Curry[edit]

BCKW System[edit]

The Pathetic History[edit]

~~Delete.~~ I don't see reasons above, but it looks like original research and is not very encyclopedic. It also fails to explain just what this system is. =b Fennec 05:22, 26 Mar 2004 (UTC)
- Clean up. Seriously. Fennec 13:49, 26 Mar 2004 (UTC)
Was originally listed by User:Ihcoyc, don't know the reasons, no comment at this time. - Hephaestos|§ 05:25, 26 Mar 2004 (UTC)
I can't seem to find anything about this on google, but I don't actually know anything about combinatorics, so I will abstain. It certainly seems to be idiosyncratic. -- Friedo 06:12, 26 Mar 2004 (UTC)
It appears to be genuine, but I don't know if it's original or not. Dysprosia 09:15, 26 Mar 2004 (UTC)
Keep. This looks credible to me, and the works quoted are well-known references in the field. See Talk:B,C,K,W System for (many) suggestions for improvement, but none of them are VfD matters. Andrewa 10:09, 26 Mar 2004 (UTC)
I tried to list this, but was stymied by the new format. At any rate, it seems to be: 1.) about some guy's thesis; 2.) so lacking in context as to be completely unintelligible; and 3.) full of dubious looking "equations" with terms like BIIIIIII. Smerdis of Tlön 12:49, 26 Mar 2004 (UTC)~
- Hmmm. 1.) It's not original research, see the talk page. 2.) and 3.) Is this just that you're not familiar with similar topics? Some quite valid treatments (not ours) of the halting problem look very like this, for example. Andrewa 23:28, 26 Mar 2004 (UTC)
- You get a lot of things that look like that in logic-related math work... Based on the discussion that has gone on at the talk page, I say keep now. Dysprosia 23:33, 26 Mar 2004 (UTC)
  - It strikes me that, at least as the article is currently written, only those who already know what it is talking about can make heads or tails of it. They don't need the information it contains; and for those who lack that background, it is gibberish. Smerdis of Tlön 01:52, 27 Mar 2004 (UTC)
    - Hmmmm. Good points. I'd agree that to those who lack sufficient background it would look like gibberish, and I guess you're saying you are a case of this. But that's true of any written material, even writing itself. I don't agree it's useless to the point that anyone who can read it already knows it all, and I'd be counter example to this myself. And I certainly agree there is lots of improvement possible, see the talk page. That's all part of the project, and not a reason for deletion. Andrewa 02:59, 27 Mar 2004 (UTC)
  - FWIW, I will withdraw this request for deletion. I am persuaded at least that it is not idiosyncratic, original research, or vanity; and that others recognise the subject itself as valid. While all by its ownself, the text on the page is pretty hopeless, that text may not be utterly useless to someone who wants to work it into a semi-adequate article. I suspect that the table of "Axioms" might still belong on Wikisource rather than here; but they have an intriguing Finnegans Wake quality to them in any case. Smerdis of Tlön 04:54, 27 Mar 2004 (UTC)

The Article[edit]

Haskell Curry, in his doctoral thesis Grundlagen to der kombinatorischen Logik [GKL], already proposed a system with separated functional characteristics: association, conversion, cancellation and duplication. If in addition we request regular, proper (and between these, minimals) combinators they are, B, C, K and W (today nomenclature). As it is difficult to have the original system of combinatorial axioms we reproduce here the version given by Rosenbloom in The Elements of Mathematical Logic, where he uses application prefix which we change into usual infix notation and, in the context to recover [GKL], leave I without defining it: so, beware!.

Axioms[edit]

1) BI = I
2) C(BB(BBB))B = B(BB)B
3) C(BB(BBB))C = B(BC)(BBB)
4) C(BBB)W = B(BW)(BBB)
5) C(BBB)K = B(BK)I
6) CBI = I
7) B(B(BC)C)(BB) = BBC

8) B(B(B(B(BW)W)(BC)))(BB)(BB) = BBW
- error [EML]?
  - 8) B(B(B(B(BW)W)(BC)))B(BB)B = BBW

9) BBK =BKK
10) BCC = I
11) B(B(BC)C)(BC) = B(BC(BC))C
12) B(B(BW)C)(BC) = BCW
13) BCK = BK
14) BWC = W
15) BW(BW) = BWW
16) BWK = I

Rules[edit]

We asume the rules of the equality.

Combinatorial ones are presented like equations:

B x y z = x (y z)

C x y z = x z y

K x y = x

W x y = x y y

Works[edit]

[GKL] Curry, Haskell B.; Grundlagen to der kombinatorischen Logik; Amer. J. Math.; 52:509-536;789-834 (1930)
[EML] Rosenbloom, Paul C.; The Elements of Mathematical Logic, Dover 1950;

The Talk[edit]

Was this machine-translated from Spanish? -- Karada 12:25, 25 Mar 2004 (UTC)

the stupid part, yes!

The references quoted are well-known, the English one [EML] especially should be in many university libraries, if anyone wants to check the accuracy and preferably rewrite in better English. You'd need to be competent in ~~both combinatorics and~~ polish notation.

Strike out combinatorics as noted below, and add a strong recommendation for either mathematical logic or similar fields. I would think that if you have ever tried to read Frege's Begriffsschrift in the original notation (not necessarily the German!) and succeeded you could try, otherwise it's going to be tough going, as you probably either lack the interest (if you've never tried) or expertise (if you didn't succeed). But see below. Andrewa 18:38, 26 Mar 2004 (UTC)

The conversion of the [EML] notation from polish to standard grouping is mechanical both ways, and not original research. It's the way any undergrad course would be taught.

The quip about I (I assume that's an identity element) being undefined is unfortunate but I'm not confident I can restate it accurately off the top of my head.

An intro setting the context for lay readers would also be good. None of the weaknesses of this article are at all unusual among our math articles unfortunately, although this is one of the worst I've seen. Andrewa 10:18, 26 Mar 2004 (UTC)

"this is one of the worst I've seen" thanks, but you are probably wright.
- Sorry if it's blunt. Note the modality. It's not a very strong statement, it just looks like one. Andrewa

The expertise is in combinatory logic not (not mandatory) in combinatorics.
- Good point, see above. Andrewa

As there are serious problems with combinatory logic see Talk:Combinatory logic, I'm doing the articles that will be needed, in casual order. IF ONLY EXPERTS MAKE CORRECTIONS, finishing will be easier!
- No need to shout, see below. Andrewa

It is an standard problem in Combinatory logic misscounting of I, which is, effectively, the identity function or combinator: I x = x
- And needs a better explanation in the article. A priority I think. Andrewa

Remenber that Combinatory logic is either polish notation (unusual as [EML]) or Left associative!. Non experts can't follows the expressions!! (Sorry!)
- It's a long-standing principle of Wikipedia to write about what you know about or are prepared to learn. There seems to be a suggestion here of defending your turf, and I'd strongly recommend against this. By all means, discourage people from naive edits to the article. But try to be welcoming to all on the talk pages is my strong advice, and especially those who are prepared to learn.
- In particular, the most valuable articles are those that are readable by the most readers. Having a non-expert edit the article is by far the fastest and most efficient way of achieving this, so long as inaccuracies are not introduced. This sort of collaboration is just what a Wiki does well. Explore it and encourage it.
- Lastly, if your article ends up inaccessible to all but the expert, it is then very vulnerable to later blundering edits by non-experts who think they are, which others will not pick up. You can't be everywhere forever. The goal should be an article that the whole community can look after, or at least as big a proper subset of it as possible. Andrewa 19:14, 26 Mar 2004 (UTC)

First and last generosity:
- B x y z = x (y z)
  - is (((B x) y) z) = (x (y z)) but its sense is B(x)(y)(z) = x(y(z)) in math notation (see Currying)

- C x y z = x z y
  - (((C x) y) z) = ((x z) y) i.e. C (x) (y) (z) = x (z) (y)

- K x y = x
  - ((K x) y) = x i.e. K (x) (y) = x (quite easy: K makes constant functions, one for each x)

- W x y = x y y
  - (((W x) y) = ((x y) y) i.e. W (x) (y) = x (y) (y)

- with this take, say, 14) (very easy)
  - 14) BWC = W is
    - 14) ((BW)C) = W and means W o C = W (o functional composition) and this say: "Is useless to permute (C) after a perfect duplication (W)"

And, all this is 70 years old!.

And, "once upon a time" the article WAS orphan, you know?!!

You can name the combinators in any order, I use BCKW as reference to BCK-Algebras, a usual, but very indirect, relation. see Curry-Howard isomorphism. So BCKW-Algebra means Hilbert Algebra. see implication algebras.

DefLog 18:33, 26 Mar 2004 (UTC)

Pure combinatory logic[edit]

Most presentations of combinatory logic are versions of lambda calculus, perhaps because absolute elimination of variables is difficult to think of, but this

User:DefLog~enwiki/Sandbox