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

Model 15 (Music)
Model 15 1.0 Device: iOS iPhone Category: Music Price: $29.99, Version: 1.0 (iTunes) Description: The Moog Model 15 App is the first Moog modular synthesizer and synthesis educational tool created exclusively for iPad, iPhone and... | Read more »
How to deal with wind in Angry Birds Act...
Angry Birds Action! is a physics-based puzzler in which you're tasked with dragging and launching birds around an obstacle-littered field to achieve a set objective. It's simple enough at first, but when wind gets introduced things can get pretty... | Read more »
How to get three stars in every level of...
Angry Birds Action! is, essentially, a pinball-style take on the pull-and-fling action of the original games. When you first boot it up, you'll likely be wondering exactly what it is you have to do to get a good score. Well, never fear as 148Apps... | Read more »
The beginner's guide to Warbits
Warbits is a turn-based strategy that's clearly inspired by Nintendo's Advance Wars series. Since turn-based strategy games can be kind of tricky to dive into, see below for a few tips to help you in the beginning. Positioning is crucial [Read... | Read more »
How to upgrade your character in Spellsp...
So you’ve mastered the basics of Spellspire. By which I mean you’ve realised it’s all about spelling things in a spire. What next? Well you’re going to need to figure out how to toughen up your character. It’s all well and good being able to spell... | Read more »
5 slither.io mash-ups we'd love to...
If there's one thing that slither.io has proved, it's that the addictive gameplay of Agar.io can be transplanted onto basically anything and it will still be good fun. It wouldn't be surprising if we saw other developers jumping on the bandwagon,... | Read more »
How to navigate the terrain in Sky Charm...
Sky Charms is a whimsical match-'em up adventure that uses creative level design to really ramp up the difficulty. [Read more] | Read more »
Victorious Knight (Games)
Victorious Knight 1.3 Device: iOS Universal Category: Games Price: $1.99, Version: 1.3 (iTunes) Description: New challenges awaits you! Experience fresh RPG experience with a unique combat mechanic, packed with high quality 3D... | Read more »
Agent Gumball - Roguelike Spy Game (Gam...
Agent Gumball - Roguelike Spy Game 1.0 Device: iOS Universal Category: Games Price: $2.99, Version: 1.0 (iTunes) Description: Someone’s been spying on Gumball. What the what?! Two can play at that game! GO UNDERCOVERSneak past enemy... | Read more »
Runaway Toad (Games)
Runaway Toad 1.0 Device: iOS Universal Category: Games Price: $2.99, Version: 1.0 (iTunes) Description: It ain’t easy bein’ green! Tap, hold, and swipe to help Toad hop to safety in this gorgeous new action game from the creators of... | Read more »

Price Scanner via MacPrices.net

Price drops on clearance 2015 13-inch MacBook...
B&H Photo has dropped prices on clearance 2015 13″ MacBook Airs by up to $250. Shipping is free, and B&H charges NY sales tax only: - 13″ 1.6GHz/4GB/128GB MacBook Air (MJVE2LL/A): $799, $200... Read more
Mac minis on sale for up to $100 off MSRP
B&H Photo has Mac minis on sale for up to $100 off MSRP including free shipping plus NY sales tax only: - 1.4GHz Mac mini: $449 $50 off MSRP - 2.6GHz Mac mini: $649 $50 off MSRP - 2.8GHz Mac mini... Read more
13-inch Retina MacBook Pros on sale for up to...
B&H Photo has 13″ Retina MacBook Pros on sale for $130-$200 off MSRP. Shipping is free, and B&H charges NY tax only: - 13″ 2.7GHz/128GB Retina MacBook Pro: $1169 $130 off MSRP - 13″ 2.7GHz/... Read more
Apple price trackers, updated continuously
Scan our Apple Price Trackers for the latest information on sales, bundles, and availability on systems from Apple’s authorized internet/catalog resellers. We update the trackers continuously: - 15″... Read more
SanDisk Half-Terabyte SSD Optimized for Every...
SanDisk Corporation has announced the SanDisk Z410 SSD, a cost-competitive, half-terabyte solid state drive (SSD) that enables manufacturers to design for a broad range of desktop PCs and laptops.... Read more
Churchill Downs Racetrack Selects VenueNext t...
Churchill Downs Racetrack has announced an agreement with VenueNext to implement its technology platform for the start of Churchill Downs 2016 Spring Meet, which includes the 142nd running of the... Read more
Record 700 Million Pounds of CE Recycled in 2...
The Consumer Technology Association (CTA) reports that a record-setting 700 million pounds of consumer electronics (CE) have been recycled under the eCycling Leadership Initiative (ELI). According to... Read more
Price drops on clearance 12-inch Retina MacBo...
B&H Photo has dropped prices on leftover 2015 12″ Retina MacBooks with models now available starting at $999. Shipping is free, and B&H charges NY tax only: - 12″ 1.1GHz Gray Retina MacBook... Read more
15-inch Retina MacBook Pros available for $20...
B&H Photo has 15″ Retina MacBook Pros on sale for up to $210 off MSRP. Shipping is free, and B&H charges NY tax only: - 15″ 2.2GHz Retina MacBook Pro: $1799 $200 off MSRP - 15″ 2.5GHz Retina... Read more
Target offers Apple Watch Sport for $50 off M...
Target has Apple Watch Sports on sale for $50 off MSRP for a limited time. Choose free shipping or free local store pickup (if available). Sale prices for online orders only, in-store prices may vary... Read more

Jobs Board

*Apple* Retail - Multiple Positions - Apple,...
Job Description: Sales Specialist - Retail Customer Service and Sales Transform Apple Store visitors into loyal Apple customers. When customers enter the store, Read more
Restaurant Manager (Neighborhood Captain) - A...
…in every aspect of daily operation. WHY YOU'LL LIKE IT: You'll be the Big Apple . You'll solve problems. You'll get to show your ability to handle the stress and Read more
Restaurant Manager (Neighborhood Captain) - A...
…in every aspect of daily operation. WHY YOU'LL LIKE IT: You'll be the Big Apple . You'll solve problems. You'll get to show your ability to handle the stress and Read more
Simply Mac *Apple* Specialist- Service Repa...
Simply Mac is the largest premier retailer of Apple products in the nation. In order to support our growing customer base, we are currently looking for a driven Read more
*Apple* Retail - Multiple Positions - Apple,...
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
All contents are Copyright 1984-2011 by Xplain Corporation. All rights reserved. Theme designed by Icreon.