There was a talk using similar examples which I think is very educational.
I wrote down the examples used in the video (given the poor quality of video) to help myself learn about the Y-Combinator
Recently, I presented this topic to my team and wrote down a Python equivalent of this example.
Well, anyways, these are very interesting ideas to study and opened up some of the reasoning behind Haskell's laziness for me.
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I'd be curious to hear anyone's thoughts on the graphical notation here—I'd never seen it (or anything similar), and it's intriguing to me, but I wonder if folks have gained some specific concrete benefit from it (e.g. you can look at one of these and figure out what's happening much more quickly or something).
Here, some choice quotes from it:
"We further showed that certain variants of DOT’s recursive self types can be integrated successfully while keeping the calculus strongly normalizing. This result is surprising, as traditional recursive types are known to make a language Turing-complete."
"What is the minimum required change to achieve Turing-completeness? Consistent with our expectations from traditional models of recursive types, we demonstrate that recursive type values are enough to encode diverging computation."
"But surprisingly, with only non-recursive type values via rule (TTYP), we can still add recursive self types to the calculus and maintain strong normalization."
"Can we still do anything useful with recursive self types if the creation of proper recursive type values is prohibited? Even in this setting, recursive self types enable a certain degree of F-bounded quantification, ..."
I wish someone would try to implement Robinson arithmetic (See here: https://en.wikipedia.org/wiki/Robinson_arithmetic and here: https://ncatlab.org/nlab/show/Robinson+arithmetic ) in this. I don't think it'd work - for what I speculate as somewhat obvious reasons - but it'd seem interesting to see where exactly doing so would fail.
Aha! Now, any system with subtyping is “weakly typed”! For example, System F-sub!
Drop the "the." Just "Y combinator." It's cleaner.
Also, "the" is used to mark abstract terms, e.g. "I go on monday", but "I go on the next monday" (or just next monday).
Does that show why people care about the articles?
One explanation I found for myself why this is a problem is Normal Forms. A DB table can only have one primary key (definite). All else is secondary, n-ary or arbit-ary. The spoken language should be normalizing, too. Spoken lang... it doesn't make a difference in practice though, noone cares to be precise.