Algebraic Model Counting for Global Analysis of Optimal Decision Trees

2026-07-02

AI Summary

This work addresses the lack of a global formal analysis framework for the hypothesis space of decision trees in existing explainable AI, which hinders reliable assessment of model optimality and multi-objective trade-offs. The authors propose the Algebraic Decision Tree Counting (ADTC) framework, which, for the first time, introduces algebraic model counting to the global analysis of decision trees. By leveraging model behavior tensors and tensor semiring convolutions, ADTC unifies optimization, counting, and sampling into a single sum-product computation over semirings. A dynamic programming algorithm is devised to achieve a time complexity of $O^*(n^{O(\Delta)})$. Experiments using the emtrees software on real-world datasets demonstrate that the method efficiently characterizes the model Pareto front under multidimensional constraints such as accuracy, size, and fairness.

Abstract

Ensuring model reliability in Explainable AI requires a global assessment of the hypothesis space. We propose a formal framework for the exhaustive analysis of optimal and near-optimal decision trees, called Algebraic Decision Tree Counting (ADTC). Inspired by Algebraic Model Counting (AMC) in knowledge representation, ADTC reformulates diverse analytical tasks, such as optimization, counting, and sampling, into a unified sum-of-products computation over a semiring $R$. While the hypothesis space of decision trees is doubly exponential with respect to the maximum depth $Δ$, our dynamic programming algorithm achieves $O^*(n^{O(Δ)})$ time complexity in the number of features $n$, where $O^*$ suppresses polynomial factors. To handle complex constraints consisting of multiple tree metrics, we introduce model behavior tensors that aggregate semiring values via convolution products over a tensor semiring. This algebraic approach efficiently constructs a model profile that captures the global landscape and trade-offs between criteria such as accuracy, size, and fairness. We demonstrate the utility of our software, emtrees, on real-world datasets, illustrating how ADTC facilitates evidence-based model selection in sensitive domains.

Problem

• Explainable AI
• Decision Trees
• Model Reliability
• Global Analysis
• Hypothesis Space

Innovation

• Algebraic Model Counting
• Decision Tree Optimization
• Dynamic Programming
• Model Behavior Tensors
• Explainable AI