Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeJul 11th 2016

    I rescued combinatory logic from being a “my first slide” spam and gave it some content, mainly to record the fact (which I just learned) that under propositions as types, combinatory logic corresponds to a Hilbert system.

    I feel like there should be something semantic to say here too, like λ\lambda-calculus corresponding to a “closed, unital, cartesian multicategory” (a cartesian multicategory that is “closed and unital” as in the second example here) and combinatory logic corresponding to a closed category that is also “cartesian” in some sense. Has anyone defined such a sense?

    Relatedly, is there a notion of “linear combinatory logic” that would correspond to ordinary (symmetric) closed categories? My best guess is that instead of SS and KK you would have combinators with the following types:

    (BC)(AB)(AC) (B\to C) \to (A\to B) \to (A\to C) (A(BC))BAC (A \to (B\to C)) \to B \to A\to C

    coming from the two ways to eliminate a dependency in SS to make it linear (KK is irreducibly nonlinear). These are of course the ways that you express composition and symmetry in a closed category.

    • CommentRowNumber2.
    • CommentAuthorUlrik
    • CommentTimeJul 11th 2016

    Exactly, linear combinatory logic is BCI, where B is your first, and C is your second (and you need I for the identity as well). Affine combinatory logic is BCK.

    I don’t know about your “cartesian” closed categories question.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJul 12th 2016

    These “my first slide” pages are not spam, but the result of people clicking on the link “Make this page an S5 slideshow”, which is the most prominently placed link on the edit page. It is our fault, or that of the software anyway, that this keeps happening.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeJul 12th 2016

    Thanks Ulrik! Can you give a reference? I tried searching the Internet for “linear combinatory logic” but that didn’t seem to be a useful search term.

    Also, wouldn’t affine combinatory logic be BCKI? Or can you somehow deduce I from B, C, and K?

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 12th 2016

    The BCK combinators support affine bracket abstraction: [x]t where x occurs at most once in t.

    We can define I ≡ CKK.

    Slide 6 of Abramsky’s slides.

    • CommentRowNumber6.
    • CommentAuthorUlrik
    • CommentTimeJul 12th 2016

    The search terms seem to be BCI and BCK logic. I said “linear combinatory logic”, because that’s what you and I would understand immediately. (-:

    I guess the main results for BCI and BCK are condensed detachment (Hindley, 1993) and principal typings (Hirokawa, also 1993). I’m not that familiar with the area.

    Hindley, BCK and BCI logics, condensed detachment and the 2-property, 1993:

    Hirokawa, Principal types of BCK-lambda-terms, 1993:

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeJul 12th 2016

    Thanks! (I forgot that even in ordinary cartesian combinatory logic you can define I=SKKI=S K K; so it’s not too surprising that the same is true affinely.)

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeJul 13th 2016

    I added a mention of BCI and BCK logic to combinatory logic.

  1. the logic combinator S requires parenthesis as shown


    diff, v10, current

  2. the logic combinator S requires parenthesis as shown


    diff, v11, current

    • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 28th 2021

    Added pdf link for Freyd’s Combinators

    diff, v12, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeFeb 28th 2023

    I fixed and completed the bibdata for this item:

    diff, v13, current