Extracting analytic proofs from numerically solved Shannon-type Inequalities

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.