Pablo Arrighi and Alejandro Díaz-Caro- A System F accounting for scalars

8/3/2019 Pablo Arrighi and Alejandro Daz-Caro- A System F accounting for scalars

1/101

A S y s t e m F a c c o u n t i n g f o r s c a l a r s

a r X i v : 0 9 0 3 . 3 7 4 1

P a b l o A r r i g h i a n d A l e j a n d r o D a z - C a r o

U n i v e r s i t d e G r e n o b l e

L a b o r a t o i r e d ' I n f o r m a t i q u e d e G r e n o b l e

N o v e m b e r 1 9 t h , 2 0 0 9 . P P S ( P a r i s )


2/101

M o t i v a t i o n L i n e a r - A l g e b r a i c

- C a l c u l u s S y s t e m F T h e s c a l a r t y p e s y s t e m C o n c l u s i o n s a n d f u t u r e w o r k

G l o b a l c o n t e x t

O d d i t y o f Q u a n t u m t h e o r y = Q u a n t u m L o g i c ? 1 ( d e v e l o p e d a d h o c b e f o r e q u a n t u m c o m p u t i n g , n o c l e a r r e l a t i o n w i t h

q u a n t u m p r o g r a m s ) .

1

B i r k h o , G . a n d J . v o n N e u m a n n , T h e l o g i c o f q u a n t u m m e c h a n i c s ,

A n n a l s o f M a t h e m a t i c s 3 7 ( 1 9 3 6 ) , p p . 8 2 3 8 4 3 .

P a b l o A r r i g h i a n d A l e j a n d r o D a z - C a r o , L I G A S y s t e m F a c c o u n t i n g f o r s c a l a r s , a r X i v : 0 9 0 3 . 3 7 4 1 2 / 3 1


3/101






M o d e l s o f L i n e a r L o g i c s =

Q u a n t u m T h e o r y ? ( C o h e r e n t

s p a c e s , M i c r o m e c h a n i c s l o s e s d u p l i c a b i l i t y . )

1





4/101









C u r r y - H o w a r d : ( p r o g r a m s , t y p e s ) =( p r o o f s , l o g i c s ) . Q u a n t u m C o m p u t a t i o n : ( q u a n t u m p r o g r a m s , q u a n t u m

t y p e s ) .

C H + Q C : ( q u a n t u m t h . p r o o f s , q u a n t u m t h . l o g i c s ) ?

Q u a n t u m l o g i c s : i s o l a t i n g t h e r e a s o n i n g b e h i n d q u a n t u m

a l g o r i t h m s ?

1





5/101









C u r r y - H o w a r d : ( p r o g r a m s , t y p e s ) =( p r o o f s , l o g i c s ) . Q u a n t u m C o m p u t a t i o n : ( q u a n t u m p r o g r a m s , q u a n t u m

t y p e s ) .

C H + Q C : ( q u a n t u m t h . p r o o f s , q u a n t u m t h . l o g i c s ) ?

Q u a n t u m l o g i c s : i s o l a t i n g t h e r e a s o n i n g b e h i n d q u a n t u m

a l g o r i t h m s ?

W h a t a r e q u a n t u m t y p e s ?

1





6/101



Q u a n t u m t h e o r y

S t a t e s a r e ( n o r m a l i z e d ) v e c t o r s v .

V e c t o r s p a c e o f o . n . b . ( bi

) . T h e n v =

i

i

b

i

.



7/101






) . T h e n v =

i

i

b

i

.

E v o l u t i o n s a r e ( u n i t a r y ) l i n e a r o p e r a t o r s U .

v

=U v .



8/101






) . T h e n v =

i

i

b

i

.


v

=U v .

S y s t e m s a r e p u t n e x t t o o n e a n o t h e r w i t h

.

B i l i n e a r j u s t l i k e a p p l i c a t i o n :

u + v w = u w + v w ,u v + w = u v + u w , . . .



9/101






) . T h e n v =

i

i

b

i

.


v

=U v .

S y s t e m s a r e p u t n e x t t o o n e a n o t h e r w i t h

.

B i l i n e a r j u s t l i k e a p p l i c a t i o n :

u + v w = u w + v w ,u v + w = u v + u w , . . .

N o - c l o n i n g t h e o r e m !



10/101



N o - c l o n i n g t h e o r e m

S t a t e m e n t :

U /

v : U v

=v

v .

P r o o f :

V e c t o r s p a c e o f o . n . b . (

b

i

), s o v

=

i

i

b

i

. W e c a n h a v e

U b

i

= bi

bi

( = c o p y i n g , O K )

B u t t h e n

U v = U

i

i

b

i

=

i

i

U b

i

=

i

i

b

i

b

i

=

i j

i

j

b

i

b

j

= (i

i

b

i

) ( j

j

b

j

)

= v v ( = c l o n i n g , N o t O K )





11/101

N o - c l o n i n g t h e o r e m

S t a t e m e n t :

U /

v : U v

=v

v .

P r o o f :

V e c t o r s p a c e o f o . n . b . (

b

i

), s o v

=

i

i

b

i

. W e c a n h a v e

U b

i

= bi

bi

( = c o p y i n g , O K )

B u t t h e n

U v = U

i

i

b

i

=

i

i

U b

i

=

i

i

b

i

b

i

=

i j

i

j

b

i

b

j

= (i

i

b

i

) ( j

j

b

j

)

= v v ( = c l o n i n g , N o t O K )

C o n i c t s w i t h - r e d u c t i o n ?





12/101

L i n e a r - A l g e b r a i c - C a l c u l u s 2

T h e l a n g u a g e

H i g h e r - o r d e r c o m p u t a t i o n

t ::= x |x . t | (t t ) |

2

A r r i g h i , P . a n d G . D o w e k . L i n e a r - a l g e b r a i c - c a l c u l u s : h i g h e r - o r d e r ,

e n c o d i n g s a n d c o n u e n c e . L e c t u r e N o t e s i n C o m p u t e r S c i e n c e ( R T A ' 0 8 ) , 5 1 1 7

( 2 0 0 8 ) , p p . 1 7 3 1 .





13/101




t ::= x |x . t | (t t ) |L i n e a r a l g e b r a

t + t |. t | 0

2



( 2 0 0 8 ) , p p . 1 7 3 1 .


M o t i v a t i o n L i n e a r - A l g e b r a i c - C a l c u l u s S y s t e m F T h e s c a l a r t y p e s y s t e m C o n c l u s i o n s a n d f u t u r e w o r k


14/101




t ::= x |x . t | (t t ) |

x .t b t [ b /x ] ()

( * ) b a n a b s t r a c t i o n o r a

v a r i a b l e .

L i n e a r a l g e b r a

t + t |. t | 0

2



( 2 0 0 8 ) , p p . 1 7 3 1 .




15/101




t ::= x |x . t | (t t ) |

x .t b t [ b /x ] ()

( * ) b a n a b s t r a c t i o n o r a

v a r i a b l e .

( * * ) u c l o s e d n o r m a l .

( * * * ) u a n d u+

v c l o s e d n o r m a l .


t + t |. t | 0

E l e m e n t a r y r u l e s s u c h a s

u+

0 u a n d.( u + v ) . u + . v .

F a c t o r i s a t i o n r u l e s s u c h a s

. u +. u (+).u . ( * * )

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

u ( v + w ) ( u v ) + ( u w ) .( * * * )

2



( 2 0 0 8 ) , p p . 1 7 3 1 .




16/101

L i n e a r - A l g e b r a i c - C a l c u l u s W h y t h e r e s t r i c t i o n s : C o p y i n g v s C l o n i n g

U n t y p e d

- c a l c u l u s + l i n e a r a l g e b r a

C l o n i n g ?




17/101


U n t y p e d


C l o n i n g ?

x.(

x

x)

i

i

b

i

i

i

b

i

b

i

(

i

i

b

i

) (

i

i

b

i

)




18/101


U n t y p e d


C l o n i n g ?

x.(

x

x)

i

i

b

i

i

i

b

i

b

i

(

i

i

b

i

) (

i

i

b

i

)

N o - c l o n i n g s a y s b o t t o m r e d u c t i o n f o r b i d d e n . W e m u s t d e l a y b e t a

r e d u c t i o n t i l l a f t e r l i n e a r i t y . S o r e s t r i c t b e t a r e d u c t i o n t o b a s e

v e c t o r s i . e . a b s t r a c t i o n s o r v a r i a b l e s .




19/101

L i n e a r - A l g e b r a i c - C a l c u l u s W h y t h e r e s t r i c t i o n s : I n n i t i e s

U n t y p e d





20/101


U n t y p e d


Y b x .( b + ( x x )) x .( b + ( x x ))

Y b b + Y b




21/101


U n t y p e d


Y b x .( b + ( x x )) x .( b + ( x x ))

Y b b + Y b

B u t w h o e v e r s a y s i n n i t y s a y s t r o u b l e s a y s . . .




22/101


U n t y p e d


Y b x .( b + ( x x )) x .( b + ( x x ))

Y b b + Y b

B u t w h o e v e r s a y s i n n i t y s a y s t r o u b l e s a y s . . . i n d e n i t e f o r m s .




23/101


U n t y p e d


Y b x .( b + ( x x )) x .( b + ( x x ))

Y b b + Y b


Y b Y b b + Y b Y b b

0




24/101


U n t y p e d


Y b x .( b + ( x x )) x .( b + ( x x ))

Y b b + Y b


Y b Y b b + Y b Y b b

0

H i g h s c h o o l t e a c h e r s a y s w e m u s t r e s t r i c t f a c t o r i z a t i o n r u l e s t o

n i t e v e c t o r s i . e . c l o s e d - n o r m a l f o r m s .




25/101

S y s t e m F

S t r a i g h t f o r w a r d e x t e n s i o n o f S y s t e m F ( 2l a

)

S y s t e m F r u l e s p l u s s i m p l e r u l e s t o t y p e a l g e b r a i c t e r m s

a x

0

0

:A

u : A v : A+ I

u + v : A

t : A

I

. t : A




26/101

S y s t e m F


)


a x

0

0

:A

u : A v : A+ I

u + v : A

t : A

I

. t : A

T h e o r e m ( S t r o n g n o r m a l i z a t i o n )

t : T t i s s t r o n g l y n o r m a l i s i n g .





27/101

S y s t e m F


)


a x

0

0

:A

u : A v : A+ I

u + v : A

t : A

I

. t : A

T h e o r e m ( S t r o n g n o r m a l i z a t i o n )


P r o o f . S k e t c h : W e e x t e n d t h e n o t i o n o f s a t u r a t e d s e t s .





28/101

S y s t e m F

S t r o n g n o r m a l i z a t i o n o f 2l a

S N : S e t o f s t r o n g l y n o r m a l i s i n g t e r m s





29/101

S y s t e m F



A s u b s e t X S N i s s a t u r a t e d i f 1 n 0 , (((x t

1

) . . . ) tn

) X w h e r e ti

S N ;2

v [ b /x

]

t X

(x

v ) b

t X ;

3

t , u X

t + u X ;

4 , t X . t X ;

5 i

I ,

((ui

w

1

) . . . wn

) X

i

I

u

i

w

1

. . . w

n

X ;

6 i I , (u wi

) X u

i

I

w

i

X ;

7 .(( t1

t

2

) . . . tn

) X ((t1

t

2

) . . . . tk

) . . . tn

X (1 k n );8

0 X ;9

t

S N ,

(0

t

) X ;

1 0 t ,u S N ,

( t 0 ) u X .

X s t a b l e b y c o n s t r u c t i o n a n d a n t i - r e d u c t i o n





30/101

S y s t e m F



A s u b s e t X S N i s s a t u r a t e d i f 1 n 0 , (((x t

1

) . . . ) tn

) X w h e r e ti

S N ;2

v [ b /x

]

t X

(x

v ) b

t X ;

3

t , u X

t + u X ;

4 , t X . t X ;

5 i

I ,

((ui

w

1

) . . . wn

) X

i

I

u

i

w

1

. . . w

n

X ;

6 i I , (u wi

) X u

i

I

w

i

X ;

7 .(( t1

t

2

) . . . tn

) X ((t1

t

2

) . . . . tk

) . . . tn

X (1 k n );8

0 X ;9

t

S N ,

(0

t

) X ;

1 0 t ,u S N ,

( t 0 ) u X .

X s t a b l e b y c o n s t r u c t i o n a n d a n t i - r e d u c t i o n

S A T i s t h e s e t o f a l l s a t u r a t e d s e t s





31/101

S y s t e m F


( I I )

R e n i n g t h e s k e t c h :

T h e i d e a i s t h a t t y p e s c o r r e s p o n d t o s a t u r a t e d s e t s .

P a b l o A r r i g h i a n d A l e j a n d r o D a z - C a r o , L I G A S y s t e m F a c c o u n t i n g f o r s c a l a r s , a r X i v : 0 9 0 3 . 3 7 4 1 1 0 / 3 1




32/101

S y s t e m F


( I I )

R e n i n g t h e s k e t c h :

T h e i d e a i s t h a t t y p e s c o r r e s p o n d t o s a t u r a t e d s e t s .

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

S A T .





33/101

S y s t e m F


( I I I )

L e m m a

1

S N S A T ,

2

A , B S A T A B S A T ,

3

F o r a l l c o l l e c t i o n A

i

o f m e m b e r s o f S A T ,

iA

i

S A T ,





34/101

S y s t e m F


( I I I )

L e m m a

1

S N S A T ,

2


3


i


iA

i

S A T ,

D e n i t i o n ( M a p p i n g )

[[X ]] = (X ) ( w h e r e () : T V a r S A T )

[[A

B

]] = [[A ]] [[B ]]

[[X

.T

]] =

Y S A T [[T

]](X := Y )





35/101

S y s t e m F


( I I I )

L e m m a

1

S N S A T ,

2


3


i


iA

i

S A T ,

D e n i t i o n ( M a p p i n g )

[[X ]] = (X ) ( w h e r e () : T V a r S A T )

[[A

B

]] = [[A ]] [[B ]]

[[X

.T

]] =

Y S A T [[T

]](X := Y )

L e m m a

G i v e n a v a l u a t i o n , [[ T ]] S A T





36/101

S y s t e m F


( I V )

D e n i t i o n ( )

F o r =

x

1

:A

1

, . . . ,x

n

:A

n

, t : T m e a n s t h a t ,

x

1

[[ A1

]], . . . x n [[A n ]] t [[T ]]





37/101

S y s t e m F


( I V )

D e n i t i o n ( )

F o r =

x

1

:A

1

, . . . ,x

n

:A

n

, t : T m e a n s t h a t ,

x

1

[[ A1

]], . . . x n [[A n ]] t [[T ]]

R e n i n g t h e s k e t c h : P r o v e t h a t t : T t : T

W e p r o v e t h i s b y i n d u c t i o n o n t h e d e r i v a t i o n o f t : T ( I n f a c t

t h e d e n i t i o n o f i s s l i g h t l y d i e r e n t t o s t r e n g t h e n t h e i n d u c t i o n

h y p o t h e s i s )





38/101

S y s t e m F


( V )

T h e n , t h e p r o o f o f t h e s t r o n g n o r m a l i s a t i o n t h e o r e m i s :

L e t t : T





39/101

S y s t e m F


( V )


L e t t : T

t

:T





40/101

S y s t e m F


( V )


L e t t : T

t

:T

I f ( xi

:A

i

) , xi

[[Ai

]] t h e n t [[ T ]]





41/101

S y s t e m F


( V )


L e t t : T

t : T I f ( x

i

:A

i

) , xi

[[Ai

]] t h e n t [[ T ]]

N o t e t h a t

x

i

[[Ai

]] b e c a u s e [[A i ]] i s s a t u r a t e d ,





42/101

S y s t e m F


( V )


L e t t : T

t : T I f ( x

i

:A

i

) , xi

[[Ai

]] t h e n t [[ T ]]

N o t e t h a t

x

i

[[Ai

]] b e c a u s e [[A i ]] i s s a t u r a t e d ,

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

[[T

]] S N





43/101

S y s t e m F

L i n e a r - A l g e b r a i c - C a l c u l u s w i t h 2l a


t::=

x|

x.

t| (

t t) |

x.

t b

t[

b/

x] ()

( * ) b a n a b s t r a c t i o n o r a v a r i a b l e .


t+

t|.

t|

0


u + 0

u a n d

.( u + v ) . u + . v .


. u + .u ( + ).u .


u ( v + w ) ( u v ) + ( u w ) .




S y s t e m F


44/101



t::=

x|

x.

t| (

t t) |

x.

t b

t[

b/

x] ()


E v e r y t y p a b l e t e r m i s s t r o n g

n o r m a l i z i n g


t+

t|.

t|

0


u + 0 u a n d.( u + v ) . u + . v .


. u + .u ( + ).u .


u ( v + w ) ( u v ) + ( u w ) .




S y s t e m F


45/101



t::=

x|

x.

t| (

t t) |

x.

t b

t[

b/

x] ()




H e n c e Y b i s n o t y p a b l e !


t+

t|.

t|

0


u + 0 u a n d.( u + v ) . u + . v .


. u + .u ( + ).u .


u ( v + w ) ( u v ) + ( u w ) .




S y s t e m F


46/101



t::=

x|

x.

t| (

t t) |

x.

t b

t[

b/

x] ()




H e n c e Y b i s n o t y p a b l e !

t t 0 a l w a y s , s o i t i s n o t n e c e s a r y t o r e d u c e t r s t . w e c a n

r e m o v e t h e c l o s e d - n o r m a l

r e s t r i c t i o n s !


t+

t|.

t|

0


u + 0 u a n d.( u + v ) . u + . v .


. u + .u ( + ).u .


u ( v + w ) ( u v ) + ( u w ) .




T h es c a l a r

t y p e s y s t e m


47/101

G r a m m a r

T y p e s g r a m m a r :

T = U | X

.T | .T |0

,

U = X | U T | X .U

w h e r e S a n d (S,+,) i s a c o n m u t a t i v e r i n g .




T h es c a l a r

t y p e s y s t e m


48/101

T y p e i n f e r e n c e r u l e s

a x[

U]

,x

:U

x

:U

u :U

T

v :U

E

(u v

) :T

,x

:U

t :T

I

[U

]

x

t

:U

T

u : X.

T

E

[X

:=U

] u :

T[

U/

X]

u :T

I

[X

] w i t h X

/F V

() u :

X.

T




T h es c a l a r

t y p e s y s t e m


49/101


a x [ U ],

x:

U

x:

U

u :U

T

v :U

E

(u v

) :T

,x

:U

t :T

I

[U

]

x

t

:U

T

u : X.

T

E

[X

:=U

] u :

T[

U/

X]

u :T

I

[X

] w i t h X

/F V

() u :

X.

T

a x

0

0 :T

u : T v : T+

I

u + v : T

u : T

I

. u :T




T h es c a l a r

t y p e s y s t e m


50/101


a x [ U ],

x:

U x : U

u :.(U

T

) v :.U

E

(u v ) : ( ).T

,x

:U

t :T

I

[U

]

x

t

:U

T

u : X.

T

E

[X

:=U

] u :

T[

U/

X]

u :T

I

[X

] w i t h X

/F V

() u :

X.

T

a x

0

0 : 0

u :. T v :.T+

I

u + v : ( + ).T

u : Ts I

[] . u :. T

W h e r e U U a n d t y p e s i n c o n t e x t s a r e a r e i n U.




T h es c a l a r

t y p e s y s t e m


51/101

S t r o n g n o r m a l i s a t i o n

() : m a p t h a t t a k e t y p e s a n d r e m o v e a l l t h e s c a l a r s o n i t .




T h es c a l a r

t y p e s y s t e m


52/101


() : m a p t h a t t a k e t y p e s a n d r e m o v e a l l t h e s c a l a r s o n i t . E x a m p l e : ( U . X ) = U X




T h es c a l a r

t y p e s y s t e m


53/101



W e a l s o d e n e 0

=

T f o r s o m e T w i t h o u t s c a l a r s .




T h es c a l a r

t y p e s y s t e m


54/101




=


L e m m a ( C o r r e s p o n d e n c e w i t h 2l a

)

t : T 2

l a

t : T .




T h es c a l a r

t y p e s y s t e m


55/101




=


L e m m a ( C o r r e s p o n d e n c e w i t h 2l a

)

t : T 2

l a

t : T .

T h e o r e m ( S t r o n g n o r m a l i s a t i o n )


P r o o f . B y p r e v i o u s l e m m a 2 l a t : T

, t h e n t i s s t r o n g

n o r m a l i s i n g .




T h es c a l a r

t y p e s y s t e m


56/101

S u b j e c t r e d u c t i o n

T h e o r e m ( S u b j e c t R e d u c t i o n )

L e t t t . T h e n t : T t : T




T h es c a l a r

t y p e s y s t e m


57/101

S u b j e c t r e d u c t i o n

T h e o r e m ( S u b j e c t R e d u c t i o n )

L e t t t . T h e n t : T t : T

P r o o f . ( s k e t c h )

W e p r o o f r u l e b y r u l e t h a t i f t t u s i n g t h a t r u l e a n d t : T , t h e n t : T .

I n g e n e r a l , t h e m e t h o d i s t o t a k e t h e t e r m t , d e c o m p o s e i t

i n t o i t s s m a l l p a r t s a n d r e c o m p o s e t o t .




T h es c a l a r

t y p e s y s t e m


58/101

S u b j e c t r e d u c t i o n p r o o f : e x a m p l e ( I )

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

d e n i t i o n s :




T h es c a l a r

t y p e s y s t e m


59/101



d e n i t i o n s :

O r d e r :

W r i t e A>

B i f e i t h e r

B

X.

A o r

A

X

.C a n d B

C

[U

/X

]f o r s o m e U

U.

i s t h e r e e x i v e a n d t r a n s i t i v e c l o s u r e o f > .




T h es c a l a r

t y p e s y s t e m


60/101



d e n i t i o n s :

O r d e r :

W r i t e A>

B i f e i t h e r

B

X.

A o r

A

X

.C a n d B

C

[U

/X

]f o r s o m e U

U.

i s t h e r e e x i v e a n d t r a n s i t i v e c l o s u r e o f > .

I n s t u i t i o n o f t h i s d e n i t i o n :

T y p e s i n t h e n u m e r a t o r o f

t : AI w i t h X / F V ()

t

: X.

A

o r

t : X . C E

t:

C[

U/

X]

a r e g r e a t e r t h a n t h e t y p e s i n t h e d e n o m i n a t o r .




T h es c a l a r

t y p e s y s t e m

S u b j e c t r e d u c t i o n p r o o f : e x a m p l e ( I I )


61/101

S o m e l e m m a s n e e d e d :




T h es c a l a r

t y p e s y s t e m



62/101


1

A B a n d t : A t : B




T h es c a l a r

t y p e s y s t e m



63/101


1

A B a n d t : A t : B2

A B A B




T h es c a l a r

t y p e s y s t e m



64/101


1


A B A B3

G e n e r a t i o n l e m m a ( a p p ) : L e t ( u v ) : .T , t h e n

u : . U T

v :. UT

T=




T h es c a l a r

t y p e s y s t e m



65/101


1


A B A B3

G e n e r a t i o n l e m m a ( a p p ) : L e t ( u v ) : .T , t h e n

u : . U T

v :. UT

T=

4

G e n e r a t i o n l e m m a ( s u m ) : L e t u + v : . T , t h e n

u :. T v : . T= +




T h es c a l a r

t y p e s y s t e m

S u b j e c t r e d u c t i o n p r o o f : e x a m p l e ( I I I )


66/101

E x a m p l e : R u l e (

u+

v)

w ( u w ) + (v w ) .




T h es c a l a r

t y p e s y s t e m



67/101


u+

v)

w ( u w ) + (v w ) .

L e t (

u+

v)

w:

T .




T h es c a l a r

t y p e s y s t e m



68/101


u+

v)

w ( u w ) + (v w ) .

L e t (

u+

v)

w:

T .

U s i n g t h e p r e v i o u s l e m m a s w e c a n p r o v e t h a t

u : ( ). U T

v : ((1 ) ). U T

w :. U

w h e r e

=1 , T

T a n d

i s s o m e s c a l a r .




T h es c a l a r

t y p e s y s t e m

S u b j e c t r e d u c t i o n p r o o f : e x a m p l e ( I V )


69/101

T h e n

u : ( ).U T w :. U E

( u w ) : ( ). T .T




T h es c a l a r

t y p e s y s t e m



70/101

T h e n

u : ( ).U T w :. U E

( u w ) : ( ). T .T

A l s o ,

v

: ((1

) ).(U

T

) w

:.U

E ( v w ):(( 1 ) ). T ( 1 ). T




T h es c a l a r

t y p e s y s t e m



71/101

T h e n

u : ( ).U T w :. U E

( u w ) : ( ). T .T

A l s o ,

v

: ((1

) ).(U

T

) w

:.U

E ( v w ):(( 1 ) ). T ( 1 ). T

S o

(u w

) : .T

(v w

) : (1

).T

+ I a n d ( u w ) + ( v w ) : T




T h es c a l a r

t y p e s y s t e m

P r o b a b i l i s t i c t y p e s y s t e m : I n t u i t i o n


72/101

C o n d i t i o n a l f u n c t i o n s s a m e t y p e o n e a c h b r a n c h .




T h es c a l a r

t y p e s y s t e m



73/101


B y r e s t r i c t i n g t h e s c a l a r s t o p o s i t i v e r e a l s p r o b a b i l i s t i c t y p e s y s t e m .




T h es c a l a r

t y p e s y s t e m



74/101


B y r e s t r i c t i n g t h e s c a l a r s t o p o s i t i v e r e a l s p r o b a b i l i s t i c t y p e s y s t e m .

F o r e x a m p l e , o n e c a n t y p e f u n c t i o n s s u c h a s

x {x [ 12

.( t r u e + f a l s e )] [ 14

. t r u e + 34

. f a l s e ]} : B B

w i t h t h e t y p e s y s t e m s e r v i n g a s a g u a r a n t e e t h a t t h e f u n c t i o n

c o n s e r v e s p r o b a b i l i t i e s s u m m i n g t o o n e .




T h es c a l a r

t y p e s y s t e m

P r o b a b i l i s t i c t y p e s y s t e m : F o r m a l i s a t i o n


75/101

W e d e n e t h e p r o b a b i l i s t i c t y p e s y s t e m t o b e t h e s c a l a r t y p e

s y s t e m w i t h t h e f o l l o w i n g r e s t r i c t i o n s :




T h es c a l a r

t y p e s y s t e m



76/101



S = R+ ,




T h es c a l a r

t y p e s y s t e m



77/101



S = R+ ,

C o n t e x t s : T y p e s i n c o n t e x t s a r e c l a s s i c t y p e s ( C

) , i . e . t y p e s i n

Ue x e m p t o f a n y s c a l a r ,




T h es c a l a r

t y p e s y s t e m



78/101



S = R+ ,


) , i . e . t y p e s i n


W e c h a n g e t h e r u l e E t o

t

: X.

T

E w i t h C C t : T [C /X ]




T h es c a l a r

t y p e s y s t e m



79/101



S = R+ ,


) , i . e . t y p e s i n


W e c h a n g e t h e r u l e E t o

t

: X.

T

E w i t h C C t : T [C /X ]

T h e n a l c o n c l u s i o n m u s t b e c l a s s i c .




T h es c a l a r

t y p e s y s t e m

P r o b a b i l i s t i c t y p e s y s t e m : P r o o f


80/101

D e n i t i o n ( W e i g h t f u n c t i o n t o c h e c k p r o b a b i l i t y d i s t r i b u t i o n s

)

L e t : R+ b e a f u n c t i o n d e n e d i n d u c t i v e l y b y :

( 0 ) = 0 (t1

+ t2

) = ( t1

) + ( t2

)(

b) =

1(.

t) = (

t)

( t1

t

2

) = ( t1

) ( t2

)




T h es c a l a r

t y p e s y s t e m



81/101


)


( 0 ) = 0 (t1

+ t2

) = ( t1

) + ( t2

)(

b) =

1(.

t) = (

t)

( t1

t

2

) = ( t1

) ( t2

)

T h e o r e m ( N o r m a l t e r m s i n p r o b a b i l i s t i c h a v e w e i g h t 1 )

C t : C ( t ) = 1 .

P r o o f . W e p r o v e t h a t

t

:.C

(t

) = b y s t r u c t u r a l

i n d u c t i o n o v e r t .




T h es c a l a r

t y p e s y s t e m



82/101


)


( 0 ) = 0 (t1

+ t2

) = ( t1

) + ( t2

)(

b) =

1(.

t) = (

t)

( t1

t

2

) = ( t1

) ( t2

)


C t : C ( t ) = 1 .


t

:.C

(t


i n d u c t i o n o v e r t .E x a m p l e : 2

.(x

1

2

.x

)y




T h es c a l a r

t y p e s y s t e m



83/101


)


( 0 ) = 0 (t1

+ t2

) = ( t1

) + ( t2

)(

b) =

1(.

t) = (

t)

( t1

t

2

) = ( t1

) ( t2

)


C t : C ( t ) = 1 .


t

:.C

(t



.(x

1

2

.x

)y

(2.(

x

1

2

.x

)y

) =2




T h es c a l a r

t y p e s y s t e m



84/101


)


( 0 ) = 0 (t1

+ t2

) = ( t1

) + ( t2

)(

b) =

1(.

t) = (

t)

( t1

t

2

) = ( t1

) ( t2

)


C t : C ( t ) = 1 .


t

:.C

(t



.(x

1

2

.x

)y

y

(2.(

x

1

2

.x

)y

) =2

(y

) =1




T h es c a l a r

t y p e s y s t e m

L o g i c a l c o n t e n t : N o - c l o n i n g t h e o r e m ( I )


85/101

D e n i t i o n ( P r o o f m e t h o d o f d e p t h n )

0

(S

) =S

n

( S ) =

n 1 ( S )R

P

S

o r

k

(S

) h

R

P

S

o r

k

h

(S

)R

P

S

w h e r e

S i s a s e q u e n t ,

n

i s a c o n s t a n t d e r i v a t i o n t r e e o f s i z e n ,

m a x {k , h } = n 1 ,

R i s a t y p i n g r u l e , a n d

P

S

i s a s e q u e n t s u c h t h a t t h e r e s u l t i n g d e r i v a t i o n t r e e i s

w e l l - f o r m e d .




T h es c a l a r

t y p e s y s t e m

L o g i c a l c o n t e n t : N o - c l o n i n g t h e o r e m ( I )


86/101

D e n i t i o n ( P r o o f m e t h o d o f d e p t h n )

0

(S

) =S

n

( S ) =

n 1 ( S )R

P

S

o r

k

(S

) h

R

P

S

o r

k

h

(S

)R

P

S

w h e r e

S i s a s e q u e n t ,

n

i s a c o n s t a n t d e r i v a t i o n t r e e o f s i z e n ,

m a x {k , h } = n 1 ,

R i s a t y p i n g r u l e , a n d

P

S

i s a s e q u e n t s u c h t h a t t h e r e s u l t i n g d e r i v a t i o n t r e e i s

w e l l - f o r m e d .

C (n

(S )) d e n o t e t h e c o n c l u s i o n ( r o o t ) o f t h e t r e e n

( S ).




T h es c a l a r

t y p e s y s t e m

L o g i c a l c o n t e n t : N o - c l o n i n g t h e o r e m ( I I )


87/101

E x a m p l e s :

1

(S

) =S

I

P

S




T h es c a l a r

t y p e s y s t e m



88/101

E x a m p l e s :

1

(S

) =S

I

P

S

1

( t

:T

) = t : T

I t : X . T




T h es c a l a r

t y p e s y s t e m



89/101

E x a m p l e s :

1

(S

) =S

I

P

S

1

( t

:T

) = t : T

I t : X . T

n

( u

:.A

) = u :. An 2

v : 2 .A+

I

u + v : ( 2 + ). A

( P a r t i a l f u n c t i o n o r p a t t e r n m a t c h i n g )




T h es c a l a r

t y p e s y s t e m



90/101

E x a m p l e s :

1

(S

) =S

I

P

S

1

( t

:T

) = t : T

I t : X . T

n

( u

:.A

) = u :. A

n 2

v : 2 .A+

I

u + v : ( 2 + ). A

( P a r t i a l f u n c t i o n o r p a t t e r n m a t c h i n g ) B a s i c a l l y C(

n

(S

)) =P

S

m e a n s t h a t P

S

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

x e d p r o o f m e t h o d .




T h es c a l a r

t y p e s y s t e m

L o g i c a l c o n t e n t : N o - c l o n i n g t h e o r e m ( I I I )


91/101

T h e o r e m ( N o - c l o n i n g o f s c a l a r s )

n

s u c h t h a t , C (n

( . U )) = ( s + ). V w i t h = 0 a n d c o n s t a n t s i n S, s N>1 a n d U , V c o n s t a n t s i n U.

P r o o f . I n d u c t i o n o v e r n .




T h es c a l a r

t y p e s y s t e m



92/101


n




C o r o l l a r y ( N o - c l o n i n g T h e o r e m )

n

s u c h t h a t T , C (n

( T )) = T T .

w h e r e A B i s t h e c l a s s i c a l e n c o d i n g f o r t h e t y p e o f t u p l e s .




T h es c a l a r

t y p e s y s t e m



93/101


n




C o r o l l a r y ( N o - c l o n i n g T h e o r e m )

n

s u c h t h a t T , C (n

( T )) = T T .

w h e r e A B i s t h e c l a s s i c a l e n c o d i n g f o r t h e t y p e o f t u p l e s . P r o o f .

I t i s e a s y t o s h o w t h a t T , T . U w i t h U U



- C a l c u l u s S y s t e m F T h e s

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