arXiv:2510.19444v2 Announce Type: replace-cross 
Abstract: The analysis and control of stochastic dynamical systems rely on probabilistic models such as (continuous-space) Markov decision processes, but large or continuous state spaces make exact analysis intractable and call for principled quantitative abstraction. This work develops a unified theory of such abstraction by integrating category theory, coalgebra, quantitative logic, and optimal transport, centred on a canonical $\varepsilon$-quotient of the behavioral pseudo-metric with a universal property: among all abstractions that collapse behavioral differences below $\varepsilon$, it is the most detailed, and every other abstraction achieving the same discounted value-loss guarantee factors uniquely through it. Categorically, a quotient functor $Q_\varepsilon$ from a category of probabilistic systems to a category of metric specifications admits, via the Special Adjoint Functor Theorem, a right adjoint $R_\varepsilon$, yielding an adjunction $Q_\varepsilon \dashv R_\varepsilon$ that formalizes a duality between abstraction and realization; logically, a quantitative modal $\mu$-calculus with separate reward and transition modalities is shown, for a broad class of systems, to be expressively complete for the behavioral pseudo-metric, with a countable fully abstract fragment suitable for computation. The theory is developed coalgebraically over Polish spaces and the Giry monad and validated on finite-state models using optimal-transport solvers, with experiments corroborating the predicted contraction properties and structural stability and aligning with the theoretical value-loss bounds, thereby providing a rigorous foundation for quantitative state abstraction and representation learning in probabilistic domains.

ظهرت نظرية أساسية جديدة تعزز فهمنا للأنظمة الاحتمالية، خاصة في تحليل وتحكم الأنظمة الديناميكية العشوائية. من خلال دمج مفاهيم من نظرية الفئات، والكواليجرا، والمنطق الكمي، يتناول هذا العمل التحديات التي تطرحها المساحات الكبيرة أو المستمرة في عمليات اتخاذ القرار ماركوف. هذه التطورات مهمة لأنها توفر نهجًا مبدئيًا للتجريد الكمي، وهو أمر حاسم لتطوير نماذج أكثر فعالية في مجالات متنوعة، بما في ذلك الاقتصاد والهندسة.

Ha surgido una nueva teoría fundamental que mejora nuestra comprensión de los sistemas probabilísticos, especialmente en el análisis y control de sistemas dinámicos estocásticos. Al integrar conceptos de teoría de categorías, coalgebra y lógica cuantitativa, este trabajo aborda los desafíos que presentan los grandes espacios de estado o continuos en los procesos de decisión de Markov. Este avance es significativo ya que proporciona un enfoque fundamentado para la abstracción cuantitativa, crucial para desarrollar modelos más efectivos en diversos campos, incluyendo la economía y la ingeniería.

Une nouvelle théorie fondamentale a émergé, améliorant notre compréhension des systèmes probabilistes, en particulier dans l'analyse et le contrôle des systèmes dynamiques stochastiques. En intégrant des concepts de la théorie des catégories, de la coalgebra et de la logique quantitative, ce travail aborde les défis posés par les grands espaces d'état ou continus dans les processus de décision de Markov. Cette avancée est significative car elle fournit une approche fondée sur l'abstraction quantitative, essentielle pour développer des modèles plus efficaces dans divers domaines, y compris l'économie et l'ingénierie.

A new foundational theory has emerged that enhances our understanding of probabilistic systems, particularly in the analysis and control of stochastic dynamical systems. By integrating concepts from category theory, coalgebra, and quantitative logic, this work addresses the challenges posed by large or continuous state spaces in Markov decision processes. This advancement is significant as it provides a principled approach to quantitative abstraction, which is crucial for developing more effective models in various fields, including economics and engineering.

A Foundational Theory of Quantitative Abstraction: Adjunctions, Duality, and Logic for Probabilistic Systems

arXiv:2511.18682v1 Announce Type: new 
Abstract: Recent years have witnessed rapid advancements in 3D scanning technologies, with applications spanning VR/AR, digital human creation, and medical imaging. Structured-light scanning with phase-shifting techniques is preferred for its use of low-intensity visible light and high accuracy, making it well suited for capturing 4D facial dynamics. A key step is phase unwrapping, which recovers continuous phase values from measurements wrapped modulo 2pi. The goal is to estimate the unwrapped phase count k in the equation Phi = phi + 2pi k, where phi is the wrapped phase and Phi is the true phase. Noise, occlusions, and complex 3D geometry make recovering the true phase challenging because phase unwrapping is ill-posed: measurements only provide modulo 2pi values, and estimating k requires assumptions about surface continuity. Existing methods trade speed for accuracy: fast approaches lack precision, while accurate algorithms are too slow for real-time use. To overcome these limitations, this work proposes a phase unwrapping framework that reformulates GraphCut-based unwrapping as a pixel-labeling problem. This framework improves the estimation of the unwrapped phase count k through the invariance property of diffeomorphisms applied in image space via conformal and optimal transport (OT) maps. An odd number of diffeomorphisms are precomputed from the input phase data, and a hierarchical GraphCut algorithm is applied in each domain. The resulting label maps are fused via majority voting to robustly estimate k at each pixel. Experimental results demonstrate a 45.5x speedup and lower L2 error in real experiments and simulations, showing potential for real-time applications.

أدت التطورات الأخيرة في تقنيات المسح ثلاثي الأبعاد إلى تقديم طريقة جديدة لفك الطور تعتمد على إطار عمل عدم تغير الديفومورفيسم. تهدف هذه التقنية إلى استعادة قيم الطور المستمرة بدقة من القياسات الملفوفة، مما يعالج التحديات التي تطرحها الضوضاء والهندسة المعقدة في تطبيقات مثل الواقع الافتراضي، والواقع المعزز، والتصوير الطبي.

Los recientes avances en tecnologías de escaneo 3D han llevado a la introducción de un nuevo método de desenrollado de fase basado en el marco de invariancia de difeomorfismos. Esta técnica tiene como objetivo recuperar con precisión los valores de fase continuos a partir de mediciones envueltas, abordando los desafíos que plantean el ruido y las geometrías complejas en aplicaciones como VR, AR y la imagen médica.

Les récentes avancées dans les technologies de numérisation 3D ont conduit à l'introduction d'une nouvelle méthode de dépliage de phase basée sur le cadre d'invariance des difféomorphismes. Cette technique vise à récupérer avec précision des valeurs de phase continues à partir de mesures enveloppées, en s'attaquant aux défis posés par le bruit et les géométries complexes dans des applications telles que la réalité virtuelle, la réalité augmentée et l'imagerie médicale.

Recent advancements in 3D scanning technologies have led to the introduction of a new phase unwrapping method based on the Invariance of Diffeomorphisms Framework. This technique aims to accurately recover continuous phase values from wrapped measurements, addressing challenges posed by noise and complex geometries in applications like VR, AR, and medical imaging.

A Foundational Theory of Quantitative Abstraction: Adjunctions, Duality, and Logic for Probabilistic Systems

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