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

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

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

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

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

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

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

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

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.