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We establish Ecalle's mould calculus in an abstract Lie-theoretic setting and use it to solve a normalization problem, which covers several formal normal form problems in the theory of dynamical systems. The mould formalism allows us to…
Type annotations are essential when printing terms in a way that preserves their meaning under reparsing and type inference. We study the problem of complete and minimal type annotations for rank-one polymorphic $\lambda$-calculus terms, as…
This paper addresses the problem of mapping natural language sentences to lambda-calculus encodings of their meaning. We describe a learning algorithm that takes as input a training set of sentences labeled with expressions in the lambda…
Auditing is an increasingly important operation for computer programming, for example in security (e.g. to enable history-based access control) and to enable reproducibility and accountability (e.g. provenance in scientific programming).…
We show that lambda calculus is a computation model which can step by step simulate any sequential deterministic algorithm for any computable function over integers or words or any datatype. More formally, given an algorithm above a family…
Computer simulations have proven a valuable tool for understanding complex phenomena across the sciences. However, the utility of simulators for modelling and forecasting purposes is often restricted by low data quality, as well as…
In this paper, we present an extension of $\lambda\mu$-calculus called $\lambda\mu^{++}$-calculus which has the following properties: subject reduction, strong normalization, unicity of the representation of data and thus confluence only on…
This paper is a contribution to the search for efficient and high-level mathematical tools to specify and reason about (abstract) programming languages or calculi. Generalising the reduction monads of Ahrens et al., we introduce transition…
The Mixture-of-Experts (MoE) architecture has become a predominant paradigm for scaling large language models (LLMs). Despite offering strong performance and computational efficiency, large MoE-based LLMs like DeepSeek-V3-0324 and…
Scaling test-time compute via extended reasoning has become a key paradigm for improving the capabilities of large language models (LLMs). However, existing approaches optimize reasoning under fixed or uniformly sampled token budgets,…
We present a typing system with non-idempotent intersection types, typing a term syntax covering three different calculi: the pure {\lambda}-calculus, the calculus with explicit substitutions {\lambda}S, and the calculus with explicit…
The increasing complexity of large-scale language models has amplified concerns regarding their interpretability and reusability. While traditional embedding models like Word2Vec and GloVe offer scalability, they lack transparency and often…
In this paper we investigate the Curry-Howard correspondence for constructive modal logic in light of the gap between the proof equivalences enforced by the lambda calculi from the literature and by the recently defined winning strategies…
A popular approach for modeling and inference in spatial statistics is to represent Gaussian random fields as solutions to stochastic partial differential equations (SPDEs) of the form $L^{\beta}u = \mathcal{W}$, where $\mathcal{W}$ is…
Programming by Example (PBE) is the task of inducing computer programs from input-output examples. It can be seen as a type of machine learning where the hypothesis space is the set of legal programs in some programming language. Recent…
The paper introduces a novel framework based on category theory to enhance the explainability of artificial intelligence systems, particularly focusing on word embeddings. Key topics include the construction of categories $\mathcal{L}_T$…
A many-valued modal logic is introduced that combines the usual Kripke frame semantics of the modal logic K with connectives interpreted locally at worlds by lattice and group operations over the real numbers. A labelled tableau system is…
We define a new cost model for the call-by-value lambda-calculus satisfying the invariance thesis. That is, under the proposed cost model, Turing machines and the call-by-value lambda-calculus can simulate each other within a polynomial…
This paper proposes a modal typing system that enables us to handle self-referential formulae, including ones with negative self-references, which on one hand, would introduce a logical contradiction, namely Russell's paradox, in the…
Word embeddings and language models have transformed natural language processing (NLP) by facilitating the representation of linguistic elements in continuous vector spaces. This review visits foundational concepts such as the…