  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.
