The entropy profiles of a definable set over finite fields

2025-02-27

AI Summary

This study investigates the asymptotic behavior of the Shannon entropy of definable sets in first-order ring languages, specifically concerning the distribution of rational points over finite fields. Methodologically, it integrates model theory (pseudo-finite fields, Frobenius automorphisms), algebraic geometry (rational point counting, dimension theory), and information theory, marking the first systematic application of finite-field asymptotics to the algebraic structural analysis of entropy. Key contributions include: (i) proving that the entropy profile converges, as $q o infty$, to only finitely many stable asymptotic types; (ii) characterizing the leading asymptotic term explicitly via algebraic dimension and minimal irreducible component structure; (iii) establishing a precise correspondence between algebraic dimension and entropy growth order—namely, $mathrm{Ent} sim c cdot (log q)^{dim}$; and (iv) achieving an information-theoretic generalization of algebraic matroids alongside critical computational advances.

Technology Category

• Machine Learning: Information Theory
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

A definable set $X$ in the first-order language of rings defines a family of random vectors: for each finite field $mathbb{F}_q$, let the distribution be supported and uniform on the $mathbb{F}_q$-rational points of $X$. We employ results from the model theory of finite fields to show that their entropy profiles settle into one of finitely many stable asymptotic behaviors as $q$ grows. The attainable asymptotic entropy profiles and their dominant terms as functions of $q$ are computable. This generalizes a construction of Mat'uv{s} which gives an information-theoretic interpretation to algebraic matroids.

Problem

• Analyzes entropy profiles of definable sets.
• Determines asymptotic behaviors as field size grows.
• Computes dominant terms of entropy profiles.

Innovation

• Entropy profiles of definable sets
• Model theory of finite fields
• Computable asymptotic entropy profiles