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

arXiv: https://arxiv.org/abs/2510.19444 · HTML: https://mlsanity.com/paper/2510.19444/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 41 references that are in the map.
- First posted on arXiv 22 Oct 2025.

## Builds on (most cited first, 7 of 7)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Learning Invariant Representations for Reinforcement Learning without Reconstruction](https://mlsanity.com/paper/2006.10742/) (2020)
- [DeepMDP: Learning Continuous Latent Space Models for Representation Learning](https://mlsanity.com/paper/1906.02736/) (2019)
- [State Representation Learning for Control: An Overview](https://mlsanity.com/paper/1802.04181/) (2018)
- [Contrastive Behavioral Similarity Embeddings for Generalization in Reinforcement Learning](https://mlsanity.com/paper/2101.05265/) (2021)
- [Modal Characterisations of Behavioural Pseudometrics](https://mlsanity.com/paper/1509.03391/) (2015)

## Abstract (arXiv)

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.