> why might they be interested?

Because you get done faster.

I have no doubt any experienced C programmer could beat a k solution in run-time performance, but the k programmer will always be done faster.

> proponent give some examples of practical applications written in these array languages?

Databases are the most common: Building an API library to quickly answer questions about data is probably about the easiest thing to do in an array language, but isn't almost anything a database with a definition like that? E-commerce and storefronts, CRMs, flight trackers, and so on. In this way, I often suggest to people that the closest analogue is maybe something like SQL stored-procedures, but with a less-shit language. But this isn't exactly right.

Games are an interesting problem that aren't databases (for example), and a lot of games have vector-problems that have nice solutions in k. So let's play hangman.

If you don't know this game, we will only be playing with lowercase letters:

A:`c$"a"+!26; // A gets the characters cast [from] "a" + til 26

First, let's agree on a dictionary to use:

W:0:"/usr/share/dict/words"

and I will only choose a word with at least 7 letters in it and make it lowercase
so you have a chance:

w:_*1?W@&7<#'W; // w gets lowercase [the] first [of] 1 random-take [from the] Words [that/at] where 7< count each Words

Now when you make a guess:

G:();g:{G::G,x}

I'll now tell you what you know, but you're only allowed N guesses.

N:5
k:{@[w;&|/w=/:A^N#G;:;"."]}; // k gets a lambda, [that] amends w where the max-over [when the] word equal [to] each of-the Alphabet without [the first] N [characters] taken from the Guess

That's it. You keep guessing until you guess the word, or you run out of guesses.

g"e";k[]
"...e..e..'."
g"s";k[]
"..se..e..'s"

It's a silly game. I wrote comments above (with // characters so they should be easy to spot), and I now want you to take a look at them.

The comment is what I "read" when I am reading the k stanza. The square-brackets cover words (prepositions mostly) that help make it sound more like English, but look at the other words too: x=/:y is x equals each-of y. That's all it is. I am reading it and writing it that quickly. Yes it takes longer to write and read the English, and of course, any non-array language will be a lot more code to read and write as well.

That's what I think the value is in an array language: It's just so clear once you know how to read and write it, that you are simply faster. And being clear means the computer does what you expect (usually on the first try!) and so the programmer gets on to the next problem sooner.

I've written ad servers and presentation tools in array languages; I don't think there's anything I can't write (after all, almost everything is a database or a game if you squint hard enough), but there's a lot I wouldn't: I did an MP4 decoder in JavaScript once (to autoplay ads on iphones back when they blocked video playback in iframes) but I would never say JavaScript is the right language to write MP4 decoders. Array languages are tools in your toolbox -- like JavaScript or any other -- and you should try to put as many tools in your toolbox so that you have the best chance of using the right tool for the job.