arXiv subjects

Alexandre Bouayad

Publications and source records attributed to Alexandre Bouayad.

4 recordsLinked to original sources

arXiv2026

Weave of Formal Thought

Large language models (LLMs) attain remarkable surface fluency on code, yet they neither formally guarantee the syntactic validity of their output nor leverage the hierarchical structure defining the target language. While existing constrained-decoding frameworks address the former, they operate under rigid assumptions that preclude critical lexical mechanisms -- including context-sensitive lexing, maximal-munch tokenization, and keyword extraction -- and only approximate vocabulary masking, sacrificing completeness. For the latter, code LLMs typically inject grammatical structure via predetermined policies rather than learning which structural information to expose. In this work, we introduce Weave of Formal Thought (WoFT), a paradigm uniting rigorous syntactic validation with learned structural representations. First, we present a formal engine and constrained decoder that is sound and complete with respect to the full Tree-sitter specification. By augmenting generalized LR (GLR) parsing with a speculative-lexing construction that maintains concurrent lexer-state hypotheses synchronized with a GLR graph-structured stack, our decoder admits every subword token extending to a valid program prefix and rejects all others. Second, we present a latent-variable fine-tuning method training the language model to interleave non-terminal grammar symbols directly into generation. Utilizing the reweighted wake-sleep (RWS) algorithm to optimize the importance-weighted evidence lower bound (IW-ELBO) of the surface text, the model learns to selectively retain formal derivations as an adaptive structural scratchpad. For Python, fine-tuning StarCoder2-3B with our RWS objective reduces per-token cross-entropy by 14.3% relative to a text-only SFT baseline, demonstrating that discretionary latent syntax recovers critical structural information that flat autoregressive training discards.

cs.CL↗

arXiv2022

Generalised Quantum Enveloping Algebras, Coloured Kac-Moody Algebras, and Langlands Interpolation

We propose in this thesis a new deformation process of Kac-Moody algebras and their representations. The direction of deformation is given by a collection of numbers, called a colouring. The natural numbers lead for example to the classical algebras, while the quantum numbers lead to the associated quantum algebras. We first establish sufficient and necessary conditions on colourings to allow the process depend polynomially on a formal parameter and to provide the generalised quantum enveloping (GQE) algebras. We then lift the restrictions and show that the process still exists via the coloured Kac Moody algebras. We formulate the GQE conjecture which predicts that every representation in the category Oint of a Kac-Moody algebra can be deformed into a representation of an associated GQE algebra. We give various evidences for this conjecture and make a first step towards its resolution by proving that Kac-Moody algebras without Serre relations can be deformed into GQE algebras without Serre relations. In case the conjecture holds, we establish an analog result for coloured Kac-Moody algebras, we prove that the deformed representation theories are parallel to the classical one, we explicit a deformed Serre presentation for GQE algebras, we prove that the latter are the representatives of a natural class of formal deformations of Kac-Moody algebras and are h-trivial in finite type. As an application, we explain in terms of interpolation both classical and quantum Langlands dualities between representations of Lie algebras, and we propose a new approach which aims at proving a conjecture of Frenkel-Hernandez. In general, we prove that representations of two isogenic coloured Kac-Moody algebras can be interpolated by representations of a third one. Observing that standard quantum algebras satisfy the GQE conjecture, we give a new proof of the previously mentioned classical Langlands duality.

math.RT↗

arXiv2015

Colored Kac-Moody Algebras, Part I

We introduce a parametrization of formal deformations of Verma modules of $\mathfrak{sl}_2$. A point in the moduli space is called a colouring. We prove that for each colouring $ψ$ satisfying a regularity condition, there is a formal deformation $U_h(ψ)$ of $U(\mathfrak{sl}_2)$ acting on the deformed Verma modules. We retrieve in particular the quantum algebra $U_h(\mathfrak{sl}_2)$ from a colouring by $q$-numbers. More generally, we establish that regular colourings parametrize a broad family of formal deformations of the Chevalley-Serre presentation of $U(\mathfrak{sl}_2)$. The present paper is the first of a series aimed to lay the foundations of a new approach to deformations of Kac-Moody algebras and of their representations. We will employ in a forthcoming paper coloured Kac-Moody algebras to give a positive answer to E. Frenkel and D. Hernandez's conjectures on Langlands duality in quantum group theory.

math.QA↗

arXiv2011

Groupes Quantiques d'Interpolation de Langlands de Rang 1

Interpolating Langlands Quantum Groups of Rank 1 -- We study a certain family, parameterized by an positive integer g, of double deformations of the envelopping algebra U(sl2), in the spirit of arXiv:0809.4453. We prove that each of these double deformations simultaneously deforms two rank 1 quantum groups. We show this interpolating property explains the Langlands duality for the representations of the quantum groups in rank 1. Hence we prove a conjecture of arXiv:0809.4453 in this case : we prove for all g the existence of representations which simultaneously deform two Langlands dual representations. We also study more generaly the finite rank representation theory of this family of double deformations. ----- On étudie une certaine famille, paramétrée par un entier g strictement positif, de doubles déformations de l'algébre enveloppante U(sl2), dans l'esprit de arXiv:0809.4453. On prouve que chacune de ces doubles déformations déforme simultanément deux groupes quantiques de rang 1. On montre que cette propriété d'interpolation explique la dualité de Langlands pour les représentations des groupes quantiques en rang 1. On résout ainsi une conjecture de arXiv:0809.4453 dans ce cas : on prouve pour tout g l'existence de représentations qui déforment simultanément deux représentations Langlands duales. On étudie aussi plus généralement la théorie des représentations de rang fini de de cette famille de doubles déformations.

math.QA↗