forgetting operations according to all postulates, leading to a novel comprehensive overview highlighting differences and commonalities among the forgetting operators.

In machine teaching, a concept is represented by (and inferred from) a small number of labeled examples.

Existing post-hoc model-agnostic methods generate external explanations for opaque models, primarily by locally attributing the model output to its input features. However, they often lack an explicit and systematic framework for quantifying the contribution of individual features. Building on the Taylor expansion framework introduced by Deng et al. (2024) to unify existing local attribution methods, we propose a rigorous set of postulates -- "precision", "federation", and "zero-discrepancy" -- to govern Taylor term-specific attribution. Guided by these postulates, we introduce TaylorPODA (Taylor expansion-derived imPortance-Order aDapted Attribution), which incorporates an additional "adaptation" property. This property enables alignment with task-specific goals, especially in post-hoc settings lacking ground-truth explanations. Empirical evaluations demonstrate that TaylorPODA achieves competitive results against baseline methods, providing principled and visualization-friendly explanations. This work represents a step toward the trustworthy deployment of opaque models by offering explanations with stronger theoretical grounding.

Within the formal setting of the Lockean thesis, an agent belief set is defined in terms of degrees of confidence and these are described in probabilistic terms. This approach is of established interest, notwithstanding some limitations that make its use troublesome in some contexts, like, for instance, in belief change theory. Precisely, Lockean belief sets are not generally closed under (classical) logical deduction. The aim of the present paper is twofold: on one side we provide two characterizations of those belief sets that are closed under classical logic deduction, and on the other we propose an approach to probabilistic update that allows us for a minimal revision of those beliefs, i.e., a revision obtained by making the fewest possible changes to the existing belief set while still accommodating the new information. In particular, we show how we can deductively close a belief set via a minimal revision.

Addressing complex societal challenges, such as improving public health, fostering honesty in workplaces, or encouraging eco-friendly behaviour requires effective nudges to influence human behaviour at scale. Intervention science seeks to design such nudges within complex societal systems. While interventions primarily aim to shift the system toward a desired state, less attention is given to the sustainability of that state, which we define in terms of resilience: the system's ability to retain the desired state even under perturbations. In this work, we offer a more holistic perspective to intervention design by incorporating a nature-inspired postulate i.e., lower energy states tend to exhibit greater resilience, as a regularization mechanism within intervention optimization to ensure that the resulting state is also sustainable. Using a simple agent-based simulation where commuters are nudged to choose eco-friendly options (e.g., cycles) over individually attractive but less eco-friendly ones (e.g., cars), we demonstrate how embedding lower energy postulate into intervention design induces resilience. The system energy is defined in terms of motivators that drive its agent's behaviour. By inherently ensuring that agents are not pushed into actions that contradict their motivators, the energy-based approach helps design effective interventions that contribute to resilient behavioural states.

Despite efforts to better understand the constraints that operate on single-step parallel (aka "package", "multiple") revision, very little work has been carried out on how to extend the model to the iterated case. A recent paper by Delgrande & Jin outlines a range of relevant rationality postulates. While many of these are plausible, they lack an underlying unifying explanation. We draw on recent work on iterated parallel contraction to offer a general method for extending serial iterated belief revision operators to handle parallel change. This method, based on a family of order aggregators known as TeamQueue aggregators, provides a principled way to recover the independently plausible properties that can be found in the literature, without yielding the more dubious ones.