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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeJun 8th 2017
    • (edited Jun 8th 2017)

    The cut rule for linear logic used to be stated as

    If ΓA\Gamma \vdash A and AΔA \vdash \Delta, then ΓΔ\Gamma \vdash \Delta.

    I don’t think this is general enough, so I corrected it to

    If ΓA,Φ\Gamma \vdash A, \Phi and Ψ,AΔ\Psi,A \vdash \Delta, then Ψ,ΓΔ,Φ\Psi,\Gamma \vdash \Delta,\Phi.

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 8th 2017

    No objection, but I guess it would depend on the precise rules. In MLL which has duals, if we have rules which allow us to derive

    ΓΣ,ΦΓ,Φ *Σ,Ψ,ΣΔΨΔ,Σ *\frac{\Gamma \vdash \Sigma,\Phi}{\Gamma, \Phi^\ast \vdash \Sigma}, \qquad \frac{\Psi, \Sigma \vdash \Delta}{\Psi \vdash \Delta, \Sigma^\ast}

    where Φ *\Phi^\ast is the list of duals of formulas of Φ\Phi, and A **=AA^{\ast\ast} = A, I thought we would be able to derive your more general cut rule from the more specific cut rule:

    ΓA,ΦΨ,AΔ Γ,Φ *AAΨ *,Δ Γ,Φ *Ψ *,Δ Ψ,ΓΔ,Φ \array{ \arrayopts{\rowlines{solid}} \Gamma \vdash A, \Phi\;\;\;\;\;\;\;\;\; \Psi, A \vdash \Delta \\ \Gamma, \Phi^\ast \vdash A \;\;\;\;\;\;\;\;\; A \vdash \Psi^\ast, \Delta \\ \Gamma, \Phi^\ast \vdash \Psi^\ast, \Delta \\ \Psi, \Gamma \vdash \Delta, \Phi }
    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeJun 8th 2017

    Yes, you’re right. But I think it’s better to formulate the rule in a way that remains correct in fragments without involutive negation.

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 8th 2017

    Understood (and agreed); I was just explaining the likely reason the original form was the way it was (I’m guessing I wrote that).

    • CommentRowNumber5.
    • CommentAuthorTobyBartels
    • CommentTimeSep 10th 2018

    Confirm that the game semantics presented here is the same as Blass's. (I still haven't actually read Blass 1992, yet, but I read a 1994 paper by Blass that summarizes the 1992 semantics.)

    diff, v80, current

    • CommentRowNumber6.
    • CommentAuthorBenMacAdam
    • CommentTimeMar 23rd 2019

    Added differential linear logic to variants, included a reference to the paper introducing differential categories.

    diff, v82, current

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