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

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.