Jordan Meadows

Jordan Meadows, Andre Freitas

Informal mathematical text underpins real-world quantitative reasoning and communication. Developing sophisticated methods of retrieval and abstraction from this dual modality is crucial in the pursuit of the vision of automating discovery in quantitative science and mathematics. We track the development of informal mathematical language processing approaches across five strategic sub-areas in recent years, highlighting the prevailing successful methodological elements along with existing limitations.

Marco Valentino, Jordan Meadows, Lan Zhang et al.

This paper investigates the possibility of approximating multiple mathematical operations in latent space for expression derivation. To this end, we introduce different multi-operational representation paradigms, modelling mathematical operations as explicit geometric transformations. By leveraging a symbolic engine, we construct a large-scale dataset comprising 1.7M derivation steps stemming from 61K premises and 6 operators, analysing the properties of each paradigm when instantiated with state-of-the-art neural encoders. Specifically, we investigate how different encoding mechanisms can approximate expression manipulation in latent space, exploring the trade-off between learning different operators and specialising within single operations, as well as the ability to support multi-step derivations and out-of-distribution generalisation. Our empirical analysis reveals that the multi-operational paradigm is crucial for disentangling different operators, while discriminating the conclusions for a single operation is achievable in the original expression encoder. Moreover, we show that architectural choices can heavily affect the training dynamics, structural organisation, and generalisation of the latent space, resulting in significant variations across paradigms and classes of encoders.

Jordan Meadows, Marco Valentino, Andre Freitas

This paper investigates how hallucination rates in Large Language Models (LLMs) may be controlled via a symbolic data generation framework, exploring a fundamental relationship between the rate of certain mathematical errors and types of input intervention. Specifically, we systematically generate data for a derivation generation task using a symbolic engine, applying targeted interventions to prompts to perturb features of mathematical derivations such as the surface forms of symbols, equational tree structures, and mathematical context. We then evaluate the effect of prompt interventions across a range of LLMs including fine-tuned T5 models, GPT, and LLaMa-based models. Our experiments suggest that T5-Large can outperform the few-shot performance of GPT-4 on various evaluation sets generated via the framework. However, an extensive evaluation based on human analysis, template-based error detection, and text generation metrics reveals model weaknesses beyond what the reference-based metrics singularly describe. We use these results to tie characteristic distributional footprints of interventions to the human evaluation of LLM derivation quality, potentially leading to significant control over fine-grained mathematical capabilities of language models with respect to specific types of errors.

Jordan Meadows, Lan Zhang, Andre Freitas

Formalising informal mathematical reasoning into formally verifiable code is a significant challenge for large language models. In scientific fields such as physics, domain-specific machinery (\textit{e.g.} Dirac notation, vector calculus) imposes additional formalisation challenges that modern LLMs and agentic approaches have yet to tackle. To aid autoformalisation in scientific domains, we present FormalScience; a domain-agnostic human-in-the-loop agentic pipeline that enables a single domain expert (without deep formal language experience) to produce \textit{syntactically correct} and \textit{semantically aligned} formal proofs of informal reasoning for low economic cost. Applying FormalScience to physics, we construct FormalPhysics, a dataset of 200 university-level (LaTeX) physics problems and solutions (primarily quantum mechanics and electromagnetism), along with their Lean4 formal representations. Compared to existing formal math benchmarks, FormalPhysics achieves perfect formal validity and exhibits greater statement complexity. We evaluate open-source models and proprietary systems on a statement autoformalisation task on our dataset via zero-shot prompting, self-refinement with error feedback, and a novel multi-stage agentic approach, and explore autoformalisation limitations in modern LLM-based approaches. We provide the first systematic characterisation of semantic drift in physics autoformalisation in terms of concepts such as notational collapse and abstraction elevation which reveals what formal language verifies when full semantic preservation is unattainable. We release the codebase together with an interactive UI-based FormalScience system which facilitates autoformalisation and theorem proving in scientific domains beyond physics.https://github.com/jmeadows17/formal-science

Jordan Meadows, Marco Valentino, Damien Teney et al.

This paper proposes a methodology for generating and perturbing detailed derivations of equations at scale, aided by a symbolic engine, to evaluate the generalisability of Transformers to out-of-distribution mathematical reasoning problems. Instantiating the framework in the context of sequence classification tasks, we compare the capabilities of GPT-4, GPT-3.5, and a canon of fine-tuned BERT models, exploring the relationship between specific operators and generalisation failure via the perturbation of reasoning aspects such as symmetry and variable surface forms. Surprisingly, our empirical evaluation reveals that the average in-distribution performance of fine-tuned models surpasses GPT-3.5, and rivals GPT-4. However, perturbations to input reasoning can reduce their performance by up to 80 F1 points. Overall, the results suggest that the in-distribution performance of smaller open-source models may potentially rival GPT by incorporating appropriately structured derivation dependencies during training, and highlight a shared weakness between BERT and GPT involving a relative inability to decode indirect references to mathematical entities. We release the full codebase, constructed datasets, and fine-tuned models to encourage future progress in the field.

Jordan Meadows, Tamsin James, Andre Freitas

Language models (LMs) can hallucinate when performing complex mathematical reasoning. Physics provides a rich domain for assessing their mathematical capabilities, where physical context requires that any symbolic manipulation satisfies complex semantics (\textit{e.g.,} units, tensorial order). In this work, we systematically remove crucial context from prompts to force instances where model inference may be algebraically coherent, yet unphysical. We assess LM capabilities in this domain using a curated dataset encompassing multiple notations and Physics subdomains. Further, we improve zero-shot scores using synthetic in-context examples, and demonstrate non-linear degradation of derivation quality with perturbation strength via the progressive omission of supporting premises. We find that the models' mathematical reasoning is not physics-informed in this setting, where physical context is predominantly ignored in favour of reverse-engineering solutions.

Jordan Meadows, Zili Zhou, Andre Freitas

In order for language models to aid physics research, they must first encode representations of mathematical and natural language discourse which lead to coherent explanations, with correct ordering and relevance of statements. We present a collection of datasets developed to evaluate the performance of language models in this regard, which measure capabilities with respect to sentence ordering, position, section prediction, and discourse coherence. Analysis of the data reveals equations and sub-disciplines which are most common in physics discourse, as well as the sentence-level frequency of equations and expressions. We present baselines that demonstrate how contemporary language models are challenged by coherence related tasks in physics, even when trained on mathematical natural language objectives.

Jordan Meadows, André Freitas

Automating the derivation of published results is a challenge, in part due to the informal use of mathematics by physicists, compared to that of mathematicians. Following demand, we describe a method for converting informal hand-written derivations into datasets, and present an example dataset crafted from a contemporary result in condensed matter. We define an equation reconstruction task completed by rederiving an unknown intermediate equation posed as a state, taken from three consecutive equational states within a derivation. Derivation automation is achieved by applying string-based CAS-reliant actions to states, which mimic mathematical operations and induce state transitions. We implement a symbolic similarity-based heuristic search to solve the equation reconstruction task as an early step towards multi-hop equational inference in physics.