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

TL;DR

This paper introduces physics-aware learnability (PL), a framework that incorporates physical constraints into learnability definitions, making certain problems decidable and providing explicit sample complexity bounds.

Contribution

It formalizes physics-aware learnability by explicitly modeling physical access protocols, connecting it to operational resources and enabling decidability and complexity analysis.

Findings

Finite-precision reduces continuum EMX to a countable problem

Explicit protocols enable provable learnability with sample complexity bounds

PL feasibility is decidable for quantum and no-signaling models

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 $(\epsilon,\delta)$ 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.