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 ∀x Px ⊢ ∃x Px, one per arbitrary witness for ∃x, there is a unique cut-free unification net, with no specified witness.
Cut elimination for unification nets is local and linear time, while Girard’s is non-local and exponential time. 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 transcend MLL1 via a methodological insight: for canonical quantifiers, the standard parallel/sequential dichotomy of proof nets is insufficient; an implicit/explicit witness dichotomy is needed. Current work extends unification nets to additives and uses them to extend combinatorial proofs [Proofs without syntax, Annals of Mathematics, 2006] to classical first-order logic.