Recent works have shown that tokenisation is NP-complete. However, these works assume tokenisation is applied to inputs with unboundedly large alphabets -- an unrealistic assumption, given that in practice tokenisers operate over fixed-size alphabets, such as bytes or Unicode characters. We close this gap by analysing tokenisation over bounded $n$-ary alphabets, considering two natural variants: bottom-up tokenisation and direct tokenisation, where we must, respectively, select a sequence of merge operations or a vocabulary whose application optimally compresses a dataset. First, we note that proving hardness results for an $n$-ary alphabet proves the same results for alphabets of any larger size. We then prove that even with binary alphabets, both variants are not only NP-complete, but admit no polynomial-time approximation scheme (unless P=NP). We further show that direct tokenisation remains NP-complete even when applied to unary alphabets. While unary alphabets may not be practically useful, this result establishes that the computational intractability of tokenisation is not an artifact of large alphabets or complex constructions, but a fundamental barrier. Overall, our results explain why practical algorithms such as BPE and UnigramLM are heuristic, and points toward approximation algorithms being an important path going forward for tokenisation research.

Modelling the complex spatiotemporal patterns of large-scale brain dynamics is crucial for neuroscience, but traditional methods fail to capture the rich structure in modalities such as magnetoencephalography (MEG). Recent advances in deep learning have enabled significant progress in other domains, such as language and vision, by using foundation models at scale. Here, we introduce MEG-GPT, a transformer based foundation model that uses time-attention and next time-point prediction. To facilitate this, we also introduce a novel data-driven tokeniser for continuous MEG data, which preserves the high temporal resolution of continuous MEG signals without lossy transformations. We trained MEG-GPT on tokenised brain region time-courses extracted from a large-scale MEG dataset (N=612, eyes-closed rest, Cam-CAN data), and show that the learnt model can generate data with realistic spatio-spectral properties, including transient events and population variability. Critically, it performs well in downstream decoding tasks, improving downstream supervised prediction task, showing improved zero-shot generalisation across sessions (improving accuracy from 0.54 to 0.59) and subjects (improving accuracy from 0.41 to 0.49) compared to a baseline methods. Furthermore, we show the model can be efficiently fine-tuned on a smaller labelled dataset to boost performance in cross-subject decoding scenarios. This work establishes a powerful foundation model for electrophysiological data, paving the way for applications in computational neuroscience and neural decoding.

Modern BPE tokenizers often split calendar dates into meaningless fragments, e.g., 20250312 $\rightarrow$ 202, 503, 12, inflating token counts and obscuring the inherent structure needed for robust temporal reasoning. In this work, we (1) introduce a simple yet interpretable metric, termed date fragmentation ratio, that measures how faithfully a tokenizer preserves multi-digit date components; (2) release DateAugBench, a suite of 6500 examples spanning three temporal reasoning tasks: context-based date resolution, format-invariance puzzles, and date arithmetic across historical, contemporary, and future time periods; and (3) through layer-wise probing and causal attention-hop analyses, uncover an emergent date-abstraction mechanism whereby large language models stitch together the fragments of month, day, and year components for temporal reasoning. Our experiments show that excessive fragmentation correlates with accuracy drops of up to 10 points on uncommon dates like historical and futuristic dates. Further, we find that the larger the model, the faster the emergent date abstraction that heals date fragments is accomplished. Lastly, we observe a reasoning path that LLMs follow to assemble date fragments, typically differing from human interpretation (year $\rightarrow$ month $\rightarrow$ day). Our datasets and code are made publicly available \href{https://github.com/gagan3012/date-fragments}{here}.

In this paper, we address the challenge of Optical Music Recognition (OMR) for handwritten jazz lead sheets, a widely used musical score type that encodes melody and chords. The task is challenging due to the presence of chords, a score component not handled by existing OMR systems, and the high variability and quality issues associated with handwritten images. Our contribution is two-fold. We present a novel dataset consisting of 293 handwritten jazz lead sheets of 163 unique pieces, amounting to 2021 total staves aligned with Humdrum **kern and MusicXML ground truth scores. We also supply synthetic score images generated from the ground truth. The second contribution is the development of an OMR model for jazz lead sheets. We discuss specific tokenisation choices related to our kind of data, and the advantages of using synthetic scores and pretrained models. We publicly release all code, data, and models.

Data representation remains a fundamental challenge in machine learning, particularly when adapting sequence-based architectures like Transformers and Large Language Models (LLMs) for structured tabular data. Existing methods often fail to cohesively encode the mix of numerical and categorical features or preserve the inherent structure of tables. This paper introduces a novel, hybrid tokenisation methodology designed to convert tabular data into a unified, sequential format suitable for LLM training. Our approach combines predefined fixed tokens to represent structural elements and low-cardinality categorical features, with a learned subword vocabulary using Byte-Pair Encoding (BPE) for high-cardinality and continuous values. We demonstrate the efficacy of this technique by applying it to a large-scale NetFlow dataset (CIDDS-001), preparing a corpus for a Network Intrusion Detection System (NIDS) foundation model. The evaluation shows that our method is highly efficient, processing over 31 million network flows in under five hours and achieving a significant data compression ratio of 6.18:1. This process resulted in a computationally manageable corpus of over one billion tokens, establishing a viable and generalisable pathway for training foundation models on structured data.

Recent dynamic tokenisation methods operate directly on bytes and pool their latent representations into patches. This bears similarities to computational models of word segmentation that determine lexical boundaries using spikes in an autoregressive model's prediction error. Inspired by this connection, we explore whether grouping predictable bytes - rather than pooling their representations - can yield a useful fixed subword vocabulary. We propose a new information-driven subword tokeniser, ByteSpan, that uses an external byte-level LM during training to identify contiguous predictable byte sequences and group them into subwords. Experiments show that ByteSpan yields efficient vocabularies with higher morphological alignment scores than BPE for English. Multilingual experiments show similar compression and Rényi efficiency for 25 languages.

Modern language models are typically trained over subword sequences, but ultimately define probabilities over character-strings. Ideally, the choice of the tokeniser -- which maps character-strings to subwords -- should not affect the probability assigned to the underlying character-string; in practice, it does. We define this mismatch as tokenisation bias. In this work, we quantify one particular type of tokenisation bias: the effect of including or not a subword (e.g., $\langle hello \rangle$) in a tokeniser's vocabulary on the probability a trained model assigns to the corresponding characters (i.e., \textit{``hello''}). Estimating this effect is challenging because each model is trained with only one tokeniser. We address this by framing tokenisation bias as a causal effect and estimating it using the regression discontinuity design. Specifically, we exploit the fact that tokenisation algorithms rank subwords and add the first $K$ to a tokeniser's vocabulary, where $K$ is an arbitrary cutoff point. As such, we can estimate a causal effect by comparing similar subwords around this cutoff. Experimentally, we find that tokenisation consistently affects models' outputs across scales, vocabularies, and tokenisers. Notably, a subword's presence in a small model's vocabulary may increase its characters' probability by up to 17 times, highlighting tokenisation as a key design choice in language modelling.

In this work, we prove the NP-completeness of two variants of tokenisation, defined as the problem of compressing a dataset to at most $\delta$ symbols by either finding a vocabulary directly (direct tokenisation), or selecting a sequence of merge operations (bottom-up tokenisation).

In natural language processing and computer vision, self-supervised pre-training on large datasets unlocks foundational model capabilities across domains and tasks. However, this potential has not yet been realised in time series analysis, where existing methods disregard the heterogeneous nature of time series characteristics. Time series are prevalent in many domains, including medicine, engineering, natural sciences, and finance, but their characteristics vary significantly in terms of variate count, inter-variate relationships, temporal dynamics, and sampling frequency. This inherent heterogeneity across domains prevents effective pre-training on large time series corpora. To address this issue, we introduce OTiS, an open model for general time series analysis, that has been specifically designed to handle multi-domain heterogeneity. We propose a novel pre-training paradigm including a tokeniser with learnable domain-specific signatures, a dual masking strategy to capture temporal causality, and a normalised cross-correlation loss to model long-range dependencies. Our model is pre-trained on a large corpus of 640, 187 samples and 11 billion time points spanning 8 distinct domains, enabling it to analyse time series from any (unseen) domain. In comprehensive experiments across 15 diverse applications - including classification, regression, and forecasting - OTiS showcases its ability to accurately capture domain-specific data characteristics and demonstrates its competitiveness against state-of-the-art baselines. Our code and pre-trained weights are publicly available at https://github.com/oetu/otis. Self-supervised pre-training paradigms are designed to account for the specific properties of language (Radford et al., 2018; Touvron et al., 2023; Chowdhery et al., 2023) or imaging (Zhou et al., 2022; Cherti et al., 2023; Oquab et al., 2024), unlocking foundational model capabilities that apply to a wide range of domains and downstream tasks. This potential, however, remains largely unrealised in time series due to the lack of self-supervised pre-training paradigms that account for the heterogeneity of time series across domains.

Language models (LMs) estimate the probability distribution over sequences of natural language; these distributions are crucial for computing perplexity and surprisal in linguistics research. While we are usually concerned with measuring these values for words, most LMs operate over subwords. Despite seemingly straightforward, accurately computing probabilities over one unit given probabilities over the other requires care. Indeed, we show here that many recent linguistic studies have been incorrectly computing these values. This paper derives the correct methods for computing word probabilities, highlighting issues when relying on language models that use beginning-of-word (bow)-marking tokenisers, e.g., the GPT family. Empirically, we show that correcting the widespread bug in probability computations affects measured outcomes in sentence comprehension and lexical optimisation analyses.