TweetFollow Us on Twitter

Daze Y
Volume Number:8
Issue Number:1
Column Tag:Lisp Listener

Deriving Miss Daze Y

Deriving the (applicative order) Y combinator in a concrete way via fact

By André van Meulebrouck, Chatsworth, California: Internet: vanMeule@cup.portal.com

“Deriving Miss Daze Y”

“The utmost abstractions are the true weapons with which to control our thought of concrete fact.” - Alfred North Whitehead

This article will seek to derive the (applicative order) Y combinator in a concrete way via fact.

Definition: The Y combinator is a function which, when applied to the abstracted version of a recursive function, is the equivalent of the (original) recursive function. [vanMeule May 1991]

Definition: fact is that pedagogical function of obsessive interest, the factorial function.

Abstracted fact

In [vanMeule May 1991], the Y combinator was motivated by a desire to convert everything into a combinator (a lambda expression which has no free variables). In “combinatorizing” everything we found the following definition in need of abstrac-tion (the process whereby we get rid of free variables by making them bound in an outer lambda expression, then promising to pass in “the right thing” when invoking the outer lambda expression).

(* 1 *)

(define fact
 (lambda (n)
 (if (zero? n)
 1
 (* n (fact (1- n))))))

In the definition of fact above, the variable is a free variable. (Such recursive definitions rely on free variables being resolved in an odd, not-purely-lexical way.) The definition for abstracted-fact looks like the following.

(* 2 *)

(define abstracted-fact
 (lambda (fact)
 (lambda (n)
 (if (zero? n)
 1
 (* n (fact (1- n)))))))

The free variable is gone, but we are not home and dry be-cause we now have to pass in the definition of fact. In fact, we have to have a mechanism that is capable of providing them on demand!

Recursionless fact

In [vanMeule Jun 1991], what is perhaps the simplest trick for getting rid of recursion was shown: passing the would be recursive function as an argument!

(* 3 *)

>>>
(define pass-fact
 (lambda (f n)
 (if (zero? n)
 1
 (* n (f f (1- n))))))
pass-fact
>>> (pass-fact pass-fact 5)
120

Notice what happened to the original definition of fact: it was changed! In abstracted-fact, we did not change the definition at all - we merely wrapped a lambda form around the untampered-with-definition of fact.

Merging facts

What we really want is a way to get rid of recursion without modifying the definition of the function we’re ridding the recursion from. In other words, we want to have the best of the two different approaches: abstracted-fact gets rid of the free variable yet keeps the definition intact; pass-fact seems to have captured a recursive mechanism without using recursion.

Theoretically, it should be possible to start from pass-fact and massage it into two parts; a “recursionless recursion mechanism” (the Y combinator), and abstracted-fact.

To Be or to Let it Be

In the discussion that follows, we will use let, which hasn’t been “properly” introduced yet. So, let’s take a look at let via the following example.

(* 4 *)

(* (+ 3 2)
 (+ 3 2))

The expression (+ 3 2) is being recomputed. Alternatively, we can compute the value of (+ 3 2) once, and hold onto the result via let.

(* 5 *)

(let ((three-plus-two (+ 3 2)))
 (* three-plus-two three-plus-two))

While the main motivation behind let is to avoid recomp-utations, it can be used purely for the sake of readability (i.e. even if the value being leted will only be used once).

Our use of let herein will be purely syntactic sugar for a more (syntactically) cumbersome looking (but semantically equivalent) lambda expression. For instance, our example would look like the following if we were to use lambda instead of let.

(* 6 *)

(lambda (three-plus-two)
 (* three-plus-two three-plus-two))
 (+ 3 2))

[Rees et al. 1986] gives a more rigorous and precise Scheme definition for let.

Getting the facts straight

In the style of [Gabriel 1988], let’s start with pass-fact and try to massage it into what we want.

Since one of the rules of our “minimalist” game [vanMeule Jun 1991] was to stick to combinators and l-calculus, we are compelled to curry (a requirement of l-calculus). Also, since there are cases where currying gains expressive power that we would otherwise have to simulate, it seems natural to curry as the first step.

(* 7 *)

>>>
(define pass-fact
 (lambda (f)
 (lambda (n)
 (if (zero? n)
 1
 (* n ((f f) (1- n)))))))
pass-fact
>>>
((pass-fact pass-fact) 5)
120

Notice how the invocation of the new version of fact is more complicated than the recursive 
version. That can be fixed by tucking the invocation, which passes the function as an argument, 
inside the new definition of fact.

(* 8 *)

>>>
(define fact
 (let ((g (lambda (f)
 (lambda (n)
 (if (zero? n)
 1
 (* n ((f f) (1- n))))))))
 (g g)))
fact
>>>
(fact 5)
120

(Note that we would have looked like the Department of Redundancy Department had we not curried - parameter n would have to be bound twice.)

(* 9 *)

(define redundant-fact
 (lambda (n)
 (let ((g (lambda (f n)
 (if (zero? n)
 1
 (* n (f f (1- n)))))))
 (g g n))))

Recalling that our game plan was to separate out abstracted-fact and Y from pass-fact, it would be interesting to see how close the definitional part of fact (the part that has the if) now is to abstracted-fact.

(* 10 *)

(lambda (n)
 (if (zero? n)
 1
 (* n ( (ff) (1-n)))))

As can be seen above, we’re actually quite close already! If the (f f) part in the box were abstracted out we’d be there!

(* 11 *)

(lambda (F)
 (lambda (n)
 (if (zero? n)
 1
 (* n (F (1- n))))))

(Note: the name of the parameters in the above expression are not significant because there are no free variables in the expression. For instance, parameter F could be renamed to fact or any other name we want other than n.)

After abstracting out the (f f) part and invoking it on the argument it needs, we have the following.

(* 12 *)

>>>
(define fact
 (let ((g (lambda (f)
 (lambda (n)
 ((lambda (func)
 (if (zero? n)
 1
 (* n (func (1- n)))))
 (f f))))))
 (g g)))
fact
>>> (fact 5)
120

(Question for the Überprogrammer: Why couldn’t we do the abstraction and invocation as in the following?)

(* 13 *)

(define dont-try-this-at-home-fact
 (let ((g (lambda (f)
 ((lambda (func)
 (lambda (n)
 (if (zero? n)
 1
 (* n (func (1- n))))))
 (f f)))))
 (g g)))

Now, massage the definitional part of fact some more so that it looks just like abstracted-fact.

(* 14 *)

>>>
(define fact
 (let ((g (lambda (f)
 (lambda (n)
 (((lambda (func)
 (lambda (n)
 (if (zero? n)
 1
 (* n (func (1- n))))))
 (f f))
 n)))))
 (g g)))
fact
>>> (fact 5)
120

Using a gratuitous let, we can pull out the definition of abstracted-fact and name it locally.

(*  15 *)

>>>
(define fact
 (let ((abstracted-fact
 (lambda (f)
 (lambda (n)
 (if (zero? n)
 1
 (* n (f (1- n))))))))
 (let ((g (lambda (f)
 (lambda (n)
 ((abstracted-fact (f f)) n)))))
 (g g))))
fact
>>> (fact 5)
120

Notice that in doing the above, a free variable was introduced into g in the second let. (abstracted-fact is free with respect to g and bound with respect to the outermost let.) We can fix this by abstracting out abstracted-fact from the innermost let.

(*  16 *)

>>>
(define fact
 (let ((abstracted-fact
 (lambda (f)
 (lambda (n)
 (if (zero? n)
 1
 (* n (f (1- n))))))))
 ((lambda (abstracted-function)
 (let ((g (lambda (f)
 (lambda (n)
 ((abstracted-function (f f)) n)))))
 (g g)))
 abstracted-fact)))
fact
>>> (fact 5)
120

Y is now ready to leave the nest and flY!

Notice that the last tweak to fact achieved our aim: we now have abstracted-fact totally separated out from the recursionless recursion mechanism.

We can now name the recursion mechanism and make it a function in its own right.

(*  17 *)

>>>
(define y
 (lambda (abstracted-function)
 (let ((g (lambda (f)
 (lambda (arg)
 ((abstracted-function (f f)) arg)))))
 (g g))))
y

Question: Is Y a general purpose recursion removal function? (i.e., will it remove the recursion in any arbitrary function?) Herein, I will simply claim that it is and refer the reader to [Gabriel 1988] and/or any of the many other references that address this question (some of which are are listed in [vanMeule May 1991, Jun 1991]).

Now that we’ve got Y, we can clean up the definition of fact.

(*  18 *)

>>>
(define fact
 (let ((abstracted-fact
 (lambda (f)
 (lambda (n)
 (if (zero? n)
 1
 (* n (f (1- n))))))))
 (y abstracted-fact)))
fact
>>> (fact 5)
120

We can clean up further by getting rid of the gratuitous let.

(* 19 *)

>>>
(define fact
 (y (lambda (f)
 (lambda (n)
 (if (zero? n)
 1
 (* n (f (1- n))))))))
fact
>>> (fact 5)
120

Looking ahead

There’s a type of recursion that our (applicative order) version of Y is not designed to handle. Consider the following functions.

!codeexamplestart!

(* 20 *)

>>>
(define my-even?
 (lambda (n)
 (if (zero? n)
 #t
 (my-odd? (1- n)))))
my-even?
>>>
(define my-odd?
 (lambda (n)
 (if (zero? n)
 #f
 (my-even? (1- n)))))
my-odd?
>>> (my-odd? 5)
#t
>>> (my-even? 5)
#f

These functions need to know about each other: they are mutually recursive.

We can handle this problem by coming up with a new version of Y (let’s call it Y2). Y wants one function as an argument. What happens if instead of a function, a list of functions is passed in? Such a list could contain all the functions which need to have (mutual) knowledge of each other. Accessing the different functions can then be done by using list accessing primitives. (This is the approach used to resolve the problem in l-calculus.)

Exercise for the Überprogrammer: Derive Y2. Hint for a possible game plan: starting with my-even? and my-odd? expressed via a letrec, get rid of the letrec by converting to a let and making use of a dynamic list. Then, thrash out Y2 in a similar manner as was done for Y in this article. Does Y2 turn out to be the same as Y?

Question for the Überprogrammer: if evaluation were normal order rather than applicative order, could we use the same version of Y for mutually recursive functions that we used for “regular” recursive functions (thus making a Y2 function unnecessary)?

Another question: Let’s say we have 3 or more functions which are mutually recursive. What do we need to handle this situation when evaluation is applicative order? What about in normal order?

Note: [vanMeule Jun 1991] gave enough primitives to create dynamic lists. For example, the combinator equivalent of the Scheme expression: (list ’a ’b ’c) could be built like this:

(* 21 *)

(com-cons ’a (com-cons ’b (com-cons ’c com-nil))) ,

and this same idea could be used in conjuring up a combinator version of Scheme’s list function, which could be called com-list.

“Thanks” to:

The hummingbird nest in a nearby tree which afforded much enjoyment in watching two new pilots grow up and get their wings. Bugs/infelicities due to Spring in the air.

Bibliography and References

[Gabriel 1988] Richard P. Gabriel. "The Why of Y." LISP Pointers, vol. 2, no. 2 October/November/December, 1988.

[Rees et al. 1986] Jonathan Rees and William Clinger (editors). Revised3 Report on the Algorithmic Language Scheme; AI Memo 848a. MIT Artificial Intelligence Laboratory, Cambridge, Massachusetts, USA, September 1986.

[vanMuele May 1991] André van Meulebrouck. "A Calculus for the Algebraic-like Manipulation of Computer Code" (Lambda Calculus). MacTutor, May 1991.

[vanMuele Jun 1991] André van Meulebrouck. "Going Back to Church" (Church numerals). MacTutor, June 1991.

All examples in this article were implemented in MacScheme.

 

Community Search:
MacTech Search:

Software Updates via MacUpdate

Latest Forum Discussions

See All

Top Mobile Game Discounts
Every day, we pick out a curated list of the best mobile discounts on the App Store and post them here. This list won't be comprehensive, but it every game on it is recommended. Feel free to check out the coverage we did on them in the links... | Read more »
Price of Glory unleashes its 1.4 Alpha u...
As much as we all probably dislike Maths as a subject, we do have to hand it to geometry for giving us the good old Hexgrid, home of some of the best strategy games. One such example, Price of Glory, has dropped its 1.4 Alpha update, stocked full... | Read more »
The SLC 2025 kicks off this month to cro...
Ever since the Solo Leveling: Arise Championship 2025 was announced, I have been looking forward to it. The promotional clip they released a month or two back showed crowds going absolutely nuts for the previous competitions, so imagine the... | Read more »
Dive into some early Magicpunk fun as Cr...
Excellent news for fans of steampunk and magic; the Precursor Test for Magicpunk MMORPG Crystal of Atlan opens today. This rather fancy way of saying beta test will remain open until March 5th and is available for PC - boo - and Android devices -... | Read more »
Prepare to get your mind melted as Evang...
If you are a fan of sci-fi shooters and incredibly weird, mind-bending anime series, then you are in for a treat, as Goddess of Victory: Nikke is gearing up for its second collaboration with Evangelion. We were also treated to an upcoming... | Read more »
Square Enix gives with one hand and slap...
We have something of a mixed bag coming over from Square Enix HQ today. Two of their mobile games are revelling in life with new events keeping them alive, whilst another has been thrown onto the ever-growing discard pile Square is building. I... | Read more »
Let the world burn as you have some fest...
It is time to leave the world burning once again as you take a much-needed break from that whole “hero” lark and enjoy some celebrations in Genshin Impact. Version 5.4, Moonlight Amidst Dreams, will see you in Inazuma to attend the Mikawa Flower... | Read more »
Full Moon Over the Abyssal Sea lands on...
Aether Gazer has announced its latest major update, and it is one of the loveliest event names I have ever heard. Full Moon Over the Abyssal Sea is an amazing name, and it comes loaded with two side stories, a new S-grade Modifier, and some fancy... | Read more »
Open your own eatery for all the forest...
Very important question; when you read the title Zoo Restaurant, do you also immediately think of running a restaurant in which you cook Zoo animals as the course? I will just assume yes. Anyway, come June 23rd we will all be able to start up our... | Read more »
Crystal of Atlan opens registration for...
Nuverse was prominently featured in the last month for all the wrong reasons with the USA TikTok debacle, but now it is putting all that behind it and preparing for the Crystal of Atlan beta test. Taking place between February 18th and March 5th,... | Read more »

Price Scanner via MacPrices.net

AT&T is offering a 65% discount on the ne...
AT&T is offering the new iPhone 16e for up to 65% off their monthly finance fee with 36-months of service. No trade-in is required. Discount is applied via monthly bill credits over the 36 month... Read more
Use this code to get a free iPhone 13 at Visi...
For a limited time, use code SWEETDEAL to get a free 128GB iPhone 13 Visible, Verizon’s low-cost wireless cell service, Visible. Deal is valid when you purchase the Visible+ annual plan. Free... Read more
M4 Mac minis on sale for $50-$80 off MSRP at...
B&H Photo has M4 Mac minis in stock and on sale right now for $50 to $80 off Apple’s MSRP, each including free 1-2 day shipping to most US addresses: – M4 Mac mini (16GB/256GB): $549, $50 off... Read more
Buy an iPhone 16 at Boost Mobile and get one...
Boost Mobile, an MVNO using AT&T and T-Mobile’s networks, is offering one year of free Unlimited service with the purchase of any iPhone 16. Purchase the iPhone at standard MSRP, and then choose... Read more
Get an iPhone 15 for only $299 at Boost Mobil...
Boost Mobile, an MVNO using AT&T and T-Mobile’s networks, is offering the 128GB iPhone 15 for $299.99 including service with their Unlimited Premium plan (50GB of premium data, $60/month), or $20... Read more
Unreal Mobile is offering $100 off any new iP...
Unreal Mobile, an MVNO using AT&T and T-Mobile’s networks, is offering a $100 discount on any new iPhone with service. This includes new iPhone 16 models as well as iPhone 15, 14, 13, and SE... Read more
Apple drops prices on clearance iPhone 14 mod...
With today’s introduction of the new iPhone 16e, Apple has discontinued the iPhone 14, 14 Pro, and SE. In response, Apple has dropped prices on unlocked, Certified Refurbished, iPhone 14 models to a... Read more
B&H has 16-inch M4 Max MacBook Pros on sa...
B&H Photo is offering a $360-$410 discount on new 16-inch MacBook Pros with M4 Max CPUs right now. B&H offers free 1-2 day shipping to most US addresses: – 16″ M4 Max MacBook Pro (36GB/1TB/... Read more
Amazon is offering a $100 discount on the M4...
Amazon has the M4 Pro Mac mini discounted $100 off MSRP right now. Shipping is free. Their price is the lowest currently available for this popular mini: – Mac mini M4 Pro (24GB/512GB): $1299, $100... Read more
B&H continues to offer $150-$220 discount...
B&H Photo has 14-inch M4 MacBook Pros on sale for $150-$220 off MSRP. B&H offers free 1-2 day shipping to most US addresses: – 14″ M4 MacBook Pro (16GB/512GB): $1449, $150 off MSRP – 14″ M4... Read more

Jobs Board

All contents are Copyright 1984-2011 by Xplain Corporation. All rights reserved. Theme designed by Icreon.