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We introduce and execute a transparent machine learning classification model designed for Explainable AI (XAI) that leverages expressive Boolean formulas. Potential domains of use include credit risk evaluation and medical diagnostics. The system utilizes a Boolean formula to establish a rule with adjustable complexity (or interpretability) for categorizing input data. This formula accommodates any operator applicable to single or multiple Boolean variables, offering superior expressivity relative to more rigid rule- and tree-based methodologies. Native local optimization methods are used to train the classifier, allowing for an efficient search of the feasible formula space. Fast Integer Linear Programming (ILP) or Quadratic Unconstrained Binary Optimization (QUBO) solvers—potentially enhanced by custom hardware or quantum systems—can identify shallow rules. By executing non-local moves that optimize across subtrees of the complete Boolean formula, we merge the expressivity and efficiency of the native local optimizer with the rapid processing capabilities of these devices. We present comprehensive numerical benchmarking data against multiple baseline models using recognized public datasets. The findings indicate that the native local rule classifier performs competitively with alternative models. Incorporating non-local moves yields comparable outcomes with a reduced number of iterations. Consequently, deploying specialized or quantum hardware could provide substantial acceleration by rapidly generating non-local moves.