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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 $\lambda$-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 $S$ and $K$ you would have combinators with the following types:
$(B\to C) \to (A\to B) \to (A\to C)$ $(A \to (B\to C)) \to B \to A\to C$coming from the two ways to eliminate a dependency in $S$ to make it linear ($K$ is irreducibly nonlinear). These are of course the ways that you express composition and symmetry in a closed category.
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.
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.
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?
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.
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: http://projecteuclid.org/euclid.ndjfl/1093634655
Hirokawa, Principal types of BCK-lambda-terms, 1993: http://www.sciencedirect.com/science/article/pii/030439759390171O
Thanks! (I forgot that even in ordinary cartesian combinatory logic you can define $I=S K K$; so it’s not too surprising that the same is true affinely.)
I added a mention of BCI and BCK logic to combinatory logic.
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