We define a congruence that copes with null next-symbol probabilities that
arise when the output of a language model is constrained by some means during
text generation. We develop an algorithm for efficiently learning the quotient
with respect to this congruence and evaluate it on case studies for analyzing
statistical properties of LLM.

定义了一种与在文本生成过程中通过某种方式约束语言模型输出时产生的零下一个符号概率相适应的同余。我们开发了一种算法，以高效地学习与此同余相关的商集，并在分析 LLM 的统计特性的案例研究中对其进行评估。

通过 PDFA 学习分析受限 LLM

Analyzing constrained LLM through PDFA-learning

We propose a new active learning algorithm for PDFA based on three main
aspects: a congruence over states which takes into account next-symbol
probability distributions, a quantization that copes with differences in
distributions, and an efficient tree-based data structure. Experiments showed
significant performance gains with respect to reference implementations.

我们提出了一种基于状态相似性、量化和基于树的数据结构的 PDFA 主动学习算法，并通过实验证明了其性能显著优于参考实现。

基于 PDFA 的高效主动学习

Towards Efficient Active Learning of PDFA

While the equality of differential signatures (Calabi et al, Int. J. Comput.
Vis. 26: 107-135, 1998) is known to be a necessary condition for congruence, it
is not sufficient (Musso and Nicolodi, J. Math Imaging Vis. 35: 68-85, 2009).
Hickman (J. Math Imaging Vis. 43: 206-213, 2012, Theorem 2) claimed that for
non-degenerate planar curves, equality of Euclidean signatures implies
congruence. We prove that while Hickman's claim holds for simple, closed curves
with simple signatures, it fails for curves with non-simple signatures. In the
later case, we associate a directed graph with the signature and show how
various paths along the graph give rise to a family of non-congruent,
non-degenerate curves with identical signatures. Using this additional
structure, we formulate congruence criteria for non-degenerate, closed, simple
curves and show how the paths reflect the global and local symmetries of the
corresponding curve.

该论文研究了简单封闭曲线的等度数特征相等是完全相似的必要条件，但并不是充分条件，在非正常封闭曲线的情况下，节点可以沿着多种路径移动，导致一系列不全等的封闭曲线。