Why Y works

8/18/2019 Why Y works

1/10

2

A L e c t u r e o n t h e W h y o f Y

b y M a t t h i a s F e l l e i s e n


2/10

0 . 1 . ( Y Y ) W O R K S ! 3

I s t h i s t h e f u n c t i o n l e n g t h ?

( d e n e l e n g t h

( l a m b d a ( l )

( i f ( n u l l ? l ) 0

( a d d 1 ( l e n g t h ( c d r l ) ) ) ) ) )

I t s u r e i s .

S u p p o s e ( d e n e . . . . . . ) n o l o n g e r w o r k s .

C a n y o u d e s c r i b e i n y o u r o w n w o r d s w h a t

l e n g t h d o e s ?

F o r o n e , t h e b o d y o f l e n g t h c a n n o t r e f e r t o

l e n g t h

T h e n w e m i g h t a s w e l l w r i t e s o m e t h i n g l i k e

t h i s .

( d e n e l e n g t h

( l a m b d a ( l )

( i f ( n u l l ? l ) 0

( a d d 1 ( h u k a i r s ( c d r l ) ) ) ) ) )

Y e s , e x c e p t t h a t ( d e n e . . . . . . ) d o e s n ' t

w o r k a n y m o r e .

S o p e r h a p s s o m e t h i n g m o r e l i k e t h i s ?

( l a m b d a ( l )

( i f ( n u l l ? l ) 0

( a d d 1 ( h u k a i r s ( c d r l ) ) ) ) )

Y e s , t h a t ' s b e t t e r .

B u t w h a t h a p p e n e d t o t h e f u n c t i o n ? I t i s n o l o n g e r r e c u r s i v e .

A n d w h a t d o e s i t d o ? I t m e a s u r e s t h e l e n g t h o f t h e e m p t y l i s t a n d

n o t h i n g e l s e .

A n d w h a t d o e s h u h a i r s d o ? W h o c a r e s . T h e f u n c t i o n d o e s n ' t w o r k f o r

n o n - e m p t y l i s t s i n a n y c a s e .

S u p p o s e w e c o u l d n a m e t h i s n e w f u n c t i o n .

W h a t w o u l d b e a g o o d n a m e ?

W e t h i n k l e n g t h

0

i s g r e a t b e c a u s e t h e

f u n c t i o n o n l y m e a s u r e s l i s t s o f l e n g t h 0 .


3/10

4

H o w w o u l d y o u w r i t e a f u n c t i o n t h a t

m e a s u r e s t h e l e n g t h o f l i s t s t h a t c o n t a i n o n e

i t e m ?

W e l l , w e c o u l d t r y t h e f o l l o w i n g .

( l a m b d a ( l )

( i f ( n u l l ? l ) 0

( a d d 1 ( l e n g t h

0

( c d r l ) ) ) ) )

A l m o s t , b u t ( d e n e . . . . . . ) d o e s n ' t w o r k f o r

l e n g t h

0

S o ? R e p l a c e l e n g t h

0

b y i t s d e n i t i o n .

( l a m b d a ( l )

( i f ( n u l l ? l ) 0

( a d d 1

( ( l a m b d a ( l )

( i f ( n u l l ? l ) 0

( a d d 1 ( h u k a i r s ( c d r l ) ) ) ) )

( c d r l ) ) ) ) )

A n d w h a t ' s a g o o d n a m e f o r t h i s f u n c t i o n ? T h a t ' s e a s y : l e n g t h

1

I s t h i s t h e f u n c t i o n t h a t w o u l d m e a s u r e t h e

l e n g h t o f l i s t s t h a t c o n t a i n t w o i t e m s ?

( l a m b d a ( l )

( i f ( n u l l ? l ) 0

( a d d 1

( ( l a m b d a ( l )

( i f ( n u l l ? l ) 0

( a d d 1

( ( l a m b d a ( l )

( i f ( n u l l ? l ) 0

( a d d 1

( h u k a i r s

( c d r l ) ) ) ) )

( c d r l ) ) ) ) )

( c d r l ) ) ) ) )

Y e s , t h i s i s l e n g t h

2

. W e j u s t e x p a n d t h e c a l l

t o h u k a i r s t o g e t t h e n e x t v e r s i o n o f l e n g t h

N o w , w h a t d o y o u t h i n k r e c u r s i o n i s ? W h a t d o y o u m e a n ?


4/10

0 . 1 . ( Y Y ) W O R K S ! 5

W e l l , w e h a v e s e e n h o w t o m e a s u r e t h e l i s t

w i t h n o i t e m s , w i t h o n e i t e m , w i t h t w o , a n d

s o o n . H o w c o u l d w e g e t t h e f u n c t i o n l e n g t h

b a c k ?

I f w e c o u l d w r i t e a n i n n i t e f u n c t i o n , w e

c o u l d w r i t e l e n g t h

1

B u t w e c a n ' t w r i t e a n i n n i t e f u n c t i o n . A n d w e s t i l l h a v e a l l t h e s e r e p e t i t i o n s a n d

p a t t e r n s i n t h e s e f u n c t i o n s .

A l l t h e s e p r o g r a m s c o n t a i n a f u n c t i o n t h a t

l o o k s l i k e l e n g t h , a n d t h a t ' s n o t r i g h t .

N o , l e t ' s a b s t r a c t o u t t h e s e p a t t e r n s .

I s t h i s t h e r i g h t w a y t o r e w r i t e l e n g t h

0

s o

t h a t l e n g t h r e a p p e a r s ?

( ( l a m b d a ( l e n g t h )

( l a m b d a ( l )

( i f ( n u l l ? l ) 0

( a d d 1 ( l e n g t h ( c d r l ) ) ) ) ) )

h u k a i r s )

I t ' s w o r t h a t r y .

R e w r i t e l e n g t h

1

i n t h e s a m e s t y l e .

( ( l a m b d a ( l e n g t h )

( l a m b d a ( l )

( i f ( n u l l ? l ) 0

( a d d 1 ( l e n g t h ( c d r l ) ) ) ) ) )

( ( l a m b d a ( l e n g t h )

( l a m b d a ( l )

( i f ( n u l l ? l ) 0

( a d d 1 ( l e n g t h ( c d r l ) ) ) ) ) )

h u k a i r s ) )


5/10

6

A n d l e n g t h

2

( ( l a m b d a ( l e n g t h )

( l a m b d a ( l )

( i f ( n u l l ? l ) 0

( a d d 1 ( l e n g t h ( c d r l ) ) ) ) ) )

( ( l a m b d a ( l e n g t h )

( l a m b d a ( l )

( i f ( n u l l ? l ) 0

( a d d 1 ( l e n g t h ( c d r l ) ) ) ) ) )

( ( l a m b d a ( l e n g t h )

( l a m b d a ( l )

( i f ( n u l l ? l ) 0

( a d d 1 ( l e n g t h ( c d r l ) ) ) ) ) )

h u k a i r s ) ) )

C l o s e , b u t t h e r e a r e s t i l l r e p e t i t i o n s . T r u e . L e t ' s g e t r i d o f t h e m .

W h e r e s h o u l d w e s t a r t ? N a m e t h e f u n c t i o n t h a t t a k e s l e n g t h a s a n

a r g u m e n t a n d t h a t r e t u r n s a f u n c t i o n t h a t

l o o k s l i k e l e n g t h

W h a t ' s a g o o d n a m e f o r t h i s f u n c t i o n ? W h a t a b o u t m k - l e n g t h f o r \ m a k e l e n g t h " ?

O k , d o t h i s t o l e n g t h

0

. N o p r o b l e m .

( ( l a m b d a ( m k - l e n g t h )

( m k - l e n g t h h u k a i r s ) )

( l a m b d a ( l e n g t h )

( l a m b d a ( l )

( i f ( n u l l ? l ) 0

( a d d 1 ( l e n g t h ( c d r l ) ) ) ) ) ) )


6/10

0 . 1 . ( Y Y ) W O R K S ! 7

I s t h i s l e n g t h

1

?

( ( l a m b d a ( m k - l e n g t h )

( m k - l e n g t h

( m k - l e n g t h h u k a i r s ) ) )


( l a m b d a ( l )

( i f ( n u l l ? l ) 0

( a d d 1 ( l e n g t h ( c d r l ) ) ) ) ) ) )

I t s u r e i s . A n d t h i s i s l e n g t h

2

( ( l a m b d a ( m k - l e n g t h )

( m k - l e n g t h

( m k - l e n g t h

( m k - l e n g t h h u k a i r s ) ) ) )


( l a m b d a ( l )

( i f ( n u l l ? l ) 0

( a d d 1 ( l e n g t h ( c d r l ) ) ) ) ) ) )

C a n y o u d o l e n g t h

3

? H e r e w e g o .

( ( l a m b d a ( m k - l e n g t h )

( m k - l e n g t h

( m k - l e n g t h

( m k - l e n g t h

( m k - l e n g t h h u k a i r s ) ) ) ) )


( l a m b d a ( l )

( i f ( n u l l ? l ) 0

( a d d 1 ( l e n g t h ( c d r l ) ) ) ) ) ) )

S o w h a t i s r e c u r s i o n ? I t i s l i k e a n i n n i t e t o w e r o f a p p l i c a t i o n s o f

m k - l e n g t h t o a n a r b i t r a r y f u n c t i o n .

D o w e r e a l l y n e e d a n i n n i t e t o w e r ? N o t r e a l l y o f c o u r s e . E v e r y t i m e w e u s e l e n g t h

w e o n l y n e e d a n i t e n u m b e r , b u t w e n e v e r

k n o w h o w m a n y .

C o u l d w e g u e s s h o w m a n y w e n e e d ? S u r e , b u t w e m a y n o t g u e s s a l a r g e e n o u g h

n u m b e r .

W h e n d o w e n d o u t t h a t w e d i d n ' t g u e s s a

l a r g e e n o u g h n u m b e r ?

W h e n w e a p p l y t h e f u n c t i o n h u k a i r s t h a t i s

p a s s e d t o t h e r s t m k - l e n g t h

W h a t i f w e c o u l d c r e a t e a n o t h e r a p p l i c a t i o n

o f m k - l e n g t h t o h u h a i r s a t t h i s p o i n t ?

T h a t w o u l d p o s t p o n e t h e p r o b l e m b y o n e ,

a n d b e s i d e s , h o w c o u l d w e d o t h a t ?


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8

W e l l , s i n c e n o b o d y c a r e s w h a t f u n c t i o n w e

p a s s t o m k - l e n g t h , w e c o u l d p a s s i t m k - l e n g t h

i n i t i a l l y .

T h a t ' s t h e r i g h t i d e a . A n d t h e n w e i n v o k e

m k - l e n g t h o n h u h a i r s a n d t h e r e s u l t o f t h i s

o n t h e c d r s o t h a t w e g e t o n e m o r e p i e c e o f

t h e t o w e r .

T h e n t h i s i s s t i l l l e n g t h

0

?

( ( l a m b d a ( m k - l e n g t h )

( m k - l e n g t h m k - l e n g t h ) )

( l a m b d a ( m k - l e n g t h )

( l a m b d a ( l )

( i f ( n u l l ? l ) 0

( a d d 1 ( m k - l e n g t h ( c d r l ) ) ) ) ) ) )

Y e s . A n d w h e n w e a p p l y m k - l e n g t h o n c e , w e

g e t l e n g t h

1

( ( l a m b d a ( m k - l e n g t h )



( l a m b d a ( l )

( i f ( n u l l ? l ) 0

( a d d 1 ( ( m k - l e n g t h h u k a i r s )

( c d r l ) ) ) ) ) ) )

C o u l d w e d o t h i s m o r e t h a n o n c e ? Y e s , j u s t k e e p p a s s i n g m k - l e n g t h t o i t s e l f ,

a n d w e c a n d o t h i s a s o f t e n a s w e n e e d t o !

W h a t w o u l d y o u c a l l t h i s f u n c t i o n ?

( ( l a m b d a ( m k - l e n g t h )



( l a m b d a ( l )

( i f ( n u l l ? l ) 0

( a d d 1 ( ( m k - l e n g t h m k - l e n g t h )

( c d r l ) ) ) ) ) ) )

I t i s l e n g t h , o f c o u r s e .

E x c e p t t h a t i t n o l o n g e r c o n t a i n s t h e

f u n c t i o n t h a t l o o k s l i k e l e n g t h . C a n w e x

t h a t ?

W e c o u l d e x t r a c t t h e s e l f - a p p l i c a t i o n o f

m k - l e n g t h a n d c a l l i t l e n g t h

W h y ? B e c a u s e i t r e a l l y m a k e s t h e f u n c t i o n l e n g t h

B u t ( m k - l e n g t h m k - l e n g t h ) i s n o t a f u n c t i o n .

I t o n l y r e t u r n s a f u n c t i o n .

S o w h a t d o w e d o ?


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0 . 1 . ( Y Y ) W O R K S ! 9

M a k e t h e s e l f - a p p l i c a t i o n o f m k - l e n g t h i n t o a

f u n c t i o n .

N o p r o b l e m , w e j u s t u s e t h e o l d t r i c k o f

w r a p p i n g a l a m b d a a r o u n d t h e a p p l i c a t i o n .

A f t e r a l l , t h e s e l f - a p p l i c a t i o n d o e s r e t u r n a

f u n c t i o n !

W h i c h f u n c t i o n ? T h e f u n c t i o n l e n g t h . R e m e m b e r ?

O k , d o i t !

( ( l a m b d a ( m k - l e n g t h )



( l a m b d a ( l )

( i f ( n u l l ? l ) 0

( a d d 1

( ( l a m b d a ( x )

( ( m k - l e n g t h m k - l e n g t h ) x ) )

( c d r l ) ) ) ) ) ) )

M o v e o u t t h e n e w f u n c t i o n s o t h a t w e g e t

l e n g t h b a c k .

( ( l a m b d a ( m k - l e n g t h )



( ( l a m b d a ( l e n g t h )

( l a m b d a ( l )

( i f ( n u l l ? l ) 0

( a d d 1 ( l e n g t h ( c d r l ) ) ) ) ) )

( l a m b d a ( x )

( ( m k - l e n g t h m k - l e n g t h ) x ) ) ) ) )

I s t h i s o k t o d o ? Y e s . T h i n k a b o u t i t . W e a l w a y s d i d t h e

r e v e r s e : W h e n w e k n e w w h a t t h e a r g u m e n t

t o a f u n c t i o n w a s , w e p r o c e e d e d w i t h t h e

f u n c t i o n b o d y a n d u s e d t h e a r g u m e n t v a l u e

w h e n e v e r w e s a w t h e p a r a m e t e r n a m e .

C a n w e e x t r a c t t h e f u n c t i o n t h a t l o o k s l i k e

l e n g t h a n d g i v e i t a n a m e ?

Y e s , i t d o e s n o t d e p e n d o n m k - l e n g t h a t a l l !


9/10

1 0

I s t h i s t h e r i g h t f u n c t i o n ?

( ( l a m b d a ( l e )

( ( l a m b d a ( m k - l e n g t h )



( l e ( l a m b d a ( x )

( ( m k - l e n g t h m k - l e n g t h ) x ) ) ) ) ) )


( l a m b d a ( l )

( i f ( n u l l ? l ) 0

( a d d 1 ( l e n g t h ( c d r l ) ) ) ) ) ) )

Y e s .

W h a t d i d w e a c t u a l l y g e t b a c k ? W e e x t r a c t e d t h e o l d f u n c t i o n m k - l e n g t h

L e t ' s s e p a r a t e t h e f u n c t i o n t h a t m a k e s l e n g t h

f r o m t h e f u n c t i o n t h a t l o o k s l i k e l e n g t h

T h a t ' s e a s y .

( l a m b d a ( l e )

( ( l a m b d a ( m k - l e n g t h )



( l e ( l a m b d a ( x )

( ( m k - l e n g t h m k - l e n g t h ) x ) ) ) ) ) )

D o e s t h i s f u n c t i o n h a v e a n a m e ? Y e s , i t i s c a l l e d t h e a p p l i c a t i v e - o r d e r Y

c o m b i n a t o r . A t l e a s t , w h e n w e r e w r i t e i t a

l i t t l e b i t .

( d e n e Y

( l a m b d a ( l e )

( ( l a m b d a ( f )

( l e ( l a m b d a ( x ) ( ( f f ) x ) ) ) )

( l a m b d a ( f )

( l e ( l a m b d a ( x ) ( ( f f ) x ) ) ) ) ) ) )

D o e s ( d e n e . . . . . . ) w o r k a g a i n ?

S u r e , n o w t h a t w e k n o w w h a t r e c u r s i o n i s .


10/10

0 . 1 . ( Y Y ) W O R K S ! 1 1

D o y o u n o w k n o w w h y Y w o r k s ? R e a d t h i s c h a p t e r o n e m o r e t i m e a n d y o u

w i l l .

D o e s ( Y Y ) w o r k , t o o ?

A n d h o w i t w o r k s ! ! !

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Why Y works