Unification nets: canonical proof net quantifiers

TL;DR

Unification nets provide a canonical, efficient graphical proof representation for first-order multiplicative linear logic, overcoming redundancy and complexity issues present in Girard's proof nets.

Contribution

The paper introduces unification nets as a new canonical proof net for MLL1 that eliminates existential witness redundancy and improves cut elimination complexity.

Findings

Unification nets are unique and canonical for MLL1 proofs.

Cut elimination for unification nets is linear-time and local.

Unification nets are exponentially smaller than Girard nets in some cases.

Abstract

Proof nets for MLL (unit-free Multiplicative Linear Logic) are concise graphical representations of proofs which are canonical in the sense that they abstract away syntactic redundancy such as the order of non-interacting rules. We argue that Girard's extension to MLL1 (first-order MLL) fails to be canonical because of redundant existential witnesses, and present canonical MLL1 proof nets called unification nets without them. For example, while there are infinitely many cut-free Girard nets $\forall x Px \vdash \exists xPx$, one per arbitrary choice of witness for $\exists x$, there is a unique cut-free unification net, with no specified witness. Redundant existential witnesses cause Girard's MLL1 nets to suffer from severe complexity issues: (1) cut elimination is non-local and exponential-time (and -space), and (2) some sequents require exponentially large cut-free Girard nets. Unification nets solve both problems: (1) cut elimination is local and linear-time, and (2) cut-free unification nets grow linearly with the size of the sequent. Since some unification nets are exponentially smaller than corresponding Girard nets and sequent proofs, technical delicacy is required to ensure correctness is polynomial-time (quadratic). These results extend beyond MLL1 via a broader methodological insight: for canonical quantifiers, the standard parallel/sequential dichotomy of proof nets fails; an implicit/explicit witness dichotomy is also needed. Work in progress extends unification nets to additives and uses them to extend combinatorial proofs [Proofs without syntax, Annals of Mathematics, 2006] to classical first-order logic.