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

Duplicate Annihilator 5.7.5 - Find and d...
Duplicate Annihilator takes on the time-consuming task of comparing the images in your iPhoto library using effective algorithms to make sure that no duplicate escapes. Duplicate Annihilator... Read more
BusyContacts 1.0.2 - Fast, efficient con...
BusyContacts is a contact manager for OS X that makes creating, finding, and managing contacts faster and more efficient. It brings to contact management the same power, flexibility, and sharing... Read more
Capture One Pro 8.2.0.82 - RAW workflow...
Capture One Pro 8 is a professional RAW converter offering you ultimate image quality with accurate colors and incredible detail from more than 300 high-end cameras -- straight out of the box. It... Read more
Backblaze 4.0.0.872 - Online backup serv...
Backblaze is an online backup service designed from the ground-up for the Mac.With unlimited storage available for $5 per month, as well as a free 15-day trial, peace of mind is within reach with... Read more
Little Snitch 3.5.2 - Alerts you about o...
Little Snitch gives you control over your private outgoing data. Track background activity As soon as your computer connects to the Internet, applications often have permission to send any... Read more
Monolingual 1.6.4 - Remove unwanted OS X...
Monolingual is a program for removing unnecesary language resources from OS X, in order to reclaim several hundred megabytes of disk space. If you use your computer in only one (human) language, you... Read more
CleanApp 5.0 - Application deinstaller a...
CleanApp is an application deinstaller and archiver.... Your hard drive gets fuller day by day, but do you know why? CleanApp 5 provides you with insights how to reclaim disk space. There are... Read more
Fantastical 2.0 - Create calendar events...
Fantastical is the Mac calendar you'll actually enjoy using. Creating an event with Fantastical is quick, easy, and fun: Open Fantastical with a single click or keystroke Type in your event details... Read more
Cocktail 8.2 - General maintenance and o...
Cocktail is a general purpose utility for OS X that lets you clean, repair and optimize your Mac. It is a powerful digital toolset that helps hundreds of thousands of Mac users around the world get... Read more
Direct Mail 4.0.4 - Create and send grea...
Direct Mail is an easy-to-use, fully-featured email marketing app purpose-built for OS X. It lets you create and send great looking email campaigns. Start your newsletter by selecting from a gallery... Read more

These are All the Apple Watch Apps and G...
The Apple Watch is less than a month from hitting store shelves, and once you get your hands on it you're probably going to want some apps and games to install. Fear not! We've compiled a list of all the Apple Watch apps and games we've been able to... | Read more »
Appy to Have Known You - Lee Hamlet Look...
Being at 148Apps these past 2 years has been an awesome experience that has taught me a great deal, and working with such a great team has been a privilege. Thank you to Rob Rich, and to both Rob LeFebvre and Jeff Scott before him, for helping me... | Read more »
Hands-On With Allstar Heroes - A Promisi...
Let’s get this out of the way quickly. Allstar Heroes looks a lot like a certain other recent action RPG release, but it turns out that while it’s not yet available here, Allstar Heroes has been around for much longer than that other title. Now that... | Read more »
Macho Man and Steve Austin Join the Rank...
WWE Immortals, by Warner Bros. Interactive Entertainment and WWE, has gotten a superstar update. You'll now have access to Macho Man Randy Savage and Steve Austin. Both characters have two different versions: Macho Man Randy Savage Renegade or Macho... | Read more »
Fearless Fantasy is Fantastic for the iF...
I actually had my first look at Fearless Fantasy last year at E3, but it was on a PC so there wasn't much for me to talk about. But now that I've been able to play with a pre-release version of the iOS build, there's quite a bit for me to talk... | Read more »
MLB Manager 2015 (Games)
MLB Manager 2015 5.0.14 Device: iOS Universal Category: Games Price: $4.99, Version: 5.0.14 (iTunes) Description: Guide your favorite MLB franchise to glory! MLB Manager 2015, officially licensed by MLB.com and based on the award-... | Read more »
Breath of Light (Games)
Breath of Light 1.0.1421 Device: iOS Universal Category: Games Price: $2.99, Version: 1.0.1421 (iTunes) Description: Hold a quiet moment. Breath of Light is a meditative and beautiful puzzle game with a hypnotic soundtrack by... | Read more »
WWE WrestleMania Tags into the App Store
Are You ready to rumble? The official WWE WrestleMania app, by World Wrestling Entertainment, is now available. Now you can get all your WrestleMania info in one place before anyone else. The app offers details on superstar signings, interactive... | Read more »
Bio Inc's New Expansion is Infectin...
Bio Inc., by DryGin Studios, is the real time strategy game where you infect a human body with the worst virus your evil brain can design. Recently, the game was updated to add a whole lot of new features. Now you can play the new “Lethal”... | Read more »
The Monocular Minion is Here! Despicable...
Despicable Me: Minion Rush, by Gameloft, is introducing a new runner to the mix in their latest update. Now you can play as Carl, the prankster minion. Carl has a few new abilities to play with, including running at a higher speed from the start.... | Read more »

Price Scanner via MacPrices.net

13-inch 2.5GHz MacBook Pro (refurbished) avai...
The Apple Store has Apple Certified Refurbished 13″ 2.5GHz MacBook Pros available for $829, or $270 off the cost of new models. Apple’s one-year warranty is standard, and shipping is free: - 13″ 2.... Read more
Save up to $80 on iPad Air 2s, NY tax only, f...
 B&H Photo has iPad Air 2s on sale for $80 off MSRP including free shipping plus NY sales tax only: - 16GB iPad Air 2 WiFi: $469.99 $30 off - 64GB iPad Air 2 WiFi: $549.99 $50 off - 128GB iPad... Read more
iMacs on sale for up to $205 off MSRP
B&H Photo has 21″ and 27″ iMacs on sale for up to $205 off MSRP including free shipping plus NY sales tax only: - 21″ 1.4GHz iMac: $1019 $80 off - 21″ 2.7GHz iMac: $1189 $110 off - 21″ 2.9GHz... Read more
Färbe Technik Offers iPhone Battery Charge LI...
Färbe Technik, which manufactures and markets of mobile accessories for Apple, Blackberry and Samsung mobile devices, is offering tips on how to keep your iPhone charged while in the field: •... Read more
Electronic Recyclers International CEO Urges...
Citing a recent story on CNBC about concerns some security professionals have about the forthcoming Apple Watch, John Shegerian, Chairman and CEO of Electronic Recyclers International (ERI), the... Read more
Save up to $380 with Apple refurbished iMacs
The Apple Store has Apple Certified Refurbished iMacs available for up to $380 off the cost of new models. Apple’s one-year warranty is standard, and shipping is free: - 27″ 3.5GHz 5K iMac – $2119 $... Read more
Mac minis on sale for up to $75 off, starting...
MacMall has Mac minis on sale for up to $75 off MSRP including free shipping. Their prices are the lowest available for these models from any reseller: - 1.4GHz Mac mini: $459.99 $40 off - 2.6GHz Mac... Read more
College Student Deals: Additional $50 off Mac...
Take an additional $50 off all MacBooks and iMacs at Best Buy Online with their College Students Deals Savings, valid through April 11, 2015. Anyone with a valid .EDU email address can take advantage... Read more
Mac Pros on sale for up to $260 off MSRP
B&H Photo has Mac Pros on sale for up to $260 off MSRP. Shipping is free, and B&H charges sales tax in NY only: - 3.7GHz 4-core Mac Pro: $2799, $200 off MSRP - 3.5GHz 6-core Mac Pro: $3719.99... Read more
13-inch 2.5GHz MacBook Pro on sale for $100 o...
B&H Photo has the 13″ 2.5GHz MacBook Pro on sale for $999 including free shipping plus NY sales tax only. Their price is $100 off MSRP. Read more

Jobs Board

DevOps Software Engineer - *Apple* Pay, iOS...
**Job Summary** Imagine what you could do here. At Apple , great ideas have a way of becoming great products, services, and customer experiences very quickly. Bring 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
Sr. Technical Services Consultant, *Apple*...
**Job Summary** Apple Professional Services (APS) has an opening for a senior technical position that contributes to Apple 's efforts for strategic and transactional Read more
Lead *Apple* Solutions Consultant - Retail...
**Job Summary** Job Summary The Lead ASC is an Apple employee who serves as the Apple business manager and influencer in a hyper-business critical Reseller's store Read more
*Apple* Pay - Site Reliability Engineer - Ap...
**Job Summary** Imagine what you could do here. At Apple , great ideas have a way of becoming great products, services, and customer experiences very quickly. Bring Read more
All contents are Copyright 1984-2011 by Xplain Corporation. All rights reserved. Theme designed by Icreon.