A coding theoretic study of MLL proof nets

Abstract

In this paper we propose a novel approach for analysing proof nets of Multiplicative Linear Logic (MLL) using coding theory. We define families of proof structures called PS-families and introduce a metric space for each family. In each family: (1)an MLL proof net is a true code element; and(2)a proof structure that is not an MLL proof net is a false (or corrupted) code element. The definition of our metrics elegantly reflects the duality of the multiplicative connectives. We show that in our framework one-error-detection is always possible but one-error-correction is always impossible. We also demonstrate the importance of our main result by presenting two proof-net enumeration algorithms for a given PS-family: the first searches proof nets naively and exhaustively without help from our main result, while the second uses our main result to carry out an intelligent search. In some cases, the first algorithm visits proof structures exponentially, while the second does so only polynomially.