Concept learning becomes possible only when existing representations fail to account for experience. Most models of learning and inference, however, presuppose a fixed representational basis within which belief updating occurs. In this paper, I address a prior question: under what structural conditions can the representational basis itself expand in a principled and selective way? I propose a geometric framework in which conceptual growth is modeled as admissible basis extension evaluated under a Minimum Description Length (MDL) criterion. Experience, whether externally observed or internally simulated, is represented as vectors relative to a current conceptual subspace. Residual components capture systematic representational failure, and candidate conceptual extensions are restricted to low-rank, admissible transformations. I show that any MDL-accepted extension can be chosen so that its novel directions lie entirely within the residual span induced by experience, while extensions orthogonal to this span strictly increase description length and are therefore rejected. This yields a conservative account of imagination and conceptual innovation. Internally generated counterfactual representations contribute to learning only insofar as they expose or amplify structured residual error, and cannot introduce arbitrary novelty. I further distinguish representational counterfactuals--counterfactuals over an agent's conceptual basis--from causal or value-level counterfactuals, and show how MDL provides a normative selection principle governing representational change. Overall, the framework characterizes conceptual development as an error-driven, geometry-constrained process of basis extension, clarifying both the role and the limits of imagination in learning and theory change.

Multilingual large language models (LLMs) are expected to recall factual knowledge consistently across languages. However, the factors that give rise to such crosslingual consistency -- and its frequent failure -- remain poorly understood. In this work, we hypothesize that these inconsistencies may arise from failures in entity alignment, the process of mapping subject and object entities into a shared conceptual space across languages. To test this, we assess alignment through entity-level (subject and object) translation tasks, and find that consistency is strongly correlated with alignment across all studied models, with misalignment of subjects or objects frequently resulting in inconsistencies. Building on this insight, we propose SubSub and SubInj, two effective methods that integrate English translations of subjects into prompts across languages, leading to substantial gains in both factual recall accuracy and consistency. Finally, our mechanistic analysis reveals that these interventions reinforce the entity representation alignment in the conceptual space through model's internal pivot-language processing, offering effective and practical strategies for improving multilingual factual prediction.

Conceptual spaces represent entities and concepts using cognitively meaningful dimensions, typically referring to perceptual features. Such representations are widely used in cognitive science and have the potential to serve as a cornerstone for explainable AI. Unfortunately, they have proven notoriously difficult to learn, although recent LLMs appear to capture the required perceptual features to a remarkable extent. Nonetheless, practical methods for extracting the corresponding conceptual spaces are currently still lacking. While various methods exist for extracting embeddings from LLMs, extracting conceptual spaces also requires us to encode the underlying features. In this paper, we propose a strategy in which features (e.g. sweetness) are encoded by embedding the description of a corresponding prototype (e.g. a very sweet food). To improve this strategy, we fine-tune the LLM to align the prototype embeddings with the corresponding conceptual space dimensions. Our empirical analysis finds this approach to be highly effective.

Intelligence-biological, artificial, or collective-requires structural coherence across recursive reasoning processes to scale effectively. As complex systems grow, coherence becomes fragile unless a higher-order structure ensures semantic consistency. This paper introduces the Recursive Coherence Principle (RCP): a foundational constraint stating that for any reasoning system of order N, composed of systems operating over conceptual spaces of order N-1, semantic coherence is preserved only by a recursively evaluable generalization operator that spans and aligns those lower-order conceptual spaces. Crucially, this coherence enables structural alignment. Without recursive coherence, no system can reliably preserve goals, meanings, or reasoning consistency at scale. We formally define the Functional Model of Intelligence (FMI) as the only known operator capable of satisfying the RCP at any scale. The FMI is a minimal, composable architecture with internal functions (evaluation, modeling, adaptation, stability, decomposition, bridging) and external functions (storage, recall, System 1 and System 2 reasoning) vital for preserving semantic structure across inference and coordination layers. We prove that any system lacking the FMI will experience recursive coherence breakdown as it scales, arguing that common AI issues like misalignment, hallucination, and instability are symptoms of this structural coherence loss. Unlike other foundational principles, RCP uniquely captures the internal, recursive dynamics needed for coherent, alignable intelligence, modeling semantic coherence under recursion. This work significantly impacts AI alignment, advocating a shift from behavioral constraints to structural coherence, and offers a pathway for safely generalizable, robustly coherent AI at scale.

Creative processes are typically divided into three types: combinatorial, exploratory, and transformational. Here, we provide a graphical theory of transformational scientific creativity, synthesizing Boden's insight that trans-formational creativity arises from changes in the "enabling constraints" of a conceptual space (Boden 1992) and Kuhn's structure of scientific revolutions as resulting from paradigm shifts (Kuhn 1962). We prove that modifications made to axioms of our graphical model have the most transformative potential and then illustrate how several historical instances of transforma-tional creativity can be captured by our framework.

Expressive range analysis is a visualization-based technique used to evaluate the performance of generative models, particularly in game level generation. It typically employs two quantifiable metrics to position generated artifacts on a 2D plot, offering insight into how content is distributed within a defined metric space. In this work, we use the expressive range of a generator as the conceptual space of possible creations. Inspired by the quality diversity paradigm, we explore this space to generate levels. To do so, we use a constraint-based generator that systematically traverses and generates levels in this space. To train the constraint-based generator we use different tile patterns to learn from the initial example levels. We analyze how different patterns influence the exploration of the expressive range. Specifically, we compare the exploration process based on time, the number of successful and failed sample generations, and the overall interestingness of the generated levels. Unlike typical quality diversity approaches that rely on random generation and hope to get good coverage of the expressive range, this approach systematically traverses the grid ensuring more coverage. This helps create unique and interesting game levels while also improving our understanding of the generator's strengths and limitations.

The opaque nature of Large Language Models (LLMs) has led to significant research efforts aimed at enhancing their interpretability, primarily through post-hoc methods. More recent in-hoc approaches, such as Concept Bottleneck Models (CBMs), offer both interpretability and intervenability by incorporating explicit concept representations. However, these methods suffer from key limitations, including reliance on labeled concept datasets and significant architectural modifications that challenges re-integration into existing system pipelines. In this work, we introduce a new methodology for incorporating interpretability and intervenability into an existing model by integrating Concept Layers (CLs) into its architecture. Our approach projects the model's internal vector representations into a conceptual, explainable vector space before reconstructing and feeding them back into the model. Furthermore, we eliminate the need for a human-selected concept set by algorithmically searching an ontology for a set of concepts that can be either task-specific or task-agnostic. We evaluate CLs across multiple tasks, demonstrating that they maintain the original model's performance and agreement while enabling meaningful interventions. Additionally, we present a proof of concept showcasing an intervenability interface, allowing users to adjust model behavior dynamically, such as mitigating biases during inference.

A conceptual space represents concepts as nodes and semantic relatedness as edges. Word embeddings, combined with a similarity metric, provide an effective approach to constructing such a space. Typically, embeddings are derived from traditional distributed models or encoder-only pretrained models, whose objectives directly capture the meaning of the current token. In contrast, decoder-only models, including large language models (LLMs), predict the next token, making their embeddings less directly tied to the current token's semantics. Moreover, comparative studies on LLMs of different scales remain underexplored. In this paper, we construct a conceptual space using word embeddings from LLMs of varying scales and comparatively analyze their properties. We establish a network based on a linguistic typology-inspired connectivity hypothesis, examine global statistical properties, and compare LLMs of varying scales. Locally, we analyze conceptual pairs, WordNet relations, and a cross-lingual semantic network for qualitative words. Our results indicate that the constructed space exhibits small-world properties, characterized by a high clustering coefficient and short path lengths. Larger LLMs generate more intricate spaces, with longer paths reflecting richer relational structures and connections. Furthermore, the network serves as an efficient bridge for cross-lingual semantic mapping.

Semantic map models (SMMs) construct a network-like conceptual space from cross-linguistic instances or forms, based on the connectivity hypothesis. This approach has been widely used to represent similarity and entailment relationships in cross-linguistic concept comparisons. However, most SMMs are manually built by human experts using bottom-up procedures, which are often labor-intensive and time-consuming. In this paper, we propose a novel graph-based algorithm that automatically generates conceptual spaces and SMMs in a top-down manner. The algorithm begins by creating a dense graph, which is subsequently pruned into maximum spanning trees, selected according to metrics we propose. These evaluation metrics include both intrinsic and extrinsic measures, considering factors such as network structure and the trade-off between precision and coverage. A case study on cross-linguistic supplementary adverbs demonstrates the effectiveness and efficiency of our model compared to human annotations and other automated methods. The tool is available at \url{https://github.com/RyanLiut/SemanticMapModel}.

Conceptual spaces represent entities in terms of their primitive semantic features. Such representations are highly valuable but they are notoriously difficult to learn, especially when it comes to modelling perceptual and subjective features. Distilling conceptual spaces from Large Language Models (LLMs) has recently emerged as a promising strategy, but existing work has been limited to probing pre-trained LLMs using relatively simple zero-shot strategies. We focus in particular on the task of ranking entities according to a given conceptual space dimension. Unfortunately, we cannot directly fine-tune LLMs on this task, because ground truth rankings for conceptual space dimensions are rare. We therefore use more readily available features as training data and analyse whether the ranking capabilities of the resulting models transfer to perceptual and subjective features. We find that this is indeed the case, to some extent, but having at least some perceptual and subjective features in the training data seems essential for achieving the best results.