{"ID":2851655,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.19444","arxiv_id":"2510.19444","title":"A Foundational Theory of Quantitative Abstraction: Adjunctions, Duality, and Logic for Probabilistic Systems","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 $μ$-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.","short_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 in...","url_abs":"https://arxiv.org/abs/2510.19444","url_pdf":"https://arxiv.org/pdf/2510.19444v3","authors":"[\"Nivar Anwer\",\"Ezequiel López-Rubio\",\"David Elizondo\",\"Rafael M. Luque-Baena\"]","published":"2025-10-22T10:16:24Z","proceeding":"cs.LO","tasks":"[\"cs.LO\",\"cs.AI\",\"cs.LG\"]","methods":"[]","has_code":false}