用范畴论统一神经符号系统中的逻辑语义，实现灵活扩展与转换。
NeSyCat: A Monad-Based Categorical Semantics of the Neurosymbolic ULLER Framework
- 基于单子的范畴框架统一了经典、模糊和概率三种逻辑语义
- 支持在任意（包括无限）域上扩展逻辑张量网络的广义量化机制
- 可模块化实现于Python和Haskell，便于集成到不同神经符号系统
ULLER（统一学习与推理语言）提供了一种统一的一阶逻辑（FOL）语法，使其知识库可直接应用于多种神经符号系统。原始规范为该语法赋予了三种相互独立的语义：经典、模糊和概率，每种均配有专用语义规则。本文表明，这些看似不同的语义均可视为同一范畴框架（基于单子）的实例，而单子正是函数式编程中建模副作用的核心构造。这一统一框架支持新语义的模块化添加及语义间的系统性转换。例如，我们通过将吉里单子（Giry monad）扩展至概率空间，实现了逻辑张量网络（LTN）在任意（包括无限）域上的广义量化。此外，本方法支持在Python和Haskell中模块化实现ULLER，目前已在GitHub发布初步版本。
原文摘要 · Abstract (English)
ULLER (Unified Language for LEarning and Reasoning) offers a unified first-order logic (FOL) syntax, enabling its knowledge bases to be used directly across a wide range of neurosymbolic systems. The original specification endows this syntax with three pairwise independent semantics: classical, fuzzy, and probabilistic, each accompanied by dedicated semantic rules. We show that these seemingly disparate semantics are all instances of one categorical framework based on monads, the very construct that models side effects in functional programming. This enables the modular addition of new semantics and systematic translations between them. As example, we outline the addition of generalised quantification in Logic Tensor Networks (LTN) to arbitrary (also infinite) domains by extending the Giry monad to probability spaces. In particular, our approach allows a modular implementation of ULLER in Python and Haskell, of which we have published initial versions on GitHub.
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