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

• URL: http://arxiv.org/abs/2510.19444v2
• Date: Wed, 05 Nov 2025 01:26:07 GMT
• Title: A Foundational Theory of Quantitative Abstraction: Adjunctions, Duality, and Logic for Probabilistic Systems
• Authors: Nivar Anwer, Ezequiel López-Rubio, David Elizondo, Rafael M. Luque-Baena,
• Abstract summary: Large or continuous state spaces make exact analysis intractable and call for principled quantitative abstraction.<n>This work develops a unified theory of such abstraction by integrating category theory, coalgebra, quantitative logic, and optimal transport.
• Score: 2.362412515574206
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

Related papers

• SIGMA: Scalable Spectral Insights for LLM Collapse [51.863164847253366]
  We introduce SIGMA (Spectral Inequalities for Gram Matrix Analysis), a unified framework for model collapse.<n>By utilizing benchmarks that deriving and deterministic bounds on the matrix's spectrum, SIGMA provides a mathematically grounded metric to track the contraction of the representation space.<n>We demonstrate that SIGMA effectively captures the transition towards states, offering both theoretical insights into the mechanics of collapse.
  arXiv  Detail & Related papers  (2026-01-06T19:47:11Z)
• Sparse Knowledge Distillation: A Mathematical Framework for Probability-Domain Temperature Scaling and Multi-Stage Compression [0.0]
  We develop a unified theoretical framework for sparse knowledge distillation based on probability-domain softening operators.<n>We introduce an axiomatic definition of probability-domain softening operators based on ranking preservation, continuity, entropy monotonicity, identity, and boundary behavior.<n>Results provide theoretical grounding for black-box teacher distillation, partial-access settings such as top-$k$ truncation and text-only outputs, and privacy-equivalent model compression.
  arXiv  Detail & Related papers  (2026-01-06T17:17:24Z)
• A Mean-Field Theory of $Θ$-Expectations [2.1756081703276]
  We develop a new class of calculus for such non-linear models.<n>Theta-Expectation is shown to be consistent with the axiom of subaddivity.
  arXiv  Detail & Related papers  (2025-07-30T11:08:56Z)
• A Theory of $θ$-Expectations [2.1756081703276]
  We develop a framework for a class of differential equations where the driver is a pointwise geometry.<n>The system's tractability is predicated on a global existence of a unique and globally globally.<n> Lipschitz maximizer map for the driver function.
  arXiv  Detail & Related papers  (2025-07-27T16:56:01Z)
• Transfinite Fixed Points in Alpay Algebra as Ordinal Game Equilibria in Dependent Type Theory [0.0]
  This paper contributes to the Alpay Algebra by demonstrating that the stable outcome of a self referential process is identical to the unique equilibrium of an unbounded revision dialogue between a system and its environment.<n>By unifying concepts from fixed point theory, game semantics, ordinal analysis, and type theory, this research establishes a broadly accessible yet formally rigorous foundation for reasoning about infinite self referential systems.
  arXiv  Detail & Related papers  (2025-07-25T13:12:55Z)
• Loss-Complexity Landscape and Model Structure Functions [53.92822954974537]
  We develop a framework for dualizing the Kolmogorov structure function $h_x(alpha)$.<n>We establish a mathematical analogy between information-theoretic constructs and statistical mechanics.<n>We explicitly prove the Legendre-Fenchel duality between the structure function and free energy.
  arXiv  Detail & Related papers  (2025-07-17T21:31:45Z)
• Bound by semanticity: universal laws governing the generalization-identification tradeoff [8.437463955457423]
  We show that finite-resolution similarity is a fundamental emergent informational constraint, not merely a toy-model artifact.<n>These results provide an exact theory of the generalization-identification trade-off and clarify how semantic resolution shapes the representational capacity of deep networks and brains alike.
  arXiv  Detail & Related papers  (2025-06-01T15:56:26Z)
• Conditional Distribution Quantization in Machine Learning [83.54039134248231]
  Conditional expectation mathbbE(Y mid X) often fails to capture the complexity of multimodal conditional distributions mathcalL(Y mid X)<n>We propose using n-point conditional quantizations--functional mappings of X that are learnable via gradient descent--to approximate mathcalL(Y mid X)
  arXiv  Detail & Related papers  (2025-02-11T00:28:24Z)
• Enriching Disentanglement: From Logical Definitions to Quantitative Metrics [59.12308034729482]
  Disentangling the explanatory factors in complex data is a promising approach for data-efficient representation learning. We establish relationships between logical definitions and quantitative metrics to derive theoretically grounded disentanglement metrics. We empirically demonstrate the effectiveness of the proposed metrics by isolating different aspects of disentangled representations.
  arXiv  Detail & Related papers  (2023-05-19T08:22:23Z)
• A Robustness Analysis of Blind Source Separation [91.3755431537592]
  Blind source separation (BSS) aims to recover an unobserved signal from its mixture $X=f(S)$ under the condition that the transformation $f$ is invertible but unknown. We present a general framework for analysing such violations and quantifying their impact on the blind recovery of $S$ from $X$. We show that a generic BSS-solution in response to general deviations from its defining structural assumptions can be profitably analysed in the form of explicit continuity guarantees.
  arXiv  Detail & Related papers  (2023-03-17T16:30:51Z)
• Correspondence between open bosonic systems and stochastic differential equations [77.34726150561087]
  We show that there can also be an exact correspondence at finite $n$ when the bosonic system is generalized to include interactions with the environment. A particular system with the form of a discrete nonlinear Schr"odinger equation is analyzed in more detail.
  arXiv  Detail & Related papers  (2023-02-03T19:17:37Z)
• Reinforcement Learning from Partial Observation: Linear Function Approximation with Provable Sample Efficiency [111.83670279016599]
  We study reinforcement learning for partially observed decision processes (POMDPs) with infinite observation and state spaces. We make the first attempt at partial observability and function approximation for a class of POMDPs with a linear structure.
  arXiv  Detail & Related papers  (2022-04-20T21:15:38Z)
• Logical Credal Networks [87.25387518070411]
  This paper introduces Logical Credal Networks, an expressive probabilistic logic that generalizes many prior models that combine logic and probability. We investigate its performance on maximum a posteriori inference tasks, including solving Mastermind games with uncertainty and detecting credit card fraud.
  arXiv  Detail & Related papers  (2021-09-25T00:00:47Z)
• Foundations of Reasoning with Uncertainty via Real-valued Logics [70.43924776071616]
  We give a sound and strongly complete axiomatization that can be parametrized to cover essentially every real-valued logic. Our class of sentences are very rich, and each describes a set of possible real values for a collection of formulas of the real-valued logic.
  arXiv  Detail & Related papers  (2020-08-06T02:13:11Z)
• Logical Neural Networks [51.46602187496816]
  We propose a novel framework seamlessly providing key properties of both neural nets (learning) and symbolic logic (knowledge and reasoning) Every neuron has a meaning as a component of a formula in a weighted real-valued logic, yielding a highly intepretable disentangled representation. Inference is omni rather than focused on predefined target variables, and corresponds to logical reasoning.
  arXiv  Detail & Related papers  (2020-06-23T16:55:45Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.