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Proofnets and Context Semantics for the Additives
"... We provide a context semantics for MultiplicativeAdditive Linear Logic (MALL), together with proofnets whose reduction preserves semantics, where proofnet reduction is equated with cutelimination on MALL sequents. The results extend the program of Gonthier, Abadi, and Lvy, who provided a "geom ..."
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We provide a context semantics for MultiplicativeAdditive Linear Logic (MALL), together with proofnets whose reduction preserves semantics, where proofnet reduction is equated with cutelimination on MALL sequents. The results extend the program of Gonthier, Abadi, and Lvy, who provided a "geometry of optimal hreduction" (context semantics) for hcalculus and MultiplicativeExponential Linear Logic (MELL). We integrate three features: a semantics that uses buses to implement slicing; a proofnet technology that allows multidimensional boxes and generalized garbage, preserving the linearity of additive reduction; and finally, a readback procedure that computes a cutfree proof from the semantics, which is closely related to full abstraction theorems.
Callbyvalue isn’t dual to callbyname, callbyname is dual to callbyvalue!
, 2004
"... Gentzen’s sequent calculus for classical logic shows great symmetry: for example, the rule introducing ∧ on the left of a sequent is mirror symmetric to the introduction rule for the dual operator ∨ on the right of a sequent. A consequence of this casual observation is that when Γ ⊢ ∆ is a theorem o ..."
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Gentzen’s sequent calculus for classical logic shows great symmetry: for example, the rule introducing ∧ on the left of a sequent is mirror symmetric to the introduction rule for the dual operator ∨ on the right of a sequent. A consequence of this casual observation is that when Γ ⊢ ∆ is a theorem over operators {∨,∧,¬}, then so is ∆◦ ⊢ Γ◦, where Σ◦ reverses the order of formulas in Σ, and exchanges each instance of A ∧ B (C ∨ D) in a formula of Σ with B ∨ A (D ∧ C). This symmetry of rules means that the duality of theorems is also a duality of proof structures.