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The work introduces Algebraic Decision Tree Counting (ADTC), a formal framework for exhaustive analysis of optimal and near-optimal decision trees. Inspired by Algebraic Model Counting (AMC), ADTC unifies optimization, counting, and sampling as a sum-of-products computation over a semiring. It provides efficient dynamic programming despite a doubly exponential decision-tree hypothesis space and supports complex constraint handling via model behavior tensors aggregated through convolution products over a tensor semiring. The resulting model profile captures global trade-offs among criteria like accuracy, size, and fairness, and supports evidence-based selection in sensitive domains.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/algebraic-model-counting-for-global-analysis-of-optimal-decision-trees/327417/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/algebraic-model-counting-for-global-analysis-of-optimal-decision-trees/327417.png","ImageObject",300,407,{"name":92,"@type":93},"Eliana","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-03","2026-09-21",true,{"@type":102,"interactionType":103,"userInteractionCount":34},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What problem does ADTC address in Explainable AI?","Question",{"text":112,"@type":113},"ADTC targets predictive multiplicity, where many models show similar performance but provide different explanations. It enables a global assessment of the entire hypothesis space rather than selecting a single heuristic model.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How is ADTC inspired by Algebraic Model Counting (AMC)?",{"text":117,"@type":113},"ADTC follows the algebraic computation paradigm of evaluating expressions over a semiring. It reformulates diverse tasks—optimization, counting, and sampling—into a unified sum-of-products computation over a semiring.",{"name":119,"@type":110,"acceptedAnswer":120},"How does ADTC handle complex constraints across multiple tree metrics?",{"text":121,"@type":113},"It introduces model behavior tensors that aggregate semiring values using convolution products over a tensor semiring. 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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 ∆, our dynamic programming algorithm achieves O ∗ (nO (∆)) 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.  \nKeywords: algebraic model counting · decision trees · model behavior tensors · predictive multiplicity · algorithmic fairness · global assessment  \n1 Introduction  \nExplainable AI (XAI) [28] has placed a growing emphasis on decision trees [9] due to their inherent interpretability. However, the phenomenon of predictive multiplicity [7,24], where multiple models achieve similar performance but offer different explanations, poses a significant challenge for model reliability. To address this, we propose Algebraic Decision Tree Counting (ADTC), a framework that performs a global assessment of the entire hypothesis space rather than identifying a single heuristic solution.  \nOur ADTC is inspired by algebraic model counting (AMC) proposed by De Raedt and Kimmig [22] in knowledge representation [12] . AMC is an instance of the algebraic computation paradigm (see, e.g., Aji and McEliece [2], Eiter and Kiesel [16], and Goral [17]) that generalizes logical counting by evaluating formulas over a semiring. Specifically, our framework aggregates behaviors of near-optimal prediction models based on a dataset and a constraint formula, whereas AMC aggregates weights of satisfying assignments for a given formula. In this context, near-optimal models (or good models) [13, 23, 24, 27] refer to those within the Rashomon set [8, 28, 32] that satisfy a predefined performance threshold relative to the empirical best model.  \n2 Hiroki Arimura  \n(a) Accuracy vs. F1-score. (b) Accuracy vs. DP gap.  \nFig. 1: Model profiles of 34,706 decision trees with maxdep = 5 and relminsup = 0.15 from the Rashomon set on the adult dataset generated by ADTC, showing (a) trade-off between Accuracy and F1-score, and (b) model multiplicity between Accuracy and DP-gap. See Fig. 7 for more examples and Sec. 5.4 for explanation.  \nConceptually, the framework ADTC executes an “enumerate-filter-aggregate” style query over the huge hypothesis space of decision trees, similar to analytic queries in relational databases [6] . While the number of syntactically distinct decision trees isnΘ(2∆ ), our approach achieves a time complexity of O ∗ (n∆ ), which is polynomial in data size n but exponential in query size ∆, by employing a dynamic programming scheme based on tensor operations. Consequently, our framework avoids the doubleexponential complexity of exhaustive enumeration through algebraic aggregation.  \nThe primary contributions of this paper are summarized as follows:  \n– Algebraic framework for d","cbCair6zou61OzmH","https://ap.wps.com/l/cbCair6zou61OzmH","pdf",1768588,18,"English","# Introduction\n# ADTC framework and motivation\n# Model profiling and figures\n## Accuracy vs. F1-score\n## Accuracy vs. DP gap\n# Method overview: enumerate-filter-aggregate\n# Contributions and organization of the paper","[{\"question\":\"What problem does ADTC address in Explainable AI?\",\"answer\":\"ADTC targets predictive multiplicity, where many models show similar performance but provide different explanations. It enables a global assessment of the entire hypothesis space rather than selecting a single heuristic model.\"},{\"question\":\"How is ADTC inspired by Algebraic Model Counting (AMC)?\",\"answer\":\"ADTC follows the algebraic computation paradigm of evaluating expressions over a semiring. It reformulates diverse tasks—optimization, counting, and sampling—into a unified sum-of-products computation over a semiring.\"},{\"question\":\"How does ADTC handle complex constraints across multiple tree metrics?\",\"answer\":\"It introduces model behavior tensors that aggregate semiring values using convolution products over a tensor semiring. This supports efficient computation while incorporating multiple metrics simultaneously.\"}]","Algebraic Model Counting for Global Analysis of Optimal Decision Trees | PDF",1789977160,45]