A Coding Theoretic Study on MLL proof nets

TL;DR

This paper applies coding theory to analyze proof nets of Multiplicative Linear Logic (MLL), defining metrics and demonstrating the framework's capabilities for error detection but not correction, with implications for logic selection.

Contribution

It introduces a novel coding-theoretic framework for MLL proof nets, defining metrics and analyzing error detection and correction capabilities.

Findings

One error-detecting is possible within the framework.

One error-correcting is impossible, proved via graph theory.

MLL is more suitable than affine logic and MLL + MIX for this approach.

Abstract

Coding theory is very useful for real world applications. A notable example is digital television. Basically, coding theory is to study a way of detecting and/or correcting data that may be true or false. Moreover coding theory is an area of mathematics, in which there is an interplay between many branches of mathematics, e.g., abstract algebra, combinatorics, discrete geometry, information theory, etc. In this paper we propose a novel approach for analyzing proof nets of Multiplicative Linear Logic (MLL) by coding theory. We define families of proof structures and introduce a metric space for each family. In each family, 1. an MLL proof net is a true code element; 2. a proof structure that is not an MLL proof net is a false (or corrupted) code element. The definition of our metrics reflects the duality of the multiplicative connectives elegantly. In this paper we show that in the framework one error-detecting is possible but one error-correcting not. Our proof of the impossibility of one error-correcting is interesting in the sense that a proof theoretical property is proved using a graph theoretical argument. In addition, we show that affine logic and MLL + MIX are not appropriate for this framework. That explains why MLL is better than such similar logics.