On Structuring Proof Search for First Order Linear Logic

TL;DR

This paper introduces G-Forum, a restricted yet equivalent fragment of first order linear logic, simplifying proof search and enabling proof theoretic analysis through a minimal set of inference rules.

Contribution

It proposes G-Forum, a simplified version of Forum, with restricted formulae that retains full expressiveness and facilitates proof analysis and cut elimination.

Findings

G-Forum is equivalent to full first order linear logic.

G-Forum's inference rules enable cut elimination without complex formula analysis.

Restriction to specific formula shapes does not reduce expressiveness.

Abstract

Full first order linear logic can be presented as an abstract logic programming language in Miller's system Forum, which yields a sensible operational interpretation in the 'proof search as computation' paradigm. However, Forum still has to deal with syntactic details that would normally be ignored by a reasonable operational semantics. In this respect, Forum improves on Gentzen systems for linear logic by restricting the language and the form of inference rules. We further improve on Forum by restricting the class of formulae allowed, in a system we call G-Forum, which is still equivalent to full first order linear logic. The only formulae allowed in G-Forum have the same shape as Forum sequents: the restriction does not diminish expressiveness and makes G-Forum amenable to proof theoretic analysis. G-Forum consists of two (big) inference rules, for which we show a cut elimination procedure. This does not need to appeal to finer detail in formulae and sequents than is provided by G-Forum, thus successfully testing the internal symmetries of our system.