Congruence-based Learning of Probabilistic Deterministic Finite Automata

Abstract: This work studies the question of learning probabilistic deterministic automata from LLMs. For this purpose, it focuses on analyzing the relations defined on algebraic structures over strings by equivalences and similarities on probability distributions. We introduce a congruence that extends the classical Myhill-Nerode congruence for formal languages. This new congruence is the basis for defining regularity over LLMs. We present an active learning algorithm that computes the quotient with respect to this congruence whenever the LLM is regular. The paper also defines the notion of recognizability for LLMs and shows that it coincides with regularity for congruences. For relations which are not congruences, it shows that this is not the case. Finally, it discusses the impact of this result on learning in the context of LLMs.