An Introduction to Proof Nets

We give some basic results of the theory of proof nets for multiplicative linear logic and multiplicative exponential linear logic with mix. The relation between proof nets and the lambda-calculus is precisely described. 1 Multiplicative Proof Nets 1.1 Multiplicative Linear Logic We assume given a denumerable set of atoms X, Y , . . . The formulas of multiplicative linear logic (MLL) are defined as: A,B ::= X | X⊥ | A⊗B | A`B The connective (.)⊥ is extended into an involution on all formulas by: (X⊥)⊥ = X (A⊗B)⊥ = A⊥ `B⊥ (A`B)⊥ = A⊥ ⊗B⊥ For example: (X⊥ ⊗ (X ` Y ⊥))⊥ = (X⊥ ` (X ` Y ⊥)⊥)⊥ = (X ` (X⊥ ⊗ Y ⊥))⊥ = (X ` (X⊥ ⊗ Y ))⊥ = X⊥ ⊗ (X⊥ ⊗ Y )⊥ = X⊥ ⊗ (X⊥ ` Y ⊥) = X⊥ ⊗ (X ` Y ⊥) Sequents are sequences of formulas denoted ` Γ. The sequent calculus rules of MLL are: ax ` A⊥, A ` Γ, A ` A⊥,∆ cut ` Γ,∆ ` Γ, A ` ∆, B ⊗ ` Γ,∆, A⊗B ` Γ, A,B ` ` Γ, A`B The formal definition of the system requires to add the exchange rule: ` Γ ex(ρ) ` ρ(Γ) where ρ is a permutation. However the precise use of this rule and the careful usage of the order of formulas in sequents add a lot of useless technicalities to the results we want to present here. We will thus do in the following as if sequents were multisets of formulas even if this is not strictly speaking a valid way of defining them.