TweetFollow Us on Twitter

Lambda
Volume Number:9
Issue Number:9
Column Tag:Lisp Listener

“The Lambda Lambada: Y Dance?”

Mutual Recursion

By André van Meulebrouck, Chatsworth, California

Note: Source code files accompanying article are located on MacTech CD-ROM or source code disks.

“Mathematics is thought moving in the sphere of complete abstraction from any particular instance of what it is talking about.” - Alfred North Whitehead

Welcome once again to Mutual of Omo Oz Y Old Kingdom (with apologies to the similar named TV series of yesteryears).

In this installment, Lambda, the forbidden (in conventional languages) function, does the lambada-the forbidden (in l-calculus) dance. Film at 11.

In [vanMeule Jun 91] the question was raised as to whether everything needed to create a metacircular interpreter (using combinators) has been given to the reader.

One of the last (if not the last) remaining items not yet presented is mutual recursion, which allows an interpreter’s eval and apply functions to do their curious tango (the “lambda lambada”?!?).

In this article, the derivation of a Y2 function will be shown. Y2 herein will be the sister combinator of Y, to be used for handling mutual recursion (of two functions) in the applicative order. The derivation of Y2 will be done in a similar manner as was done for deriving Y from pass-fact in [vanMeule May 92].

This exercise will hopefully give novel insights into Computer Science and the art of programming. (This is the stuff of Überprogrammers!) This exercise should also give the reader a much deeper understanding of Scheme while developing programming muscles in ways that conventional programming won’t.

Backdrop and motivation

[vanMeule Jun 91] described the minimalist game. The minimalist game is an attempt to program in Scheme using only those features of Scheme that have more or less direct counterparts in l-calculus. The aim of the minimalist game is (among other things):

1) To understand l-calculus and what it has to say about Computer Science.

2) To develop expressive skills. Part of the theory behind the minimalist game is that one’s expressive ability is not so much posited in how many programming constructs one knows, but in how cleverly one wields them. Hence, by deliberately limiting oneself to a restricted set of constructs, one is forced to exercise one’s expressive muscles in ways they would not normally get exercised when one has a large repertoire of constructs to choose from. The maxim here is: “learn few constructs, but learn them well”.

In l-calculus (and hence the minimalist game) there is no recursion. It turns out that recursion is a rather impure contortion in many ways! However, recursion can be simulated by making use of the higher order nature of l-calculus. A higher order function is a function which is either passed as an argument (to another function) or returned as a value. As thrifty as l-calculus is, it does have higher order functions, which is no small thing as very few conventional languages have such a capability, and those that do have it have only a very weak version of it. (This is one of the programming lessons to be learned from playing the minimalist game: The enormous power of higher order functions and the losses conventional languages suffer from not having them.)

Different kinds of recursion

As soon as a language has global functions or procedures and parameter passing provided via a stack discipline, you’ve got recursion! In fact, there is essentially no difference between a procedure calling itself or calling a different function-the same stack machinery that handles the one case will automatically handle the other. (There’s no need for the stack machinery to know nor care whether the user is calling other procedures or the same procedure.)

However, as soon as a language has local procedures, it makes a very big difference if a procedure calls itself! The problem is that when a local procedure sees a call to itself from within itself, by the rules of lexical scoping, it must look for its own definition outside of its own scope! This is because the symbol naming the recursive function is a free variable with respect to the context it occurs in.

; 1
>>> (let ((local-fact 
           (lambda (n)
             (if (zero? n)
                 1
                 (* n (local-fact (1- n)))))))
      (local-fact 5))
ERROR:  Undefined global variable
local-fact

Entering debugger.  Enter ? for help.
debug:> 

This is where letrec comes in.

; 2

>>> (letrec ((local-fact 
              (lambda (n)
                (if (zero? n)
                    1
                    (* n (local-fact (1- n)))))))
      (local-fact 5))
120

To understand what letrec is doing let’s translate it to its semantic equivalent. letrec can be simulated using let and set! [CR 91].

; 3
>>> (let ((local-fact ‘undefined))
      (begin
       (set! local-fact 
             (lambda (n)
               (if (zero? n)
                   1
                   (* n (local-fact (1- n))))))
       (local-fact 5)))
120

Mutual recursion is slightly different from “regular” recursion: instead of a function calling itself, it calls a different function that then calls the original function. For instance, “foo” and “fido” would be mutually recursive if foo called fido, and fido called foo. The letrec trick will work fine for mutual recursion.

; 4 

>>> (let ((my-even? ‘undefined)
          (my-odd? ‘undefined))
      (begin
       (set! my-even? 
             (lambda (n)
               (if (zero? n)
                   #t
                   (my-odd? (1- n)))))
       (set! my-odd? 
             (lambda (n)
               (if (zero? n)
                   #f
                   (my-even? (1- n)))))
       (my-even? 80)))
#t

The reason this works is because both functions that had to have mutual knowledge of each other were defined as symbols in a lexical context outside of the context in which the definitions were evaluated.

However, all the above letrec examples rely on being able to modify state. l-calculus doesn’t allow state to be modified. (An aside: since parallel machines have similar problems and restrictions in dealing with state, there is ample motivation for finding non-state oriented solutions to such problems in l-calculus.)

The recursion in local-fact can be ridded by using the Y combinator. However, in the my-even? and my-odd? example the Y trick doesn’t work because in trying to eliminate recursion using Y, the mutual nature of the functions causes us to get into a chicken-before-the-egg dilemma.

It’s clear we need a special kind of Y for this situation. Let’s call it Y2.

The pass-fact trick

[vanMeule May 92] derived the Y combinator in the style of [Gabriel 88] by starting with pass-fact (a version of the factorial function which avoids recursion by passing its own definition as an argument) and massaging it into two parts: a recursionless recursion mechanism and an abstracted version of the factorial function.

Let’s try the same trick for Y2, using my-even? and my-odd? as our starting point.

First, we want to massage my-even? and my-odd? into something that looks like pass-fact. Here’s what our “template” looks like:

; 5 

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

Here’s a version of my-even? and my-odd? modeled after the pass-fact “template”.

; 6 
>>> (define even-odd
      (cons 
       (lambda (function-list)
         (lambda (n)
           (if (zero? n)
               #t
               (((cdr function-list) function-list)
                (1- n)))))
       (lambda (function-list)
         (lambda (n)
           (if (zero? n)
               #f
               (((car function-list) function-list) 
                (1- n)))))))
even-odd
>>> (define pass-even?
      ((car even-odd) even-odd))
pass-even?
>>> (define pass-odd?
      ((cdr even-odd) even-odd))
pass-odd?
>>> (pass-even? 8)
#t

This could derive one crazy!

Now that we know we can use higher order functions to get rid of the mutual recursion in my-even? and my-odd? the next step is to massage out the recursionless mutual recursion mechanism from the definitional parts that came from my-even? and my-odd?. The following is the code of such a derivation, including test cases and comments.

; 7
(define my-even?
  (lambda (n)
    (if (zero? n)
        #t
        (my-odd? (1- n)))))
;
(define my-odd?
  (lambda (n)
    (if (zero? n)
        #f
        (my-even? (1- n)))))
;
(my-even? 5)
;
; Get out of global environment-use local environment.
;
(define mutual-even?
  (letrec 
    ((my-even? (lambda (n)
                 (if (zero? n)
                     #t
                     (my-odd? (1- n)))))
     (my-odd? (lambda (n)
                (if (zero? n)
                    #f
                    (my-even? (1- n))))))
    my-even?))
;
(mutual-even? 5)
;
; Get rid of destructive letrec.  Use let instead.
; Make a list of the mutually recursive functions.
;
(define mutual-even?
  (lambda (n)
    (let 
      ((function-list 
        (cons (lambda (functions n) ; even?
                (if (zero? n)
                    #t
                    ((cdr functions) functions 
                                     (1- n))))
              (lambda (functions n) ; odd?
                (if (zero? n)
                    #f
                    ((car functions) functions 
                                     (1- n)))))))
      ((car function-list) function-list n))))
;
(mutual-even? 5)
;
; Curry, and get rid of initial (lambda (n) ...) .
;
(define mutual-even?
  (let 
    ((function-list 
      (cons (lambda (functions) ; even?
              (lambda (n) 
                (if (zero? n)
                    #t
                    (((cdr functions) functions) 
                     (1- n)))))
            (lambda (functions) ; odd?
              (lambda (n) 
                (if (zero? n)
                    #f
                    (((car functions) functions) 
                     (1- n))))))))
    ((car function-list) function-list)))
;
(mutual-even? 5)
;
; Abstract ((cdr functions) functions) out of if, etc..
;
(define mutual-even?
  (let 
    ((function-list 
      (cons (lambda (functions) 
              (lambda (n) 
                ((lambda (f)
                   (if (zero? n)
                       #t
                       (f (1- n))))
                 ((cdr functions) functions))))
            (lambda (functions) 
              (lambda (n) 
                ((lambda (f)
                   (if (zero? n)
                       #f
                       (f (1- n))))
                 ((car functions) functions)))))))
    ((car function-list) function-list)))
;
(mutual-even? 5)
;
; Massage functions into abstracted versions of 
; originals.
;
(define mutual-even?
  (let 
    ((function-list 
      (cons (lambda (functions) 
              (lambda (n) 
                (((lambda (f)
                    (lambda (n)
                      (if (zero? n)
                          #t
                          (f (1- n)))))
                  ((cdr functions) functions))
                 n)))
            (lambda (functions) 
              (lambda (n) 
                (((lambda (f)
                    (lambda (n)
                      (if (zero? n)
                          #f
                          (f (1- n)))))
                  ((car functions) functions))
                 n))))))
    ((car function-list) function-list)))
;
(mutual-even? 5)
;
; Separate abstracted functions out from recursive 
; mechanism.
;
(define mutual-even?
  (let 
    ((abstracted-functions
      (cons (lambda (f)
              (lambda (n)
                (if (zero? n)
                    #t
                    (f (1- n)))))
            (lambda (f)
              (lambda (n)
                (if (zero? n)
                    #f
                    (f (1- n))))))))
    (let 
      ((function-list 
        (cons (lambda (functions) 
                (lambda (n) 
                  (((car abstracted-functions)
                    ((cdr functions) functions))
                   n)))
              (lambda (functions) 
                (lambda (n) 
                  (((cdr abstracted-functions)
                    ((car functions) functions))
                   n))))))
      ((car function-list) function-list))))
;
(mutual-even? 5)
;
; Abstract out variable abstracted-functions in 2nd let.
;
(define mutual-even?
  (let 
    ((abstracted-functions
      (cons (lambda (f)
              (lambda (n)
                (if (zero? n)
                    #t
                    (f (1- n)))))
            (lambda (f)
              (lambda (n)
                (if (zero? n)
                    #f
                    (f (1- n))))))))
    ((lambda (abstracted-functions)
       (let 
         ((function-list 
           (cons (lambda (functions) 
                   (lambda (n) 
                     (((car abstracted-functions)
                       ((cdr functions) functions))
                      n)))
                 (lambda (functions) 
                   (lambda (n) 
                     (((cdr abstracted-functions)
                       ((car functions) functions))
                      n))))))
         ((car function-list) function-list)))
     abstracted-functions)))
;
(mutual-even? 5)
;
; Separate recursion mechanism into separate function.
;
(define y2
  (lambda (abstracted-functions)
    (let 
      ((function-list 
        (cons (lambda (functions) 
                (lambda (n) 
                  (((car abstracted-functions)
                    ((cdr functions) functions))
                   n)))
              (lambda (functions)
                (lambda (n) 
                  (((cdr abstracted-functions)
                    ((car functions) functions))
                   n))))))
      ((car function-list) function-list))))
;
(define mutual-even? 
  (y2
   (cons (lambda (f)
           (lambda (n)
             (if (zero? n)
                 #t
                 (f (1- n)))))
         (lambda (f)
           (lambda (n)
             (if (zero? n)
                 #f
                 (f (1- n))))))))
;
(mutual-even? 5)
;
; y2 has selector built into it-generalize it!
;
(define y2-choose
  (lambda (abstracted-functions)
    (lambda (selector)
      (let 
        ((function-list 
          (cons (lambda (functions) 
                  (lambda (n) 
                    (((car abstracted-functions)
                      ((cdr functions) functions))
                     n)))
                (lambda (functions)
                  (lambda (n) 
                    (((cdr abstracted-functions)
                      ((car functions) functions))
                     n))))))
        ((selector function-list) function-list)))))
;
; Now we can achieve the desired result-defining 
; both mutual-even? and mutual-odd? without recursion.
;
(define mutual-even-odd?
  (y2-choose
   (cons (lambda (f)
           (lambda (n)
             (if (zero? n)
                 #t
                 (f (1- n)))))
         (lambda (f)
           (lambda (n)
             (if (zero? n)
                 #f
                 (f (1- n))))))))
;
(define mutual-even? 
  (mutual-even-odd? car))
;
(define mutual-odd?
  (mutual-even-odd? cdr))  
;
(mutual-even? 5)
(mutual-odd? 5)
(mutual-even? 4)
(mutual-odd? 4)

Deriving Mutual Satisfaction

Notice that mutual-even? and mutual-odd? could have been defined using y2 instead of y2-choose, however, the definitional bodies of my-even? and my-odd? would have been repeated in defining mutual-even? and mutual-odd?.

Exercises for the Reader

• Herein Y2 was derived from mutual-even?. Try deriving it instead from pass-even?.

• 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: evaluation in l-calculus is normal order.)

Looking Ahead

Creating a “minimalist” (i.e., combinator based) metacircular interpreter might now be possible if we can tackle the problem of manipulating state!

Thanks to:

The local great horned owls that watch over everything from on high; regularly letting fellow “night owls” know that all is well by bellowing their calming, reassuring “Who-w-h-o-o” sounds.

Bugs/infelicities due to: burning too much midnite oil!

Bibliography and References

[CR 91] William Clinger and Jonathan Rees (editors). “Revised4 Report on the Algorithmic Language Scheme”, LISP Pointers, SIGPLAN Special Interest Publication on LISP, Volume IV, Number 3, July-September, 1991. ACM Press.

[Gabriel 88] Richard P. Gabriel. “The Why of Y”, LISP Pointers, Vol. II, Number 2, October-November-December, 1988.

[vanMeule May 91] André van Meulebrouck. “A Calculus for the Algebraic-like Manipulation of Computer Code” (Lambda Calculus), MacTutor, Anaheim, CA, May 1991.

[vanMeule Jun 91] André van Meulebrouck. “Going Back to Church” (Church numerals.), MacTutor, Anaheim, CA, June 1991.

[vanMeule May 92] André van Meulebrouck. “Deriving Miss Daze Y”, (Deriving Y), MacTutor, Los Angeles, CA, April/May 1992.

 

Community Search:
MacTech Search:

Software Updates via MacUpdate

Bookends 13.2.6 - Reference management a...
Bookends is a full-featured bibliography/reference and information-management system for students and professionals. Bookends uses the cloud to sync reference libraries on all the Macs you use.... Read more
BusyContacts 1.4.0 - 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
Chromium 77.0.3865.75 - Fast and stable...
Chromium is an open-source browser project that aims to build a safer, faster, and more stable way for all Internet users to experience the web. Version 77.0.3865.75: A list of changes is available... Read more
DiskCatalogMaker 7.5.5 - Catalog your di...
DiskCatalogMaker is a simple disk management tool which catalogs disks. Simple, light-weight, and fast Finder-like intuitive look and feel Super-fast search algorithm Can compress catalog data for... Read more
Alfred 4.0.4 - Quick launcher for apps a...
Alfred is an award-winning productivity application for OS X. Alfred saves you time when you search for files online or on your Mac. Be more productive with hotkeys, keywords, and file actions at... Read more
A Better Finder Rename 10.45 - File, pho...
A Better Finder Rename is the most complete renaming solution available on the market today. That's why, since 1996, tens of thousands of hobbyists, professionals and businesses depend on A Better... Read more
iFinance 4.5.11 - Comprehensively manage...
iFinance allows you to keep track of your income and spending -- from your lunchbreak coffee to your new car -- in the most convenient and fastest way. Clearly arranged transaction lists of all your... Read more
OmniGraffle Pro 7.11.3 - Create diagrams...
OmniGraffle Pro helps you draw beautiful diagrams, family trees, flow charts, org charts, layouts, and (mathematically speaking) any other directed or non-directed graphs. We've had people use... Read more
BBEdit 12.6.7 - Powerful text and HTML e...
BBEdit is the leading professional HTML and text editor for the Mac. Specifically crafted in response to the needs of Web authors and software developers, this award-winning product provides a... Read more
OmniGraffle 7.11.3 - Create diagrams, fl...
OmniGraffle helps you draw beautiful diagrams, family trees, flow charts, org charts, layouts, and (mathematically speaking) any other directed or non-directed graphs. We've had people use Graffle to... Read more

Latest Forum Discussions

See All

Five Nights at Freddy's AR: Special...
Five Nights at Freddy's AR: Special Delivery is a terrifying new nightmare from developer Illumix. Last week, FNAF fans were sent into a frenzy by a short teaser for what we now know to be Special Delivery. Those in the comments were quick to... | Read more »
Rush Rally 3's new live events are...
Last week, Rush Rally 3 got updated with live events, and it’s one of the best things to happen to racing games on mobile. Prior to this update, the game already had multiplayer, but live events are more convenient in the sense that it’s somewhat... | Read more »
Why your free-to-play racer sucks
It’s been this way for a while now, but playing Hot Wheels Infinite Loop really highlights a big issue with free-to-play mobile racing games: They suck. It doesn’t matter if you’re trying going for realism, cart racing, or arcade nonsense, they’re... | Read more »
Steam Link Spotlight - The Banner Saga 3
Steam Link Spotlight is a new feature where we take a look at PC games that play exceptionally well using the Steam Link app. Our last entry talked about Terry Cavanaugh’s incredible Dicey Dungeons. Read about how it’s a great mobile experience... | Read more »
PSA: GRIS has some issues
You may or may not have seen that Devolver Digital just released GRIS on the App Store, but we wanted to do a quick public service announcement to say that you might not want to hop on buying it just yet. The puzzle platformer has come to small... | Read more »
Explore the world around you in new matc...
Got a hankering for a fresh-feeling Match-3 puzzle game that offers a unique twist? You might find exactly what you’re looking for with What a Wonderful World, a new spin on the classic mobile genre which merges entertaining puzzles with global... | Read more »
Combo Quest (Games)
Combo Quest 1.0 Device: iOS Universal Category: Games Price: $.99, Version: 1.0 (iTunes) Description: Combo Quest is an epic, time tap role-playing adventure. In this unique masterpiece, you are a knight on a heroic quest to retrieve... | Read more »
Hero Emblems (Games)
Hero Emblems 1.0 Device: iOS Universal Category: Games Price: $2.99, Version: 1.0 (iTunes) Description: ** 25% OFF for a limited time to celebrate the release ** ** Note for iPhone 6 user: If it doesn't run fullscreen on your device... | Read more »
Puzzle Blitz (Games)
Puzzle Blitz 1.0 Device: iOS Universal Category: Games Price: $1.99, Version: 1.0 (iTunes) Description: Puzzle Blitz is a frantic puzzle solving race against the clock! Solve as many puzzles as you can, before time runs out! You have... | Read more »
Sky Patrol (Games)
Sky Patrol 1.0.1 Device: iOS Universal Category: Games Price: $1.99, Version: 1.0.1 (iTunes) Description: 'Strategic Twist On The Classic Shooter Genre' - Indie Game Mag... | Read more »

Price Scanner via MacPrices.net

Sunday Sale! 2019 27″ 5K 6-Core iMacs for $20...
B&H Photo has the new 2019 27″ 5K 6-Core iMacs on stock today and on sale for up to $250 off Apple’s MSRP. Overnight shipping is free to many locations in the US. These are the same iMacs sold by... Read more
Weekend Sale! 2019 13″ MacBook Airs for $200...
Amazon has new 2019 13″ MacBook Airs on sale for $200 off Apple’s MSRP, with prices starting at $899, each including free shipping. Be sure to select Amazon as the seller during checkout, rather than... Read more
2019 15″ MacBook Pros now on sale for $350-$4...
B&H Photo has Apple’s 2019 15″ 6-Core and 8-Core MacBook Pros on sale today for $350-$400 off MSRP, starting at $2049, with free overnight shipping available to many addresses in the US: – 2019... Read more
Buy one Apple Watch Series 5 at Verizon, get...
Buy one Apple Watch Series 5 at Verizon, and get a second Watch for 50% off. Plus save $10 on your first month of service. The fine print: “Buy Apple Watch, get another up to 50% off on us. Plus $10... Read more
Sprint offers 64GB iPhone 11 for free to new...
Sprint will include the 64GB iPhone 11 for free for new customers with an eligible trade-in in of the iPhone 7 or newer through September 19, 2019. The fine print: “iPhone 11 64GB $0/mo. iPhone 11... Read more
Verizon offers new iPhone 11 models for up to...
Verizon is offering Apple’s new iPhone 11 models for $500 off MSRP to new customers with an eligible trade-in (see list below). Discount is applied via monthly bill credits over 24 months. Verizon is... Read more
AT&T offers free $300 reward card + free...
AT&T Wireless will include a second free 64GB iPhone 11 with the purchase of one eligible iPhone at full price. They will also include a free $300 rewards card. The fine print: “Buy an elig.... Read more
US Cellular offers 64GB iPhone 11 for free to...
US Cellular is offering the base 64GB iPhone 11 for free for new customers. Qualified trade-in of iPhone 7 or higher required (or a number of Android phones). Discounts are applied via monthly bill... Read more
New 7th generation 10.2″ 128GB iPads availabl...
Amazon is accepting preorders for Apple’s new 7th generation 10.2″ 128GB iPads for $399.99 each, or $30 off Apple’s MSRP for this model. Shipping is free: – 10.2″ 128GB WiFi iPad Space Gray: $399.99... Read more
Sprint has the new 7th Generation iPad on sal...
Sprint has the new 2019 7th Generation 32GB WiFi + Cellular iPad available starting at only $99.99 from 9/12/19 to 10/3/19. Their price is a $360 savings over Apple’s standard MSRP. See the deal live... Read more

Jobs Board

Geek Squad *Apple* Master Consultation Agen...
**732131BR** **Job Title:** Geek Squad Apple Master Consultation Agent **Job Category:** Services/Installation/Repair **Location Number:** 000399-Wausau-Store **Job Read more
*Apple* Mobility Pro - Best Buy (United Stat...
**723452BR** **Job Title:** Apple Mobility Pro **Job Category:** Store Associates **Location Number:** 001194-Greeley-Store **Job Description:** At Best Buy, our Read more
Best Buy *Apple* Computing Master - Best Bu...
**732027BR** **Job Title:** Best Buy Apple Computing Master **Job Category:** Store Associates **Location Number:** 002507-Alexandria-Store **Job Description:** The Read more
Best Buy *Apple* Computing Master - Best Bu...
**727669BR** **Job Title:** Best Buy Apple Computing Master **Job Category:** Sales **Location Number:** 000890-Buckhead-Store **Job Description:** **What does a Read more
Geek Squad *Apple* Master Consultation Agen...
**731944BR** **Job Title:** Geek Squad Apple Master Consultation Agent **Job Category:** Services/Installation/Repair **Location Number:** 001130-Nashville Read more
All contents are Copyright 1984-2011 by Xplain Corporation. All rights reserved. Theme designed by Icreon.