##  [Proof Nets](/proof-nets-0) 

 Definition

A graphical, canonical representation of proofs in fragments of linear logic where formulas and their connections are rendered as nodes and links; proof nets expose parallelism, collapse syntactic bureaucracy of sequent proofs, and admit graphical correctness criteria and cut-elimination as local graph transformations.

 

 

 

 

 

 





## Principle

Principle

Encode a proof as a graph-like structure (links for logical connectives and axiom links for atomic pairs) so that correctness conditions identify graphs that correspond to valid sequent proofs; cut-elimination becomes a sequence of local graph rewrites preserving correctness.

 

 

 

 

 





## Demonstration

Demonstration

In multiplicative linear logic, a derivation of A⊗B from premises can be drawn as a net with a tensor link joining the subnets for A and B and axiom links pairing dual atoms; a correctness condition (acyclicity and connectedness under switchings) certifies that the diagram represents a valid proof and that graph rewrites implement cut-elimination.

 

 

 

 

## Misapplication

Misapplication

Treating any arbitrary graph with similar degrees as a proof net without checking the prescribed correctness criterion, or applying proof-net transformations from one fragment of linear logic to another incompatible fragment, can yield unsound 'proofs' or break cut-elimination invariants.

 

 

 

 

 





## Consequence

Consequence

When correctly formed, proof nets supply a canonical, often compact witness of provability that makes parallelism explicit, simplifies equivalence of proofs, and turns cut-elimination into local, confluent graph rewriting—facilitating complexity and normalization analyses.

 

 

 

 

## Reversal

Reversal

The opposite viewpoint is the sequential sequent proof: a linearized, rule-by-rule derivation that records the order of rule applications explicitly. While sequent proofs make operational content visible, they obscure parallel structure and admit many syntactic variants of the same net.

 

 

 

 

 





## Boundary

Boundary

Proof nets are well-developed for several fragments of linear logic (multiplicative, multiplicative-exponential with care) but require tailored correctness and linking schemes per fragment; they are not a plug-and-play graphical representation for arbitrary non-linear or classical logics without modification.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension exists between proof nets and sequent-style presentations: proof nets emphasize canonical, parallel structure and canonical equivalence classes of proofs, while sequent proofs emphasize syntactic derivation order and operational interpretability; converting between them (sequentialization) is nontrivial and fragment-dependent.

 

 

 

 

 





## Synthesis

Synthesis

Proof nets compress sequent derivations into graph objects that identify the essential parallel structure and canonical content of proofs in linear logic: by imposing precise correctness checks and interpreting cut-elimination as graph rewrites, they provide a compact, semantically transparent account of proofs and normalization.