5. Parentheses Nesting

+\1 ¯1 0['()'⍳x]

   x←'⍵((∇<S),=S,(∇>S))⍵⌷⍨?≢⍵'
   x ⍪ ⍉⍪ +\1 ¯1 0['()'⍳x]
⍵ ( ( ∇ < S ) , = S , ( ∇ > S ) ) ⍵ ⌷ ⍨ ? ≢ ⍵
0 1 2 2 2 2 1 1 1 1 1 2 2 2 2 1 0 0 0 0 0 0 0

The phrase has an ancient and storied history. As recounted by Alan Perlis [22]:

I was at a meeting in Newcastle, England, where I’d been invited to give a talk, as had Don Knuth of Stanford, Ken Iverson from IBM, and a few others as well. I was sitting in the audience sandwiched between two very esteemed people in computer science and computing — Fritz Bauer, who runs computing in Bavaria from his headquarters in Munich, and Edsger Dijkstra, who runs computing all over the world from his headquarters in Holland.

Ken was showing some slides — and one of his slides had something on it that I was later to learn was an APL one-liner. And he tossed this off as an example of the expressiveness of the APL notation. I believe the one-liner was one of the standard ones for indicating the nesting level of the parentheses in an algebraic expression. But the one-liner was very short — ten characters, something like that — and having been involved with programming things like that for a long time and realizing that it took a reasonable amount of code to do, I looked at it and said, “My God, there must be something in this language.” Bauer, on my left, didn’t see that. What he saw or heard was Ken’s remark that APL is an extremely appropriate language for teaching algebra, and he muttered under his breath to me, in words I will never forget, “As long as I am alive, APL will never be used in Munich.” And Dijkstra, who was sitting on my other side, leaned toward Bauer and said, “Nor in Holland.” The three of us were listening to the same lecture, but we obviously heard different things.