The Y Combinator, or How to Succeed at Recursion Without Really Recursing(mvanier.livejournal.com)
mvanier.livejournal.com
The Y Combinator, or How to Succeed at Recursion Without Really Recursing
http://mvanier.livejournal.com/2897.html
19 comments
"Recursing" and "recurse" aren't words. Recursion comes from the word recur. A function recurs, not recurses. Got taught that during the infamous CS1321 Scheme course at Georgia Tech in 2001.
They are now. Words become a part of the language because enough people use them.
Math envy.
Why do hackers focus on it so much? It has only very remote relevance to normal programming.
The beautiful idea I see behind the Y combinator is that of formalism. Once you describe something in mathematical terms, magic can happen. But this would be much better illustrated by a good introduction to analysis or algebra. Even number theory has some cool results, and it's easier to grasp.
In my second course at university, the teacher proved by contradiction that 1 >= 0:
(that proof above is not really satisfactory, especially without the context and the axioms, but it served its purpose well five years ago)
Why do hackers focus on it so much? It has only very remote relevance to normal programming.
The beautiful idea I see behind the Y combinator is that of formalism. Once you describe something in mathematical terms, magic can happen. But this would be much better illustrated by a good introduction to analysis or algebra. Even number theory has some cool results, and it's easier to grasp.
In my second course at university, the teacher proved by contradiction that 1 >= 0:
Assume 1 is negative, 1 < 0
Multiply both sides by 1, so that 1*1 > 0*1 (multiplication by a negative flips the inequality)
1 being an identity, this implies 1 > 0, contradiction
(that proof above is not really satisfactory, especially without the context and the axioms, but it served its purpose well five years ago)
1. Conceptual elegance. It seems to me that recursion and the complexity that emerges from uses it fascinates many people. GEB comes to mind.
2. Directly dealing with these mathematical formalisms (the Y combinator, SKI combinators (http://en.wikipedia.org/wiki/SKI_combinator_calculus), etc.) can become really important in programs such as interpreters.
2. Directly dealing with these mathematical formalisms (the Y combinator, SKI combinators (http://en.wikipedia.org/wiki/SKI_combinator_calculus), etc.) can become really important in programs such as interpreters.
1. I was writing and rewriting a rebuttal, but you're right... There is something cool about achieving self-reference without a name.
2. Really? If you feel like elaborating / providing link, I'd appreciate.
2. Really? If you feel like elaborating / providing link, I'd appreciate.
For example, recently I wrote a simple interpreter for a stack-based / postfix language kind of like Forth and Joy (http://en.wikipedia.org/wiki/Joy_programming_language) because I'm trying to get my mind around that language family* . It's still very basic and doesn't have much more built into it than the core Lisp PG discusses in The Roots of Lisp does (http://paulgraham.com/rootsoflisp.html), but I really noticed how the specific interplay of those sorts of forms end up setting much of the subsequent language. Lambda calculus, at least the passing acquaintance I have with it, has given me a conceptual vocabulary with which to think about evaluation, scoping, and which aspects of a language are redundant, can themselves be used to derive others, etc. I guess #2 could have been better said, "Understanding these mathematical formalisms ... can be really helpful when ...".
As a more concrete example, I originally had words (functions, so to speak) defined Forth-style, e.g.
By the way, if you'd like to write a quick interpreter as a learning experience, Forth has my recommendation. It's small and (aggressively) simple, but still supports an impressive amount of abstraction. The parser for it is even simpler than for a Lisp, and that's really saying something. (Here is a nice example of one written in heavily documented i386 assembly: http://www.annexia.org/_file/jonesforth.s.txt )
Second, OCaml seems to be particularly well-suited for writing interpreters (and compilers, I'm sure), though in this case it was serious overkill.
* I'm kind of whining here, but AFAICT it isn't possible to install a working version of Factor (http://factorcode.org/) on OpenBSD (or, probably, any Unix) without downgrading to a version of the OS using the same dynamic libraries, installing their binary version, and then progressively updating both in tandem until it's running with current libraries. Maybe I just don't have enough experience bootstrapping new language environments, but that was a disappointment -- it looks like a cool language.
(Also, 1 isn't really an argument so much as an observation.)
As a more concrete example, I originally had words (functions, so to speak) defined Forth-style, e.g.
: avg + 2 / ;
[+ 2 /] "avg" def
By the way, if you'd like to write a quick interpreter as a learning experience, Forth has my recommendation. It's small and (aggressively) simple, but still supports an impressive amount of abstraction. The parser for it is even simpler than for a Lisp, and that's really saying something. (Here is a nice example of one written in heavily documented i386 assembly: http://www.annexia.org/_file/jonesforth.s.txt )
Second, OCaml seems to be particularly well-suited for writing interpreters (and compilers, I'm sure), though in this case it was serious overkill.
* I'm kind of whining here, but AFAICT it isn't possible to install a working version of Factor (http://factorcode.org/) on OpenBSD (or, probably, any Unix) without downgrading to a version of the OS using the same dynamic libraries, installing their binary version, and then progressively updating both in tandem until it's running with current libraries. Maybe I just don't have enough experience bootstrapping new language environments, but that was a disappointment -- it looks like a cool language.
(Also, 1 isn't really an argument so much as an observation.)
Thanks a lot.
One more --
Also see chapter 22 in Shriram Krishanamurthi's _Programming Languages: Application and Interpretation_. (http://www.cs.brown.edu/~sk/Publications/Books/ProgLangs/200...)
The goal of the chapter is essentially, "Now that we've built our own small programming language on top of Scheme, how can we reduce the number of Scheme elements that it relies on directly?" Among other things, he shows how to use the Y combinator to allow recursive function definitions without having to use Scheme's define.
Also see chapter 22 in Shriram Krishanamurthi's _Programming Languages: Application and Interpretation_. (http://www.cs.brown.edu/~sk/Publications/Books/ProgLangs/200...)
The goal of the chapter is essentially, "Now that we've built our own small programming language on top of Scheme, how can we reduce the number of Scheme elements that it relies on directly?" Among other things, he shows how to use the Y combinator to allow recursive function definitions without having to use Scheme's define.
Because describing something formally means you have a deep understanding of it. Formal descriptions require synthesizing the important concepts from the whole.
Understanding a new concept feels good intellectually, but it also gives you more power: now you can recognize that pattern elsewhere and perhaps apply new techniques.
Understanding a new concept feels good intellectually, but it also gives you more power: now you can recognize that pattern elsewhere and perhaps apply new techniques.
God, if I see another code example using Factorial I am going to hurl.
What are some other simple examples of recursion, though?
Quicksort would be pretty good for this, but the straightforward recursive version has several edge cases that lead to bad performance.
Getting the length of a linked list is probably a better example:
Any other ideas?
Quicksort would be pretty good for this, but the straightforward recursive version has several edge cases that lead to bad performance.
Getting the length of a linked list is probably a better example:
(* ocaml *)
let rec length l = match l with
| [] -> 0
| h::t -> 1 + length t
; scheme
(define (length l)
(if (null? l)
0
(+ 1 (len (cdr l))))
Any other ideas?
[deleted]
Does performance matter when seeking "simple" examples?
I vote for quicksort - not because it's quick, but because its implementation has the same form as its proof. It's simple if you use the Erlang-style version.
I vote for quicksort - not because it's quick, but because its implementation has the same form as its proof. It's simple if you use the Erlang-style version.
Factorial is the hello world of functional programming. The fact that everyone knows it inside out aids the example, because the reader can focus on the important details (the Y Combinator).
Agreed, but in a parallel universe where the parent post actually asked the implied question, what other functions would you suggest? I try to explain these things to other people enough that I'd love to hear other suggestions.
Map might be another good one, although for better and worse it expects the reader to follow the idea of first class functions. Something like
Map might be another good one, although for better and worse it expects the reader to follow the idea of first class functions. Something like
map square [1, 2, 3, 4, 5] ---> [1, 4, 9, 16, 25]
Agreed, map is a good example only if the person reading it gets first-class functions.
I can only think of a couple other suggestions for simple recursive functions. Euclid's algorithm for GCD would be a good one. Fibonacci numbers are easily expressed with recursion, but the most obvious solution is unfortunately not tail recursive and inefficient. My favorite example of mutual recursion is "even" and "odd":
even 0 = True even x = odd (x - 1)
odd 0 = False odd x = even (x - 1)
Inefficient, of course, but easy to prove correctness even to someone who's brain hasn't yet clicked into understanding recursion. Length of a list is as boring as factorials, but I guess that would be one too. Same with generating a list of counting numbers.
I can only think of a couple other suggestions for simple recursive functions. Euclid's algorithm for GCD would be a good one. Fibonacci numbers are easily expressed with recursion, but the most obvious solution is unfortunately not tail recursive and inefficient. My favorite example of mutual recursion is "even" and "odd":
even 0 = True even x = odd (x - 1)
odd 0 = False odd x = even (x - 1)
Inefficient, of course, but easy to prove correctness even to someone who's brain hasn't yet clicked into understanding recursion. Length of a list is as boring as factorials, but I guess that would be one too. Same with generating a list of counting numbers.
I wonder if not getting first class functions is a consequence of already knowing too much, though. If you're thinking in C, you might get bogged down in thinking about passing around function pointers, but aside from implementation details, first class functions are not intimidating when the functions applied are familiar: "Take this list and do X to everything in it: square each number in a list of numbers, or convert every word to lowercase in a string list, or negate every bool in a bool list, etc.".
Euclid's algorithm is a really good example, by the way. Thanks.
Euclid's algorithm is a really good example, by the way. Thanks.