[PPT] - R e l a t i o n a l R e a s o n i n g f o r M PowerPoint Presentation

SLIDE 1

R e l a t i

n

a l R e a s

n

i n g f

r

M a r k

v

C h a i n s i n a P r

b

a b i l i s t i c G u a r d e d L a mb d a C a l c u l u s

Alejandro Aguirre, Gilles Barthe, Lars Birkedal, Aleš Bizjak, Marco Gaboardi and Deepak Garg

Imdea Software, Aarhus University, University at Buffalo SUNY, MPI-SWS

SLIDE 2

n

e

t

i me p a d

1 1 1 1 1 1 1 1 1 1 1 1 1 1 message key (uniformly sampled) ciphertext K e y p r

p

e r t y : p e r f e c t s e c r e c y

SLIDE 3

r e l a t i

n

a l p r

p

e r t i e s

t

w

e

x e c u t i

n

s

f

t h e s a me

r

d i f f e r e n t p r

g

r a ms

o

n e c

u

l d c

mp

u t e b

t

h a n d c

mp

a r e t h e m

i

s t h e r e a b e t t e r w a y t

r

e a s

n

?

two executions

SLIDE 4

r e l a t i

n

a l l

g

i c

SLIDE 5

r e l a t i

n

a l l

g

i c

ICFP’17

SLIDE 6

r e l a t i

n

a l l

g

i c

ICFP’17

SLIDE 7

r e l a t i

n

a l l

g

i c

ICFP’17 T wo terms

SLIDE 8

r e l a t i

n

a l l

g

i c

ICFP’17 T wo terms related by a property

SLIDE 9

r e l a t i

n

a l l

g

i c

ICFP’17 T wo terms related by a property

SLIDE 10

r e l a t i

n

a l l

g

i c

How to express this exactly?

SLIDE 11

r e mi n d e r : p r

b

a b i l i t y

( d i s c r e t e ) d i s t r i b u t i

n

ma r g i n a l s f

r

x y

SLIDE 12

c

u

p l i n g s 1 1

l

e t

l

e t

i s a c

u

p l i n g & i s a R

c
u

p l i n g & & U s e f u l c a s e s : ( D e n

t

e d )

SLIDE 13

e x a mp l e : c

i

n fl i p

S
me

w a y s

f

c

u

p l i n g t w

f

a i r c

i

n s :

H T H 1/2 T 1/2 H T H 1/2 T 1/2 H T H 1/4 1/4 T 1/4 1/4

equality product inequality

H T H p ½ - p T ½ - p p

general

SLIDE 14

f u n d a me n t a l l e mma

I d e a : T

p

r

v

e a r e l a t i

n

a l p r

p

e r t y a b

u

t d i s t r i b u t i

n

s , w e “ s y n c r a n d

mn

e s s ” t

b

u i l d t h e a p p r

p

r i a t e c

u

p l i n g I n p a r t i c u l a r :

(

s t

c

h a s t i c d

mi

n a n c e )

I f i s a n R

c
u

p l i n g

f

a n d ,

SLIDE 15

n

e

t

i me p a d r e v i s i t e d

1 1 1 1 1 1 1 1 1

SLIDE 16

n

e

t

i me p a d r e v i s i t e d

1 1 1 1 1 1 1 1 1 1 1 1 1 1

SLIDE 17

n

e

t

i me p a d r e v i s i t e d

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

SLIDE 18

n

e

t

i me p a d r e v i s i t e d

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

SLIDE 19

n

e

t

i me p a d r e v i s i t e d

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

SLIDE 20

n

e

t

i me p a d r e v i s i t e d

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 T h e r e f

r

e ,

SLIDE 21

n

e

t

i me p a d r e v i s i t e d

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 T h e r e f

r

e , O n e c a n a l s

s

h

w

,

SLIDE 22

( d i s c r e t e t i me ) ma r k

v

c h a i n s

d

i s c r e t e s e t

f

s t a t e s

p

r

b

a b i l i s t i c t r a n s i t i

n

f u n c t i

n

S1 S2 S3 1/2 1/2 1 1

SLIDE 23

r e l a t i

n

a l p r

p

e r t i e s

r e l a t e t w

M

a r k

v

c h a i n s

r

t w

r

u n s

f

t h e s a me

n

e

1 2 ½ ½ ½ . . . ½ ½ ½ 1 2 ⅔ ⅔ ⅔ . . . ⅓ ⅓ ⅓ w h i c h

n

e i s f a s t e r ?

SLIDE 24

e x a mp l e : s t

c

h a s t i c d

mi

n a n c e

1 2 ½ ½ ½ . . . ½ ½ ½ 1 2 ⅔ ⅔ ⅔ . . . ⅓ ⅓ ⅓ n

n+1 1/2

m

m+1

1/2

SLIDE 25

e x a mp l e : s t

c

h a s t i c d

mi

n a n c e

1 2 ½ ½ ½ . . . ½ ½ ½ 1 2 ⅔ ⅔ ⅔ . . . ⅓ ⅓ ⅓ n

n+1 1/2

m

m+1

1/2

n n

1/6

m

m+1

1/6

SLIDE 26

e x a mp l e : s t

c

h a s t i c d

mi

n a n c e

1 2 ½ ½ ½ . . . ½ ½ ½ 1 2 ⅔ ⅔ ⅔ . . . ⅓ ⅓ ⅓ n

n+1 1/2

m

m+1

1/2

n n

1/3

m m

1/3

n n

1/6

m

m+1

1/6

SLIDE 27

e x a mp l e : s t

c

h a s t i c d

mi

n a n c e

1 2 ½ ½ ½ . . . ½ ½ ½ 1 2 ⅔ ⅔ ⅔ . . . ⅓ ⅓ ⅓ n

n+1 1/2

m

m+1

1/2

n n

1/3

m m

1/3

n n

1/6

m

m+1

1/6

SLIDE 28

s

f

a r

P

r

b

a b i l i t i e s & M a r k

v

c h a i n s a r e a u s e f u l mo d e l i n g t

l
S
me

i n t e r e s t i n g p r

p

e r t i e s a r e r e l a t i

n

a l

T

h e s e c a n b e s t u d i e d w i t h c

u

p l i n g s

SLIDE 29

r e p r e s e n t i n g M a r k

v

C h a i n s

i

n fi n i t e s e q u e n c e ( s t r e a m )

f

d i s t r i b u t i

n

s

SLIDE 30

r e p r e s e n t i n g M a r k

v

C h a i n s

i

n fi n i t e s e q u e n c e ( s t r e a m )

f

d i s t r i b u t i

n

s

SLIDE 31

r e p r e s e n t i n g M a r k

v

C h a i n s

i

n fi n i t e s e q u e n c e ( s t r e a m )

f

d i s t r i b u t i

n

s

SLIDE 32

r e p r e s e n t i n g M a r k

v

C h a i n s

i

n fi n i t e s e q u e n c e ( s t r e a m )

f

d i s t r i b u t i

n

s

SLIDE 33

r e p r e s e n t i n g M a r k

v

C h a i n s

i

n fi n i t e s e q u e n c e ( s t r e a m )

f

d i s t r i b u t i

n

s

. . .

SLIDE 34

r e p r e s e n t i n g M a r k

v

C h a i n s

i

n fi n i t e s e q u e n c e ( s t r e a m )

f

d i s t r i b u t i

n

s

P r

b

l e m : 1 ) N

t

e x p r e s s i v e e n

u

g h !

. . .

SLIDE 35

r e p r e s e n t i n g M a r k

v

C h a i n s

i

n fi n i t e s e q u e n c e ( s t r e a m )

f

d i s t r i b u t i

n

s

P r

b

l e m : 1 ) N

t

e x p r e s s i v e e n

u

g h ! 2 ) P r

b

a b i l i s t i c d e p e n d e n c e

. . .

SLIDE 36

r e p r e s e n t i n g M a r k

v

C h a i n s

a

s d i s t r i b u t i

n

s

v

e r s t r e a ms

1 2 1 2 1 . . .

1

1

1
2

. . .

1
2
3
2
3
4

. . .

SLIDE 37

r e p r e s e n t i n g M a r k

v

C h a i n s

a

s d i s t r i b u t i

n

s

v

e r s t r e a ms

1 2 1 2 1 . . .

1

1

1
2

. . .

1
2
3
2
3
4

. . .

SLIDE 38

r e p r e s e n t i n g M a r k

v

C h a i n s

a

s d i s t r i b u t i

n

s

v

e r s t r e a ms

1 2 1 2 1 . . .

1

1

1
2

. . .

1
2
3
2
3
4

. . .

P r

b

l e m : N

t

d i s c r e t e !

SLIDE 39

t h e p r

b

l e m w i t h s t r e a m d e fi n i t i

n

s

A

s t r e a m i s p r

d

u c t i v e i f i t

u

t p u t s e v e r y fi n i t e p r e fi x i n fi n i t e t i me

E

x a mp l e

f

n

n
p

r

d

u c t i v e s t r e a m:

F

r

i n s t a n c e ,

SLIDE 40

g u a r d e d l a mb d a c a l c u l u s

T

e r ms :

T

y p e s :

[1] Clouston, R., Bizjak, A., Grathwohl, H.B., Birkedal, L.: The guarded lambda-calculus: Programming and reasoning with guarded recursion for coinductive types. ( LMCS ‘16 )

An element now A stream “later”

SLIDE 41

s t r e a ms i n g l c

1 1 2

. . .

time 2 time 1 time 0

SLIDE 42

s t r e a ms i n g l c

1 1 2

. . .

time 2 time 1 time 0 All prefxes are discrete!

SLIDE 43

p r

b

a b i l i s t i c g l c

T

e r ms :

T

y p e s :

SLIDE 44

s t r e a m d i s t r i b u t i

n

s i n g l c

1 1 2

. . .

time 2 time 1 time 0

1 2 1 2 2 1

SLIDE 45

s t r e a m d i s t r i b u t i

n

s i n g l c

1 1 2

. . .

time 2 time 1 time 0

1 2 1 2 2 1

D i s c r e t e P r

b

a b i l i t i e s !

SLIDE 46

( t w

)

r e l a t i

n

a l l

g

i c s f

r

g l c

T wo terms related by a property/coupling

SLIDE 47

p r

v

i n g p r

p

e r t i e s

f

M a r k

v

c h a i n s

SLIDE 48

p r

v

i n g p r

p

e r t i e s

f

M a r k

v

c h a i n s

1 ) P r

v

e l

c

a l p r

p

e r t y a b

u

t p a i r s

f

s t a t e s

1)

SLIDE 49

p r

v

i n g p r

p

e r t i e s

f

M a r k

v

c h a i n s

1 ) P r

v

e l

c

a l p r

p

e r t y a b

u

t p a i r s

f

s t a t e s 2 ) S h

w

t h a t t h e t r a n s i t i

n

f u n c t i

n

s p r e s e r v e s i t

1) 2)

SLIDE 50

p r

v

i n g p r

p

e r t i e s

f

M a r k

v

c h a i n s

1 ) P r

v

e l

c

a l p r

p

e r t y a b

u

t p a i r s

f

s t a t e s 2 ) S h

w

t h a t t h e t r a n s i t i

n

f u n c t i

n

s p r e s e r v e s i t 3 ) L i f t t h e p r

p

e r t y t

a

g l

b

a l p r

p

e r t y a b

u

t p a i r s

f

s t r e a ms

1) 2) 3)

SLIDE 51

e x a mp l e : s t

c

h a s t i c d

mi

n a n c e

1 2 ½ ½ ½ . . . ½ ½ ½ 1 2 ⅔ ⅔ ⅔ . . . ⅓ ⅓ ⅓ I d e a :

SLIDE 52

e x a mp l e : s t

c

h a s t i c d

mi

n a n c e

1 2 ½ ½ ½ . . . ½ ½ ½ 1 2 ⅔ ⅔ ⅔ . . . ⅓ ⅓ ⅓

SLIDE 53

e x a mp l e : s t

c

h a s t i c d

mi

n a n c e

1 2 ½ ½ ½ . . . ½ ½ ½ 1 2 ⅔ ⅔ ⅔ . . . ⅓ ⅓ ⅓

1 )

SLIDE 54

e x a mp l e : s t

c

h a s t i c d

mi

n a n c e

1 2 ½ ½ ½ . . . ½ ½ ½ 1 2 ⅔ ⅔ ⅔ . . . ⅓ ⅓ ⅓

1 ) 2 )

SLIDE 55

e x a mp l e : s t

c

h a s t i c d

mi

n a n c e

1 2 ½ ½ ½ . . . ½ ½ ½ 1 2 ⅔ ⅔ ⅔ . . . ⅓ ⅓ ⅓

1 ) 2 ) 3 )

SLIDE 56

i n t h e p a p e r

d

e n

t

a t i

n

a l s e ma n t i c s

r

e s t

f

t h e p r

f

r u l e s

r

e l a t i

n

a l l

g

i c f

r

d e t e r mi n i s t i c s t r e a ms

mo

r e e x a mp l e s

– R

a n d

m

w a l k s ( l a z y v s n

n
l

a z y , 3 D v s 4 D )

– A

p p r

x

i ma t i

n

s e r i e s

SLIDE 57

c

n

c l u s i

n
a

l

g

i c t

r

e a s

n

a b

u

t p r

b

a b i l i s t i c i n fi n i t e d a t a s t r u c t u r e s

ma

r k

v

c h a i n s a r e r e p r e s e n t e d a s d i s t r i b u t i

n
v

e r s t r e a ms

p

r

p

e r t i e s a r e p r

v

e n v i a c

u

p l i n g s

f

u t u r e w

r

k :

e

x t e n d t h e l a n g u a g e

c
n

t i n u

u

s d i s t r i b u t i

n

s

SLIDE 58

Thanks!

SLIDE 59

R e l a t i

n

a l R e a s

n

i n g f

r

M a r k

v

C h a i n s i n a P r

b

a b i l i s t i c G u a r d e d L a mb d a C a l c u l u s

Alejandro Aguirre, Gilles Barthe, Lars Birkedal, Aleš Bizjak, Marco Gaboardi and Deepak Garg

Imdea Software, Aarhus University, University at Buffalo SUNY, MPI-SWS