Unification nets: canonical proof net quantifiers

by   Dominic J. D. Hughes, et al.

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 ∃ xPx, one per arbitrary choice of witness for ∃ 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.


page 1

page 2

page 3

page 4


Taylor expansion in linear logic is invertible

Each Multiplicative Exponential Linear Logic (MELL) proof-net can be exp...

Exponentially Handsome Proof Nets and Their Normalization

Handsome proof nets were introduced by Retoré as a syntax for multiplica...

An application of parallel cut elimination in multiplicative linear logic to the Taylor expansion of proof nets

We examine some combinatorial properties of parallel cut elimination in ...

Proof nets, coends and the Yoneda isomorphism

Proof nets provide permutation-independent representations of proofs and...

Graphical Proof Theory I: Multiplicative Linear Logic Beyond Cographs

Cographs are a class of (undirected) graphs, characterized by the absenc...

Logic and computation as combinatorics

The syntactic nature of logic and computation separates them from other ...

Redundant Sudoku Rules

The rules of Sudoku are often specified using twenty seven all_different...

Please sign up or login with your details

Forgot password? Click here to reset