Proof net

In proof theory, proof nets are a geometrical method of representing proofs that eliminates two forms of bureaucracy that differentiates proofs: (A) irrelevant syntactical features of regular proof calculi such as the natural deduction calculus and the sequent calculus, and (B) the order of rules applied in a derivation. In this way, the formal properties of proof identity correspond more closely to the intuitively desirable properties. Proof nets were introduced by Jean-Yves Girard.

For instance, these two linear logic proofs are “morally” identical:

${\displaystyle \vdash }$ A, B, C, D ${\displaystyle \vdash }$ A ⅋ B, C, D ${\displaystyle \vdash }$ A ⅋ B, C ⅋ D

${\displaystyle \vdash }$ A, B, C, D ${\displaystyle \vdash }$ A, B, C ⅋ D ${\displaystyle \vdash }$ A ⅋ B, C ⅋ D

And their corresponding nets will be the same.

Correctness criteria

Several correctness criteria are known to check if a sequential proof structure (i.e. something which seems to be a proof net) is actually a concrete proof structure (i.e. something which encodes a valid derivation in linear logic). The first such criterion is the long-trip criterion[1] which was described by Jean-Yves Girard.

See also

• Linear logic
• Ludics
• Geometry of interaction
• Coherent space
• Deep inference
• Interaction nets

References

1. Girard, Jean-Yves. Linear logic, Theoretical Computer Science, Vol 50, no 1, pp. 1–102, 1987

Sources

• Proofs and Types. Girard J-Y, Lafont Y, and Taylor P. Cambridge Press, 1989.
• Roberto Di Cosmo and Vincent Danos, The Linear Logic Primer
• Sean A. Fulop, A survey of proof nets and matrices for substructural logics