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

VueScan 9.7.96 - Scanner software with a...
VueScan is a scanning program that works with most high-quality flatbed and film scanners to produce scans that have excellent color fidelity and color balance. VueScan is easy to use, and has... Read more
FileMaker Pro 19.6.1 - Quickly build cus...
FileMaker Pro is the tool you use to create a custom app. You also use FileMaker Pro to access your app on a computer. Start by importing data from a spreadsheet or using a built-in Starter app to... Read more
Duet 3.1.0.0 - Use your iPad as an exter...
Duet is the first app that allows you to use your iDevice as an extra display for your Mac using the Lightning or 30-pin cable. Note: This app requires a iOS companion app. Release notes were... Read more
Firefox 107.0.1 - Fast, safe Web browser...
Firefox offers a fast, safe Web browsing experience. Browse quickly, securely, and effortlessly. With its industry-leading features, Firefox is the choice of Web development professionals and casual... Read more
War Thunder 2.21.1.91 - Multiplayer war...
In War Thunder, aircraft, attack helicopters, ground forces and naval ships collaborate in realistic competitive battles. You can choose from over 1,500 vehicles and an extensive variety of combat... Read more
Numbers 12.2.1 - Apple's spreadshee...
With Apple Numbers, sophisticated spreadsheets are just the start. The whole sheet is your canvas. Just add dramatic interactive charts, tables, and images that paint a revealing picture of your data... Read more
DEVONthink Pro 3.8.7 - Knowledge base, i...
DEVONthink is DEVONtechnologies' document and information management solution. It supports a large variety of file formats and stores them in a database enhanced by artificial intelligence (AI). Many... Read more
Drive Genius 6.2.3 - $79.00
Drive Genius features a comprehensive Malware Scan. Automate your malware protection. Protect your investment from any threat. The Malware Scan is part of the automated DrivePulse utility. DrivePulse... Read more
VLC Media Player 3.0.18 - Popular multim...
VLC Media Player is a highly portable multimedia player for various audio and video formats (MPEG-1, MPEG-2, MPEG-4, DivX, MP3, OGG, ...) as well as DVDs, VCDs, and various streaming protocols. It... Read more
Apple Pages 12.2.1 - Apple's word p...
Apple Pages is a powerful word processor that gives you everything you need to create documents that look beautiful. And read beautifully. It lets you work seamlessly between Mac and iOS devices, and... Read more

Latest Forum Discussions

See All

‘Railbound’ Update Now Available Adding...
One of our favorite puzzlers released this year is Railbound from Afterburn Games, which hit in early September and earned our Game of the Week recognition for being an absolutely ace logic puzzler. The goal is to place rail pieces down in order to... | Read more »
The Seven Deadly Sins: Grand Cross celeb...
Netmarble Corporation has pulled out all the stops to celebrate the 3 and a half year anniversary of The Seven Deadly Sins: Grand Cross. The Grand Cross 3.5th Year Anniversary the Ultimate One, a rather wordy title, brings with it a brand new... | Read more »
‘Skullgirls Mobile’ Major Update 5.2 Out...
Developer Hidden Variable pushed out a major update for Skullgirls Mobile (Free) a few hours ago. The version 5.2 update brings in Black Dahlia (before the console and PC game), Retakes, XP Treats, free gifts, and more. Since launch, Skullgirls... | Read more »
Out Now: ‘Disgaea 4’, ‘Romancing SaGa: M...
Each and every day new mobile games are hitting the App Store, and so each week we put together a big old list of all the best new releases of the past seven days. Back in the day the App Store would showcase the same games for a week, and then... | Read more »
SwitchArcade Round-Up: ‘Elevator Action...
Hello gentle readers, and welcome to the SwitchArcade Round-Up for November 30th, 2022. We’re finishing up the month on a quiet note, friends. There are five new releases to look at today, with a few notables amongst them. We’ve got summaries for... | Read more »
Midijiwan AB announce a new 5-year deal...
Developers Midjiwan AB has revealed a new 5-year deal with global esports platform Challengermode to develop the competitive side of The Battle of Polytopia. Joining forces with an esport giant like this shows that Midijiwan is taking serious... | Read more »
‘Genshin Impact’ Genius Invokation TCG G...
Last week, HoYoverse revealed the release date and more information detailing the upcoming Genshin Impact (Free) version 3.3 ‘All Senses Clear, All Existence Void’ update. | Read more »
The Latest Update for ‘Marvel Snap’ Has...
You know, it’s kind of funny in hindsight that Marvel Snap (Free) got to its global launch without having Thanos in there. The game’s name and a major game mechanic are references to his iconic finger-flick, after all.  I guess it could not live... | Read more »
Side-Scrolling Shooter ‘Pulstar’ ACA Neo...
Last week, Top Hunter Roddy and Cathy hit iOS and Android as Hamster and SNK’s newest ACA NeoGeo release. Read Shaun’s review of it here. Today, the side-scrolling shooter Pulstar has hit mobile platforms. Pulstar debuted in 1995, and it has... | Read more »
‘Disgaea 4: A Promise Revisited’ Gets Su...
NIS just surprise launched Disgaea 4: A Promise Revisited on iOS and Android as a premium release (via Gematsu). Disgaea 4: A Promise Revisited is an enhanced version of Disgaea 4. The original game debuted on PS3 before it was ported to PS Vita... | Read more »

Price Scanner via MacPrices.net

13″ M1 MacBook Pros available starting at $10...
Apple has clearance 13″ M1 MacBook Pros available, Certified Refurbished, starting at $1059 and ranging up to $270 off original MSRP. These are the cheapest 13″ MacBook Pros for sale today at Apple,... Read more
Amazon’s $79 Apple AirPods deal is back in st...
Amazon has 2nd generation Apple AirPods back on sale for only $79 as part of their Holiday 2022 sale. That’s a whopping $50 off Apple’s MSRP. Shipping is free. Their price is the lowest we’ve ever... Read more
Get last year’s Apple AirPods Pro for only $1...
Verizon has first-generation Apple AirPods Pro on sale for $159.99 on their online store as part of their continuing Holiday sale. Their price is $90 off Apple’s original MSRP, and it’s the lowest... Read more
12.9″ M1 128GB WiFi + Cellular iPad Pro on cl...
Amazon has clearance, previous-generation 2021 12″ 128GB WiFi + Cellular M1-powered iPad Pros in stock and on sale today for $999.99 shipped. That’s $250 off MSRP. Be sure to make your purchase from... Read more
Back in stock: Mac minis at Apple starting at...
Apple has a full line of M1-powered Mac minis available in their Certified Refurbished section starting at only $589 and ranging up to $140 off MSRP. Each mini comes with Apple’s one-year warranty,... Read more
Season’s best 14″ MacBook Pro discount still...
Amazon is offering $500 off MSRP discounts on select Apple 14-inch MacBook Pros with M1 Pro CPUs as part of their continuing Holiday sale. Their price is the lowest available for this models from any... Read more
The cheapest new Mac for sale this Holiday se...
B&H Photo has previous-generation Intel-based 3.6GHz 4-core Mac minis on clearance sale for only $499. Their price is $300 off original MSRP for this mini, and it’s the lowest price we can find... Read more
13″ Apple MacBook Pros with M2 CPUs on sale f...
B&H Photo has new 13″ MacBook Pros with Apple M2 processors in stock and on sale today for $150 off MSRP as part of their Holiday sale. Their prices are the lowest currently available for M2... Read more
Apple MagSafe iPhone Battery on sale for 25%...
Amazon has Apple’s MagSafe iPhone Battery on sale for 25% off MSRP today. Shipping is free: – MagSafe Battery: $74.99 $25 off MSRP For the latest prices and sales, see our Apple MagSafe Charger Price... Read more
Apple has 13″ M2 MacBook Pros in stock today...
Apple standard-configuration 13″ MacBook Pros with M2 CPUs in stock and available today starting at $1169, Certified Refurbished, and ranging up to $150 off original MSRP. These are the cheapest 13″... Read more

Jobs Board

Support Technician II - *Apple* Support - O...
…problems and acting as a liaison between customers and resolving groups. As an Apple Technical Specialist, you will be supporting many of our popular Apple Read more
*Apple* Electronic Repair Technician - PlanI...
…a highly motivated individual to join our Production Department as an Apple Electronic Repair Technician. The computer repair technician will diagnose, assemble, Read more
Lead Developer - *Apple* tvOS - Rumble (Uni...
…earnings, and positive sentiment About the role: We are looking for a Lead Apple tvOS Developer to join our application engineering team to expand our video centric Read more
Tier 1 Endpoint Engineer - *Apple* - Red Ri...
…Desk on site, at our Client's location, with a focus on support to Apple products. This position will handle technical support requests directly from customers and Read more
Product Manager II - *Apple* - DISH (United...
…you will be doing We seek an ambitious, data-driven thinker to assist the Apple Product Development team as our new Retail Wireless division continues to grow and Read more
All contents are Copyright 1984-2011 by Xplain Corporation. All rights reserved. Theme designed by Icreon.