Tsetlin Machines (TMs) have garnered increasing interest for their ability to learn concepts via propositional formulas and their proven efficiency across various application domains. Despite this, the convergence proof for the TMs, particularly for the AND operator (\emph{conjunction} of literals), in the generalized case (inputs greater than two bits) remains an open problem. This paper aims to fill this gap by presenting a comprehensive convergence analysis of Tsetlin automaton-based Machine Learning algorithms. We introduce a novel framework, referred to as Probabilistic Concept Learning (PCL), which simplifies the TM structure while incorporating dedicated feedback mechanisms and dedicated inclusion/exclusion probabilities for literals. Given $n$ features, PCL aims to learn a set of conjunction clauses $C_i$ each associated with a distinct inclusion probability $p_i$. Most importantly, we establish a theoretical proof confirming that, for any clause $C_k$, PCL converges to a conjunction of literals when $0.5<p_k<1$. This result serves as a stepping stone for future research on the convergence properties of Tsetlin automaton-based learning algorithms. Our findings not only contribute to the theoretical understanding of Tsetlin Machines but also have implications for their practical application, potentially leading to more robust and interpretable machine learning models.

翻译：Tsetlin机器因其通过命题公式学习概念的能力以及在各种应用领域中的高效性而日益受到关注。然而，在广义情形（输入超过两比特）下，针对Tsetlin机器（特别是其AND算子——即文字合取）的收敛性证明仍是一个未解决问题。本文旨在填补这一空白，提出一种基于Tsetlin自动机的机器学习算法的综合收敛性分析。我们引入一个名为概率化概念学习的新框架，该框架简化了Tsetlin机器结构，同时融入了专用反馈机制以及文字特有的包含/排除概率。给定n个特征，PCL旨在学习一组合取子句C_i，每个子句关联一个不同的包含概率p_i。最重要的是，我们建立了一个理论证明，确认对于任意子句C_k，当0.5<p_k<1时，PCL将收敛到文字的合取。该结果为未来研究Tsetlin自动机学习算法的收敛特性奠定了基础。我们的发现不仅有助于加深对Tsetlin机器的理论理解，还对其实际应用具有启示意义，有望推动构建更鲁棒且可解释的机器学习模型。