Extracting analytic proofs from numerically solved Shannon-type Inequalities

TL;DR

This paper presents an algorithm to extract formal, analytic proofs from numerical solutions of Shannon-type inequalities, enhancing understanding of complex information theoretic constraints.

Contribution

The authors develop a method to derive human-readable proofs from numerical solutions of Shannon-type inequalities, bridging the gap between computation and formal proof.

Findings

Algorithm successfully extracts formal proofs from numerical solutions

Applicable to complex inequalities involving multiple constraints

Improves interpretability of information theoretic bounds

Abstract

A class of information inequalities, called Shannon-type inequalities (STIs), can be proven via a computer software called ITIP. In previous work, we have shown how this technique can be utilized to Fourier-Motzkin elimination algorithm for Information Theoretic Inequalities. Here, we provide an algorithm for extracting \emph{analytic} proofs of information inequalities. Shannon-type inequalities are proven by solving an optimization problem. We will show how to extract a \emph{formal} proof of numerically solved information inequality. Such proof may become useful when an inequality is implied by several constraints due to the PMF, and the proof is not apparent easily. More complicated are cases where an inequality holds due to both constraints from the PMF and due to other constraints that arise from the statistical model. Such cases include information theoretic capacity regions, rate-distortion functions and lossless compression rates. We begin with formal definition of Shannon-type information inequalities. We then review the optimal solution of the optimization problem and how to extract a proof that is readable to the user.