Sergey Trifonov | Searcharxiv

arXiv2026

An Ontology-Based Approach to Optimizing Geometry Problem Sets for Skill Development

Euclidean geometry has historically played a central role in cultivating logical reasoning and abstract thinking within mathematics education, but has experienced waning emphasis in recent curricula. The resurgence of interest, driven by advances in artificial intelligence and educational technology, has highlighted geometry's potential to develop essential cognitive skills and inspired new approaches to automated problem solving and proof verification. This article presents an ontology-based framework for annotating and optimizing geometry problem sets, originally developed in the 1990s. The ontology systematically classifies geometric problems, solutions, and associated skills into interlinked facts, objects, and methods, supporting granular tracking of student abilities and facilitating curriculum design. The core concept of 'solution graphs': directed acyclic graphs encoding multiple solution pathways and skill dependencies enables alignment of problem selection with instructional objectives. The framework has been tested in practice through the annotation of thousands of problems over three decades. We contend that our approach addresses longstanding challenges in representing dynamic, procedurally complex mathematical knowledge. We conclude by articulating a research agenda: the open problems of automated problem annotation and solution validation, whose resolution would reduce the time teachers spend validating student work and enable interactive feedback for self-learners.

math.HO↗

arXiv2026

Trustworthy Clinical Decision Support Using Meta-Predicates and Domain-Specific Languages

\textbf{Background:} Regulatory frameworks for AI in healthcare, including the EU AI Act and FDA guidance on AI/ML-based medical devices, require clinical decision support to demonstrate not only accuracy but auditability. Existing formal languages for clinical logic validate syntactic and structural correctness but not whether decision rules use epistemologically appropriate evidence. \textbf{Methods:} Drawing on design-by-contract principles, we introduce meta-predicates -- predicates about predicates -- for asserting epistemological constraints on clinical decision rules expressed in a DSL. An epistemological type system classifies annotations along four dimensions: purpose, knowledge domain, scale, and method of acquisition. Meta-predicates assert which evidence types are permissible in any given rule. The framework is instantiated in AnFiSA, an open-source platform for genetic variant curation, and demonstrated using the Brigham Genomics Medicine protocol on 5.6 million variants from the Genome in a Bottle benchmark. \textbf{Results:} Decision trees used in variant interpretation can be reformulated as unate cascades, enabling per-variant audit trails that identify which rule classified each variant and why. Meta-predicate validation catches epistemological errors before deployment, whether rules are human-written or AI-generated. The approach complements post-hoc methods such as LIME and SHAP: where explanation reveals what evidence was used after the fact, meta-predicates constrain what evidence may be used before deployment, while preserving human readability. \textbf{Conclusions:} Meta-predicate validation is a step toward demonstrating not only that decisions are accurate but that they rest on appropriate evidence in ways that can be independently audited. While demonstrated in genomics, the approach generalises to any domain requiring auditable decision logic.

cs.AI↗