Another example is juggling notation, that allowed not only the sharing of patterns but the discovery of new ones [2].
Its modern descendent are https://en.wikipedia.org/wiki/J_(programming_language) & https://en.wikipedia.org/wiki/K_(programming_language).
There's also bqn[0], among others.
Have you seen the version used by dzaima/apl[1]? The equivalent of '(-&.:{:) i.5' works and results in 0 1 2 3 _4.
> APL has some really dumb but cherished-by-the-community ideas that make the language less expressive and much more awkward to learn, e.g. the idea of replicating the terrible defect of normal mathematical notation where - is overloaded for negation and subtraction to every other function
Klong[2] is a partial attempt to resolve this. I won't repeat the arguments in favour of ambivalent functions, as I guess you've heard them a dozen times before
> u(f,g) = x => f^-1(g(f(x)).
Other way round; it's g^-1(f(g(x)))
Tricky.
[0] https://www.numberphile.com/videos/juggling-by-numbers
[1] As far as we know. Without the notation we don't actually know what had been done, but when I took the new patterns to juggling conventions, no one knew them[2].
[2] Actually it's stronger than that. I showed people some of the new patterns at the British Juggling Convention in 1985 and no one knew them. Then at the European Juggling Convention just 4 months later, people from the USA were proclaiming them as the latest patterns that they had just learned, and were perplexed at how I not only knew them, but knew many, many more.
[3] And actually Paul Klimek had beaten us to it, but hadn't been able to get others interested in the notation. As far as we can tell, Paul was the first to get the notation.
Yes. Many built-in words have inverses assigned, and you can assign inverse functions to your own words with :. https://code.jsoftware.com/wiki/Vocabulary/codot
EDIT: and here's a table with predefined inverses: https://code.jsoftware.com/wiki/Vocabulary/Inverses
In that case you'll be wanting ngn/apl[1], which runs in a browser and compiles to js.
> ambivalent
The arguments are mostly linguistic. Natural language is also context-sensitive, so we are well-equipped to parse such formations; and they allow us to reuse information. The monadic and dyadic forms of '-' are related, so it's less cognitive overhead to recognize its meaning.
J for C programmers is a good book - https://www.jsoftware.com/docs/help807/jforc/contents.htm - and it could take significantly less than many weeks to "get" some important ideas. Specifically, make sure you understand ranks at chapters 5 and 6.
After you understand how +/ with different ranks can sum along different axis, you're well on the way.
I mean, here is a cube of numbers:
i. 2 3 4
0 1 2 3
4 5 6 7
8 9 10 11
12 13 14 15
16 17 18 19
20 21 22 23
+/ i. 2 3 4
12 14 16 18
20 22 24 26
28 30 32 34
0 1 2 3
4 5 6 7
8 9 10 11
12 13 14 15
16 17 18 19
20 21 22 23
If you specify +/"0 - this sets the rank of the verb (function) to 0 - then +/ will be applied to each number separately and results will be combined. Adding a single number (not with itself - just as it is, without the other argument for summation) makes the same number, so +/"0 doesn't change the result - it's the same cube as in i. 2 3 4
Trying with +/"1 gives
+/"1 i. 2 3 4
6 22 38
54 70 86
Finally,
+/"2 i. 2 3 4
12 15 18 21
48 51 54 57
The book tells it better, of course.
dzaima/APL being written in Java means getting it to run in a browser would be a bit hard, and ngn has given up on ngn/apl, but BQN[0] could definitely get a web canvas based graphics interface.
Somewhat interesting to add to the conversation about Under is that, in my impl, calling a function, calling its inverse, or doing something under it (i.e. structural under) are all equally valid ways to "use" a function, it's just a "coincidence" that there's direct syntax for invoking only one. (Dyalog does not yet have under, but it definitely is planned.)