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

Tor Browser Bundle 7.0.7 - Anonymize Web...
The Tor Browser Bundle is an easy-to-use portable package of Tor, Vidalia, Torbutton, and a Firefox fork preconfigured to work together out of the box. It contains a modified copy of Firefox that... Read more
Data Rescue 5.0.1 - Powerful hard drive...
Data Rescue’s new and improved features let you scan, search, and recover your files faster than ever before. We have modernized the file-preview capabilities, added new files types to the recovery... Read more
Alfred 3.5.1 - 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
Tunnelblick 3.7.3 - GUI for OpenVPN.
Tunnelblick is a free, open source graphic user interface for OpenVPN on OS X. It provides easy control of OpenVPN client and/or server connections. It comes as a ready-to-use application with all... Read more
DEVONthink Pro 2.9.16 - Knowledge base,...
Save 10% with our exclusive coupon code: MACUPDATE10 DEVONthink Pro is your essential assistant for today's world, where almost everything is digital. From shopping receipts to important research... Read more
AirRadar 4.0 - $9.95
With AirRadar, scanning for wireless networks is now easier and more personalized! It allows you to scan for open networks and tag them as favourites or filter them out. View detailed network... Read more
ForkLift 3.0.8 Beta - Powerful file mana...
ForkLift is a powerful file manager and ferociously fast FTP client clothed in a clean and versatile UI that offers the combination of absolute simplicity and raw power expected from a well-executed... Read more
Opera 48.0.2685.50 - High-performance We...
Opera is a fast and secure browser trusted by millions of users. With the intuitive interface, Speed Dial and visual bookmarks for organizing favorite sites, news feature with fresh, relevant content... Read more
FotoMagico 5.5 - Powerful slideshow crea...
FotoMagico lets you create professional slideshows from your photos and music with just a few, simple mouse clicks. It sports a very clean and intuitive yet powerful user interface. High image... Read more
Adobe Audition CC 2018 11.0.0 - Professi...
Audition CC 2018 is available as part of Adobe Creative Cloud for as little as $19.99/month (or $9.99/month if you're a previous Audition customer). Adobe Audition CC 2018 empowers you to create and... Read more

Wheels of Aurelia (Games)
Wheels of Aurelia 1.0.1 Device: iOS Universal Category: Games Price: $3.99, Version: 1.0.1 (iTunes) Description: | Read more »
Halcyon 6: Starbase Commander guide - ti...
Halcyon 6 is a well-loved indie RPG with stellar tactical combat and some pretty good writing, too. It's now landed on the App Store, so mobile fans, if you're itching for a good intergalactic adventure, here's your game. Being a strategy RPG, the... | Read more »
Game of Thrones: Conquest guide - how to...
Fans of base building games might be excited to know that yet another entry in the genre has materialized - Game of Thrones: Conquest. Yes, you can now join the many kingdoms of the famed book series, or create your own, as you try to conquer... | Read more »
Halcyon 6: Starbase Commander (Games)
Halcyon 6: Starbase Commander 1.4.2.0 Device: iOS Universal Category: Games Price: $6.99, Version: 1.4.2.0 (iTunes) Description: An epic space strategy RPG with base building, deep tactical combat, crew management, alien diplomacy,... | Read more »
Legacy of Discord celebrates its 1 year...
It’s been a thrilling first year for fans of Legacy of Discord, the stunning PvP dungeon-crawling ARPG from YOOZOO Games, and now it’s time to celebrate the game’s first anniversary. The developers are amping up the festivities with some exciting... | Read more »
3 reasons to play Thunder Armada - the n...
The bygone days of the Battleship board game might have past, but naval combat simulators still find an audience on mobile. Thunder Armada is Chinese developer Chyogames latest entry into the genre, drawing inspiration from the explosive exchanges... | Read more »
Experience a full 3D fantasy MMORPG, as...
Those hoping to sink their teeth into a meaty hack and slash RPG that encourages you to fight with others might want to check out EZFun’s new Eternity Guardians. Available to download for iOS and Android, Eternity Guardians is an MMORPG that lets... | Read more »
Warhammer Quest 2 (Games)
Warhammer Quest 2 1.0 Device: iOS Universal Category: Games Price: $4.99, Version: 1.0 (iTunes) Description: Dungeon adventures in the Warhammer World are back! | Read more »
4 of the best Halloween updates for mobi...
Halloween is certainly one of our favorite times for mobile game updates. Many popular titles celebrate this spooky season with fun festivities that can stretch from one week to even the whole month. As we draw closer and closer to Halloween, we'... | Read more »
Fire Rides guide - how to swing to succe...
It's another day, which means another Voodoo game has come to glue our hands to our mobile phones. Yes, it's been an especially prolific month for this particular mobile publisher, but we're certainly not complaining. Fire Rides is yet another... | Read more »

Price Scanner via MacPrices.net

Apple restocks full line of refurbished 13″ M...
Apple has restocked a full line of Apple Certified Refurbished 2017 13″ MacBook Pros for $200-$300 off MSRP. A standard Apple one-year warranty is included with each MacBook, and shipping is free.... Read more
13″ 3.1GHz/256GB MacBook Pro on sale for $167...
Amazon has the 2017 13″ 3.1GHz/256GB Space Gray MacBook Pro on sale today for $121 off MSRP including free shipping: – 13″ 3.1GHz/256GB Space Gray MacBook Pro (MPXV2LL/A): $1678 $121 off MSRP Keep an... Read more
13″ MacBook Pros on sale for up to $120 off M...
B&H Photo has 2017 13″ MacBook Pros in stock today and on sale for up to $120 off MSRP, each including free shipping plus NY & NJ sales tax only: – 13-inch 2.3GHz/128GB Space Gray MacBook... Read more
15″ MacBook Pros on sale for up to $200 off M...
B&H Photo has 15″ MacBook Pros on sale for up to $200 off MSRP. Shipping is free, and B&H charges sales tax in NY & NJ only: – 15″ 2.8GHz MacBook Pro Space Gray (MPTR2LL/A): $2249, $150... Read more
Roundup of Apple Certified Refurbished iMacs,...
Apple has a full line of Certified Refurbished 2017 21″ and 27″ iMacs available starting at $1019 and ranging up to $350 off original MSRP. Apple’s one-year warranty is standard, and shipping is free... Read more
Sale! 27″ 3.8GHz 5K iMac for $2098, save $201...
Amazon has the 27″ 3.8GHz 5K iMac (MNED2LL/A) on sale today for $2098 including free shipping. Their price is $201 off MSRP, and it’s the lowest price available for this model (Apple’s $1949... Read more
Sale! 10″ Apple WiFi iPad Pros for up to $100...
B&H Photo has 10.5″ WiFi iPad Pros in stock today and on sale for $50-$100 off MSRP. Each iPad includes free shipping, and B&H charges sales tax in NY & NJ only: – 10.5″ 64GB iPad Pro: $... Read more
Apple iMacs on sale for up to $130 off MSRP w...
B&H Photo has 21-inch and 27-inch iMacs in stock and on sale for up to $130 off MSRP including free shipping. B&H charges sales tax in NY & NJ only: – 27″ 3.8GHz iMac (MNED2LL/A): $2179 $... Read more
2017 3.5GHz 6-Core Mac Pro on sale for $2799,...
B&H Photo has the 2017 3.5GHz 6-Core Mac Pro (MD878LL/A) on sale today for $2799 including free shipping plus NY & NJ sales tax only . Their price is $200 off MSRP. Read more
12″ 1.2GHz Space Gray MacBook on sale for $11...
Amazon has the 2017 12″ 1.2GHz Space Gray Retina MacBook on sale for $100 off MSRP. Shipping is free: 12″ 1.2GHz Space Gray MacBook: $1199.99 $100 off MSRP Read more

Jobs Board

Product Manager - *Apple* Pay on the *Appl...
Job Summary Apple is looking for a talented product manager to drive the expansion of Apple Pay on the Apple Online Store. This position includes a unique Read more
*Apple* Retail - Multiple Positions - Farmin...
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
Frameworks Engineer, *Apple* Watch - Apple...
Job Summary Join the team that is shaping the future of software development for Apple Watch! As a software engineer on the Apple Watch Frameworks team you will Read more
*Apple* News Product Marketing Mgr., Publish...
Job Summary The Apple News Product Marketing Manager will work closely with a cross-functional group to assist in defining and marketing new features and services. Read more
Fraud Analyst, *Apple* Advertising Platform...
Job Summary Apple Ad Platforms has an opportunity to redefine advertising on mobile devices. Apple reaches hundreds of millions of iPhone, iPod touch, and iPad Read more
All contents are Copyright 1984-2011 by Xplain Corporation. All rights reserved. Theme designed by Icreon.