Not signed in (Sign In)

Start a new discussion

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-categories 2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry beauty bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology combinatorics complex-geometry computable-mathematics computer-science connection constructive constructive-mathematics cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry differential-topology digraphs duality education elliptic-cohomology enriched fibration finite foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck 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 infinity integration integration-theory k-theory lie lie-theory limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal-logic model model-category-theory monad monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pasting philosophy physics planar 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 simplicial space spin-geometry stable-homotopy-theory stack string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory 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.

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)