Physics-Aware Learnability: From Set-Theoretic Independence to Operational Constraints

2026-02-27

AI Summary

Traditional learnability theory in continuous domains—such as the EMX problem—relies on set-theoretic assumptions, leading to conclusions that are inconsistent across ZFC models and lack physical realizability. This work proposes Physical-aware Learnability (PL), a novel framework that explicitly incorporates physical constraints such as finite precision and quantum POVMs into learning theory. By leveraging coarse-graining, pushforward/pullback reductions, and semidefinite programming, PL establishes a computationally tractable and physically consistent foundation for learnability. The framework eliminates the logical fragility arising from set-theoretic independence, proving that continuous EMX becomes learnable under finite precision with explicit $(\varepsilon,\delta)$-sample complexity. Furthermore, PL feasibility is decidable in both quantum and no-signaling models, and it satisfies a Helstrom-type lower bound.

Technology Category

• Machine Learning: Learning Theory
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Search and Optimization: Learning to Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Security and Privacy: Security and privacy of machine learning and AI applications

Abstract

Beyond binary classification, learnability can become a logically fragile notion: in EMX, even the class of all finite subsets of $[0,1]$ is learnable in some models of ZFC and not in others. We argue the paradox is operational. The standard definitions quantify over arbitrary set-theoretic learners that implicitly assume non-operational resources (infinite precision, unphysical data access, and non-representable outputs). We introduce physics-aware learnability (PL), which defines the learnability relative to an explicit access model -- a family of admissible physical protocols. Finite-precision coarse-graining reduces continuum EMX to a countable problem, via an exact pushforward/pullback reduction that preserves the EMX objective, making the independence example provably learnable with explicit $(ε,δ)$ sample complexity. For quantum data, admissible learners are exactly POVMs on $d$ copies, turning sample size into copy complexity and yielding Helstrom(-type) lower bounds. For finite no-signaling and quantum models, PL feasibility becomes linear or semidefinite and is therefore decidable.

Problem

• learnability
• EMX
• physics-aware
• set-theoretic independence
• operational constraints

Innovation

• Physics-Aware Learnability
• Operational Constraints
• EMX Problem
• Quantum POVM
• Sample Complexity