概率系统定量抽象的奠基理论：伴随、对偶与逻辑
A Foundational Theory of Quantitative Abstraction: Adjunctions, Duality, and Logic for Probabilistic Systems
AI总结：
本文提出概率系统定量抽象的理论框架，结合范畴论、余代数和最优传输，通过伴随关系和定量逻辑实现抽象与实现的对偶性，为概率领域中的状态抽象和表示学习提供严格基础。
摘要：
随机动态系统的分析与控制依赖于概率模型，如连续空间马尔可夫决策过程，但大或连续状态空间使精确分析不可行，需要原理性的定量抽象。本文通过整合范畴论、余代数、定量逻辑和最优传输，发展了一种统一的抽象理论，以一个规范的ε-商作为行为伪度量的中心，具有普遍性质：在所有将行为差异压缩到ε以下的抽象中，它是最详细的，且任何其他实现相同折扣价值损失保证的抽象都唯一通过它。范畴论中，从概率系统范畴到度量规范范畴的商 functor Q_ε 通过特殊伴随 functor 定理具有右伴随 R_ε，从而得到伴随 Q_ε⊣R_ε，形式化了抽象与实现之间的对偶关系；逻辑上，一个带有单独奖励和转移模态的定量模态 μ-算术被展示为在广泛系统类中对行为伪度量的表达完整，具有可数的完全抽象片段，适用于计算。该理论在波兰空间和吉里单子上以余代数方式发展，并在有限状态模型上用最优传输求解器验证，实验支持预测的收缩性质和结构稳定性，并与理论价值损失界限一致，从而为概率领域中的定量状态抽象和表示学习提供了严格的基础。
英文摘要：
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.
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