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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

  1. Link to game semantics article

    Ammar Husain

    diff, v84, current

  2. Copycat example.

    Ammar Husain

    diff, v84, current

  3. Caires, Pfenning

    Ammar Husain

    diff, v85, current

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