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.

 
AAPL
$118.93
Apple Inc.
-0.07
MSFT
$47.81
Microsoft Corpora
+0.06
GOOG
$541.83
Google Inc.
+1.46

MacTech Search:
Community Search:

Software Updates via MacUpdate

Adobe Photoshop Elements 13.0 - Consumer...
Adobe Photoshop Elements 12--the #1 selling consumer photo editing software--helps you edit pictures with powerful, easy-to-use options and share them via print, the web, Facebook, and more.Version... Read more
Skype 7.2.0.412 - Voice-over-internet ph...
Skype allows you to talk to friends, family and co-workers across the Internet without the inconvenience of long distance telephone charges. Using peer-to-peer data transmission technology, Skype... Read more
HoudahSpot 3.9.6 - Advanced file search...
HoudahSpot is a powerful file search tool built upon MacOS X Spotlight. Spotlight unleashed Create detailed queries to locate the exact file you need Narrow down searches. Zero in on files Save... Read more
RapidWeaver 6.0.3 - Create template-base...
RapidWeaver is a next-generation Web design application to help you easily create professional-looking Web sites in minutes. No knowledge of complex code is required, RapidWeaver will take care of... Read more
iPhoto Library Manager 4.1.10 - Manage m...
iPhoto Library Manager lets you organize your photos into multiple iPhoto libraries. Separate your high school and college photos from your latest summer vacation pictures. Or keep some photo... Read more
iExplorer 3.5.1.9 - View and transfer al...
iExplorer is an iPhone browser for Mac lets you view the files on your iOS device. By using a drag and drop interface, you can quickly copy files and folders between your Mac and your iPhone or... Read more
MacUpdate Desktop 6.0.3 - Discover and i...
MacUpdate Desktop 6 brings seamless 1-click installs and version updates to your Mac. With a free MacUpdate account and MacUpdate Desktop 6, Mac users can now install almost any Mac app on macupdate.... Read more
SteerMouse 4.2.2 - Powerful third-party...
SteerMouse is an advanced driver for USB and Bluetooth mice. It also supports Apple Mighty Mouse very well. SteerMouse can assign various functions to buttons that Apple's software does not allow,... Read more
iMazing 1.1 - Complete iOS device manage...
iMazing (was DiskAid) is the ultimate iOS device manager with capabilities far beyond what iTunes offers. With iMazing and your iOS device (iPhone, iPad, or iPod), you can: Copy music to and from... Read more
PopChar X 7.0 - Floating window shows av...
PopChar X helps you get the most out of your font collection. With its crystal-clear interface, PopChar X provides a frustration-free way to access any font's special characters. Expanded... Read more

Latest Forum Discussions

See All

Mystery Case Files: Dire Grove, Sacred G...
Mystery Case Files: Dire Grove, Sacred Grove HD Review By Jennifer Allen on November 28th, 2014 Our Rating: iPad Only App - Designed for the iPad A decent new installment for the popular Mystery Case Files series.   | Read more »
Castaway Paradise – Tips, Tricks, and St...
Ahoy there, castaways: Were you curious about our own thoughts regarding this pristine shipwreck? Check out our Castaway Paradise review! Castaway Paradise is out for iOS, finally giving mobile gamers the opportunity to enjoy the idyllic lifestyle... | Read more »
Castaway Paradise VIP Subs are on Sale f...
Castaway Paradise VIP Subs are on Sale for a Limited Time, and a Special Holiday Update is Coming Soon Posted by Rob Rich on November 28th, 2014 [ | Read more »
Primitive Review
Primitive Review By Jordan Minor on November 28th, 2014 Our Rating: :: FOLK ARTUniversal App - Designed for iPhone and iPad True to its name, Primitive is about as straightforward as runners get.   | Read more »
7 tips to get ahead of the competition i...
7 tips to get ahead of the competition in Dynasty of Dungeons Posted by Simon Reed on November 28th, 2014 [ permalink ] Playcrab has launched their action-packed new dungeon crawler, Dynasty of Dungeons, today. | Read more »
Master of Tea Kung Fu Review
Master of Tea Kung Fu Review By Jordan Minor on November 28th, 2014 Our Rating: :: ONE DROP RULESUniversal App - Designed for iPhone and iPad Master of Tea Kung Fu is a creative and complex caffeinated brawler.   | Read more »
Monster Strike Review
Monster Strike Review By Campbell Bird on November 28th, 2014 Our Rating: :: BILLIARD STRATEGYUniversal App - Designed for iPhone and iPad Collect monsters and battle by flinging them across the battlefield in this strangely... | Read more »
Proun+ Review
Proun+ Review By Jennifer Allen on November 28th, 2014 Our Rating: :: TWITCHY RACINGUniversal App - Designed for iPhone and iPad Twitchy racing aplenty in Proun+, an enjoyably tricky title.   | Read more »
Lucha Amigos (Games)
Lucha Amigos 1.0 Device: iOS Universal Category: Games Price: $1.99, Version: 1.0 (iTunes) Description: Forget Ninja Turtles, and meet Wrestlers Turtles! Crazier, Spicier and…Bouncier! Sling carapaces of 7 Luchadores to knock all... | Read more »
Record of Agarest War Zero (Games)
Record of Agarest War Zero 1.0 Device: iOS Universal Category: Games Price: $7.99, Version: 1.0 (iTunes) Description: HyperDevbox Holiday Turkey Black Friday Special Pricing! To celebrate the opening of the holiday season HyperDevbox... | Read more »

Price Scanner via MacPrices.net

Best Black Friday Deal: 15-inch Retina MacBoo...
 B&H Photo has the new 2014 15″ Retina MacBook Pros on sale for $300 off MSRP as part of their Black Friday sale. Shipping is free, and B&H charges NY sales tax only: - 15″ 2.2GHz Retina... Read more
Up To 75% Off Infovole Text Apps Over Black F...
Infovole’s entire range of apps, including the Textkraft family of word processors for iPads and iPhones, is being offered at 50-75% off over the Black Friday and Cyber Monday weekend. The five-day... Read more
Black Friday: Up to $60 off Mac minis, NY tax...
 B&H Photo has new 2014 Mac minis on sale for up to $60 off MSRP as part of their Black Friday sale. Shipping is free, and B&H charges NY sales tax only: - 1.4GHz Mac mini: $449.99 $50 off... Read more
Black Friday: 27-inch 5K iMac for $2299, save...
 B&H Photo continues to offer Black Friday sale prices on the 27″ 3.5GHz 5K iMac, in stock today and on sale for $2299 including free shipping plus NY sales tax only. Their price is $200 off MSRP... Read more
Karalux Announces 24K Gold-Plated iPhone 6
Karalux, a Vietnam-based jewellery firm, has launched a unique 24 karat gold-plated iPhone 6 version with gold-cast monolithic dragon on its back panel. The real 24 karat gold plated enclosure doesn’... Read more
Black Friday: 13-inch 2.6GHz Retina MacBook P...
 B&H Photo has lowered their price for the 13″ 2.6GHz/128GB Retina MacBook Pro to $1159 for Black Friday. That’s $140 off MSRP, and it’s the lowest price for this model (except for Apple’s $1099... Read more
View all the Black Friday sales on our Mac Pr...
We’ve updated our Mac Price Trackers with the latest information on prices, bundles, and availability on systems from Apple’s authorized internet/catalog resellers. View Black Friday sale prices at a... Read more
Black Friday: 11-inch MacBook Air for $779, s...
 Best Buy has lowered their price for the 2014 11″ 1.4GHz/128GB MacBook Air to $779.99 for Black Friday. That’s $120 off MSRP. Choose free shipping or free local store pickup (if available). Sale... Read more
Apple Store Black Friday sale for 2014: $100...
BLACK FRIDAY The Apple Store has posted their Black Friday deals for 2014. Receive a $100 PRODUCT(RED) branded iTunes gift card with the purchase of select Macs, $50 with iPads, and $25 with iPods,... Read more
Black Friday: 15% off iTunes Gift Cards
Staples is offering 15% off $50 and $100 iTunes Gift Cards on their online store as part of their Black Friday sale. Click here for more information. Shipping is free. Best Buy is offering $100... Read more

Jobs Board

*Apple* Solutions Consultant (ASC) - Apple (...
**Job Summary** The ASC is an Apple employee who serves as an Apple brand ambassador and influencer in a Reseller's store. The ASC's role is to grow Apple Read more
Senior Event Manager, *Apple* Retail Market...
…This senior level position is responsible for leading and imagining the Apple Retail Team's global event strategy. Delivering an overarching brand story; in-store, Read more
*Apple* Retail - Multiple Positions (US) - A...
Sales Specialist - Retail Customer Service and Sales Transform Apple Store visitors into loyal Apple customers. When customers enter the store, you're also the Read more
*Apple* Solutions Consultant (ASC) - Apple (...
**Job Summary** The ASC is an Apple employee who serves as an Apple brand ambassador and influencer in a Reseller's store. The ASC's role is to grow Apple Read more
*Apple* Solutions Consultant (ASC) - Apple (...
**Job Summary** The ASC is an Apple employee who serves as an Apple brand ambassador and influencer in a Reseller's store. The ASC's role is to grow Apple Read more
All contents are Copyright 1984-2011 by Xplain Corporation. All rights reserved. Theme designed by Icreon.