[PPT] - Coherence Spaces for Resource-Sensitive Computation in Analysis K PowerPoint Presentation

SLIDE 1

1

Coherence Spaces for Resource-Sensitive Computation in Analysis

K e i M a t s u mo t

(

R I M S , K y

t
U

n i v e r s i t y )

SLIDE 2

2

Background

C
mp

u t a b l e a n a l y s i s s t u d i e s c

mp

u t a t i

n
v

e r t

p
l
g

i c a l s p a c e s , b y g i v i n g r e p r e s e n t a t i

n

s .

– T

y p e t wo t h e

r

y

f

E f f e c t i v i t y

– D

ma

i n r e p r e s e n t a t i

n

s

T

h e i r a p p r

a

c h e s a r e t

t

r a c k c

mp

u t a t i

n

b y c

n

t i n u

u

s ma p s

v

e r “ s y mb

l

i c ” s p a c e s .

T h e p r i n c i p l e : C

mp

u t a b l e ⇒ C

n

t i n u

u

s

B a i r e s p . , S c

t

t d

ma

i n s , . . .

SLIDE 3

3

Our Proposal

O

u r p r i n c i p l e : C

mp

u t a b l e ⇒ S t a b l e [ B e r r y ' 7 8 ]

U s i n g i n s t e a d

f

S c

t

t

d
ma

i n s c

h

e r e n c e s p a c e s [ G i r a r d ' 8 6 ] .

B

T W,

t w

mo

r p h i s ms coexists i n c

h

e r e n c e s p a c e s : s t a b l e & l i n e a r ma p s .

A

n e w q u e s t i

n

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

Y X F

X Y

f

T

p
l
g

i c a l s p . C

h

e r e n c e s p .

T r a c k e d b y s t a b l e ma p .

Wh a t a r e L i n e a r C

mp

u t a t i

n

s i n T

p
l
g

y ?

SLIDE 4

Girard's Linear Logic

O n e

f

t h e mo s t i n f l u e n t i a l p a p e r s i n 8 ' s i n b

t

h l

g

i c a n d c

mp

u t e r s c i e n c e .

R

e s t r u c t u r i n g b

t

h C l a s s i c a l & I n t u i t i

n

i s t i c L

g

i c

P

r

f

N e t s

R

e s

u

r c e

C
n

c i

u

s n e s s

A t t r a c t i v e I d e a s :

SLIDE 5

Girard's Linear Logic

R

e s t r u c t u r i n g b

t

h C l a s s i c a l & I n t u i t i

n

i s t i c L

g

i c

P

r

f

N e t s

R

e s

u

r c e

C
n

c i

u

s n e s s

A t t r a c t i v e I d e a s :

O n e

f

t h e mo s t i n f l u e n t i a l p a p e r s i n 8 ' s i n b

t

h l

g

i c a n d c

mp

u t e r s c i e n c e .

SLIDE 6

Girard's Linear Logic

R

e s t r u c t u r i n g b

t

h C l a s s i c a l & I n t u i t i

n

i s t i c L

g

i c

P

r

f

N e t s

R

e s

u

r c e

C
n

c i

u

s n e s s

A t t r a c t i v e I d e a s :

O n e

f

t h e mo s t i n f l u e n t i a l p a p e r s i n 8 ' s i n b

t

h l

g

i c a n d c

mp

u t e r s c i e n c e .

SLIDE 7

Girard's Linear Logic

R

e s t r u c t u r i n g b

t

h C l a s s i c a l & I n t u i t i

n

i s t i c L

g

i c

P

r

f

N e t s

R

e s

u

r c e

C
n

c i

u

s n e s s

A t t r a c t i v e I d e a s :

O n e

f

t h e mo s t i n f l u e n t i a l p a p e r s i n 8 ' s i n b

t

h l

g

i c a n d c

mp

u t e r s c i e n c e .

SLIDE 8

8

Resource-Sensitivity

Imagine: there's no resource-conciousness It isn't easy to do Nothing to comsume or lose for And no modalities too Imagine all the people Living life in Intuitionistic Logic ...

’’

SLIDE 9

9

Resource-Sensitivity

Ex.

I n I n t u i t i

n

i s t i c L

g

i c , i s t r u e .

a p e r s

n

i n t h e I n t u i t i

n

i s t i c L

g

i c wo r l d S u b s t i t u t e :

A

: = “ t

p

a y ￥4 ”

B

: = “ t

g

e t a p a c k

f

c i g a r e t t e s ”

C

: = “ t

g

e t a c u p

f

c a k e ”

SLIDE 10

10

Resource-Sensitivity

Ex.

I n I n t u i t i

n

i s t i c L

g

i c , i s t r u e .

S u b s t i t u t e :

A

: = “ t

p

a y ￥4 ”

B

: = “ t

g

e t a p a c k

f

c i g a r e t t e s ”

C

: = “ t

g

e t a c u p

f

c a k e ”

SLIDE 11

11

Resource-Sensitivity

Ex.

I n I n t u i t i

n

i s t i c L

g

i c , i s t r u e .

S u b s t i t u t e :

A

: = “ t

p

a y ￥4 ”

B

: = “ t

g

e t a p a c k

f

c i g a r e t t e s ”

C

: = “ t

g

e t a c u p

f

c a k e ”

SLIDE 12

12

Resource-Sensitivity

Ex.

I n I n t u i t i

n

i s t i c L

g

i c , i s t r u e .

S u b s t i t u t e :

A

: = “ t

p

a y ￥4 ”

B

: = “ t

g

e t a p a c k

f

c i g a r e t t e s ”

C

: = “ t

g

e t a c u p

f

c a k e ”

P a r a d

x

SLIDE 13

13

Resource-Sensitivity

Ex.

I n I n t u i t i

n

i s t i c L

g

i c , i s t r u e .

S u b s t i t u t e :

A

: = “ t

p

a y ￥4 ”

B

: = “ t

g

e t a p a c k

f

c i g a r e t t e s ”

C

: = “ t

g

e t a c u p

f

c a k e ”

L a c k

f

c

n

c i

u

s n e s s t

c
ms

u me a s s u mp t i

n

s !

A is used twice.

SLIDE 14

14

Resource-Sensitivity

Ex.

I n L i n e a r L

g

i c , i s f a l s e .

B e c a u s e :

I n L L , we mu s t u s e t h e a s s u mp t i

n

e x a c t l y

n

c e i n t h e p r

f

. C

h

e r e n c e S p a c e s a r e p r

p
s

e d a s a d e n

t

a t i

n

a l s e ma n t i c s wh i c h r e f l e c t s t h i s p r

p

e r t y . V i a t h e C u r r y

H
w

a r d i s

mo

r p h i s m, t h e y a r e a l s

a

mo d e l

f

resource-sensitive computations

f

l i n e a r f u n c t i

n

p r

g

r a ms .

N e w c

n

j u n c t i

n

/ i mp l i c a t i

n

SLIDE 15

15

Main Result from CCA’15

R e p r e s e n t a t i

n

s b a s e d

n

c

h

e r e n c e s p a c e s h a v e a n i n t e r e s t i n g f e a t u r e :

f

r

e v e r y r e a l f u n c i t

n

s , w e h a v e s h

w

n t h a t

s

t a b l y r e a l i z a b l e ⇔ c

n

t i n u

u

s

l

i n e a r l y r e a l i z a b l e ⇔ u n i f

r

ml y c

n

t i n u

u

s .

L e t u s e mp h a s i z e t h a t t h e s e c

r

r e s p

n

d e n c e s h

l

d f

r

r e a l f u n c t i

n

s .

N e x t s t e p : g e n e r a l i z e t h e m t

a

wi d e r c l a s s .

SLIDE 16

16

Ⅰ. Review: : Coherent Spa paces Ⅱ. Coherence e as Uniformity Ⅲ. Linear Admissibility Ⅳ. Concluding Comments

SLIDE 17

17

Coherence Spaces

Def. A

c

h

e r e n c e s p a c e i s a r e f l e x i v e g r a p h :

a

c

u

n t a b l e s e t

f

t

k

e n s wi t h

a

s y mme t r i c r e f l e x i v e . b i n a r y r e l .

n

Write i f f a n d (s t r i c t c

h

e r e n c e ) A c l i q u e i s a s e t

f

t

k

e n s wh i c h a r e p a i r wi s e c

h

e r e n t . A n a n t i c l i q u e i s a s e t

f

t

k

e n s i n wh i c h e v e r y p a i r i s n

t

c

h

e r e n t .

:

t h e s e t

f

a l l c l i u q e s .

:

t h e s e t

f

a l l f i n i t e c l i q u e s .

:

t h e s e t

f

a l l ma x i ma l c l i q u e s .

SLIDE 18

18

Example: Cauchy Sequences

L e t . E a c h me mb e r

f

i s i d e n t i f i e d wi t h t h e d y a d i c r a t i

n

a l a s . F

r

e a c h , d e f i n e

Ex.

D e f i n e a c

h

e r e n c e s p a c e f

r

d y a d i c C a u c h y s e q u e n c e s a s :

Ma x i ma l c l i q u e s ≈ ( r a p i d l y c

n

v e r g i n g ) C a u c h y s e q u e n c e s

R e a l i z a t i

n
f

R e a l N u mb e r s

SLIDE 19

19

Example: Cauchy Sequences

L e t . E a c h me mb e r

f

i s i d e n t i f i e d wi t h t h e d y a d i c r a t i

n

a l a s . F

r

e a c h , d e f i n e

Ex.

D e f i n e a c

h

e r e n c e s p a c e f

r

d y a d i c C a u c h y s e q u e n c e s a s :

Ma x i ma l c l i q u e s ≈ ( r a p i d l y c

n

v e r g i n g ) C a u c h y s e q u e n c e s

R e a l i z a t i

n
f

R e a l N u mb e r s

t h e “ s t a g e ” “ s p

t

l i g h t s ”

c l a s s i f i e d b y “ c

l
r

s ”

c

mp

a c t i n t e r v a l s a r e “ p r

j

e c t e d ”

SLIDE 20

20

Example: Cauchy Sequences

L e t . E a c h me mb e r

f

i s i d e n t i f i e d wi t h t h e d y a d i c r a t i

n

a l a s . F

r

e a c h , d e f i n e

Ex.

D e f i n e a c

h

e r e n c e s p a c e f

r

d y a d i c C a u c h y s e q u e n c e s a s :

Ma x i ma l c l i q u e s ≈ ( r a p i d l y c

n

v e r g i n g ) C a u c h y s e q u e n c e s

R e a l i z a t i

n
f

R e a l N u mb e r s

t h e “ s t a g e ” “ s p

t

l i g h t s ”

c l a s s i f i e d b y “ c

l
r

s ”

c

mp

a c t i n t e r v a l s a r e “ p r

j

e c t e d ”

SLIDE 21

21

Example: Cauchy Sequences

L e t . E a c h me mb e r

f

i s i d e n t i f i e d wi t h t h e d y a d i c r a t i

n

a l a s . F

r

e a c h , d e f i n e

Ex.

D e f i n e a c

h

e r e n c e s p a c e f

r

d y a d i c C a u c h y s e q u e n c e s a s :

Ma x i ma l c l i q u e s ≈ ( r a p i d l y c

n

v e r g i n g ) C a u c h y s e q u e n c e s

R e a l i z a t i

n
f

R e a l N u mb e r s

t h e “ s t a g e ” “ s p

t

l i g h t s ”

c l a s s i f i e d b y “ c

l
r

s ”

c

mp

a c t i n t e r v a l s a r e “ p r

j

e c t e d ”

SLIDE 22

22

Example: Cauchy Sequences

L e t . E a c h me mb e r

f

i s i d e n t i f i e d wi t h t h e d y a d i c r a t i

n

a l a s . F

r

e a c h , d e f i n e

Ex.

D e f i n e a c

h

e r e n c e s p a c e f

r

d y a d i c C a u c h y s e q u e n c e s a s :

Ma x i ma l c l i q u e s ≈ ( r a p i d l y c

n

v e r g i n g ) C a u c h y s e q u e n c e s

R e a l i z a t i

n
f

R e a l N u mb e r s

t h e “ s t a g e ” “ s p

t

l i g h t s ”

c l a s s i f i e d b y “ c

l
r

s ”

L i g h t s

f

d i f f e r e n t c

l
r

s a r e

v

e r l a p p e d

SLIDE 23

23

Example: Cauchy Sequences

L e t . E a c h me mb e r

f

i s i d e n t i f i e d wi t h t h e d y a d i c r a t i

n

a l a s . F

r

e a c h , d e f i n e

Ex.

D e f i n e a c

h

e r e n c e s p a c e f

r

d y a d i c C a u c h y s e q u e n c e s a s :

Ma x i ma l c l i q u e s ≈ ( r a p i d l y c

n

v e r g i n g ) C a u c h y s e q u e n c e s

R e a l i z a t i

n
f

R e a l N u mb e r s

SLIDE 24

24

Coherence as Topology

T h e s e t

f

c l i q u e s i s

r

d e r e d b y ⊆ , e n d

we

d wi t h t h e S c

t

t t

p
l
g

y g e n e r a t e d b y t h e u p p e r s e t s

f

f i n i t e c l i q u e s :

C
h

e r e n c e s p a c e s a r e v e r y s i mp l i f i e d d

ma

i n s

C
mp

a c t E l e me n t s ＝ f i n i t e c l i q u e s

F i n i t e C l i q u e s i n d u c e T

p
l
g

y

SLIDE 25

25

Stable & Linear Maps

Def. A f u n c t i

n

i s s t a b l e i f i t i s

⊆

mo n

t
n

e a n d s a t i s f i e s Mo d e l

f

c

mp

u t a t i

n

s i n wh i c h t h e a mo u n t

f

r e s

u

r c e s t

b

e u s e d i s u n i q u e l y d e t e r mi n e d . Def. A f u n c t i

n

i s l i n e a r i f i t i s

⊆

mo n

t
n

e a n d s a t i s f i e s Mo d e l

f

c

mp

u t a t i

n

s i n wh i c h r e s

u

r c e s a r e u s e d e x a c t l y

n

c e .

S t a b l e & L i n e a r Ma p s a r e R e s

u

r c e

S

e n s i t i v e .

C

l

l e c t i

n
f

r e s

u

r c e s

SLIDE 26

26

Girard’s Formula

T wo c l

s

e d s t r u c t u r e s

f

c

h

e r e n c e s p a c e s :

T

h e c a t e g

r

y Stbl

f

c

h

. s p a c e s a n d s t a b l e ma p s i s c a r t e s i a n c l

s

e d

–

: t h e c

h

e r e n c e s p a c e f

r

s t a b l e ma p s .

T

h e c a t e g

r

y Lin

f

c

h

. s p a c e s a n d l i n e a r ma p s i s mo n

i

d a l c l

s

e d

–

: t h e c

h

e r e n c e s p a c e f

r

l i n e a r ma p s . T h e y a r e c

mb

i n e d b y i n t r

d

u c i n g t h e “

f

c

u

r s e ” mo d a l i t y : n a t u r a l l y d e f i n e d b y .

M

d

e l

f

I n t u i t . L

g

i c

Th.

L i n e a r D e c

mp
s

t i

n
f

C a r t e s i a n c l

s

e d S t r u c t u r e .

SLIDE 27

27

Ⅰ. Review: : Coherent Spa paces Ⅱ. Coherence e as Uniformity Ⅲ. Linear Admissibility Ⅳ. Concluding Comments

SLIDE 28

28

Review: Uniform Space

U n i f

r

m C

v

e r s a r e g i v e n h

r

i z

n

t a l l y

A

u n i f

r

m s p a c e i s a s e t wi t h a u n i f

r

mi t y : a c

l

l e c t i

n
f

c

v

e r i n g s

f

t h e s e t .

–

E a c h u n i f

r

m c

v

e r i s c

n

s i d e r e d t

b

e c

n

s i s t i n g

f

b a l l s

f

t h e “ s a me s i z e ”

–

T h e y a r e p a r t i a l l y

r

d e r e d b y t h e r e f i n e me n t r e l a t i

n

a n d f

r

m a f i l t e r .

T

h e y a l s

i

n d u c e a t

p
l
g

y i n t h e “ v e r t i c a l ” w a y .

–

T h e b a l l s s u r r

u

n d i n g e a c h p

i

n t f

r

m a n e i g h b

r

h

d

f i l t e r , wh i c h g e n e r a t e s t h e u n i f

r

m t

p
l
g

y .

E

v e r y u n i f

r

mi z a b l e s p a c e h a s t h e f i n e s t u n i f

r

mi t y : t h e f i n e u n i f

r

m s p a c e .

SLIDE 29

29

Review: Uniform Space

U n i f

r

m C

v

e r s a r e g i v e n h

r

i z

n

t a l l y

A

u n i f

r

m s p a c e i s a s e t wi t h a u n i f

r

mi t y : a c

l

l e c t i

n
f

c

v

e r i n g s

f

t h e s e t .

–

E a c h u n i f

r

m c

v

e r i s c

n

s i d e r e d t

b

e c

n

s i s t i n g

f

b a l l s

f

t h e “ s a me s i z e ”

–

T h e y a r e p a r t i a l l y

r

d e r e d b y t h e r e f i n e me n t r e l a t i

n

a n d f

r

m a f i l t e r .

T

h e y a l s

i

n d u c e a t

p
l
g

y i n t h e “ v e r t i c a l ” w a y .

–

T h e b a l l s s u r r

u

n d i n g e a c h p

i

n t f

r

m a n e i g h b

r

h

d

f i l t e r , wh i c h g e n e r a t e s t h e u n i f

r

m t

p
l
g

y .

E

v e r y u n i f

r

mi z a b l e s p a c e h a s t h e f i n e s t u n i f

r

mi t y : t h e f i n e u n i f

r

m s p a c e .

SLIDE 30

30

Review: Uniform Space

U n i f

r

m C

v

e r s a r e g i v e n h

r

i z

n

t a l l y

A

u n i f

r

m s p a c e i s a s e t wi t h a u n i f

r

mi t y : a c

l

l e c t i

n
f

c

v

e r i n g s

f

t h e s e t .

–

E a c h u n i f

r

m c

v

e r i s c

n

s i d e r e d t

b

e c

n

s i s t i n g

f

b a l l s

f

t h e “ s a me s i z e ”

–

T h e y a r e p a r t i a l l y

r

d e r e d b y t h e r e f i n e me n t r e l a t i

n

a n d f

r

m a f i l t e r .

T

h e y a l s

i

n d u c e a t

p
l
g

y i n t h e “ v e r t i c a l ” w a y .

–

T h e b a l l s s u r r

u

n d i n g e a c h p

i

n t f

r

m a n e i g h b

r

h

d

f i l t e r , wh i c h g e n e r a t e s t h e u n i f

r

m t

p
l
g

y .

E

v e r y u n i f

r

mi z a b l e s p a c e h a s t h e f i n e s t u n i f

r

mi t y : t h e f i n e u n i f

r

m s p a c e .

SLIDE 31

31

Review: Uniform Space

n e i g h b

u

r h

d

f i l t e r

N e i g h b

r

s a r e g i v e n v e r t i c a l l y

A

u n i f

r

m s p a c e i s a s e t wi t h a u n i f

r

mi t y : a c

l

l e c t i

n
f

c

v

e r i n g s

f

t h e s e t .

–

E a c h u n i f

r

m c

v

e r i s c

n

s i d e r e d t

b

e c

n

s i s t i n g

f

b a l l s

f

t h e “ s a me s i z e ”

–

T h e y a r e p a r t i a l l y

r

d e r e d b y t h e r e f i n e me n t r e l a t i

n

a n d f

r

m a f i l t e r .

T

h e y a l s

i

n d u c e a t

p
l
g

y i n t h e “ v e r t i c a l ” w a y .

–

T h e b a l l s s u r r

u

n d i n g e a c h p

i

n t f

r

m a n e i g h b

r

h

d

f i l t e r , wh i c h g e n e r a t e s t h e u n i f

r

m t

p
l
g

y .

E

v e r y u n i f

r

mi z a b l e s p a c e h a s t h e f i n e s t u n i f

r

mi t y : t h e f i n e u n i f

r

m s p a c e .

SLIDE 32

32

Review: Uniform Space

Th.

L e t b e t h e f i n e s p a c e c

mp

a t i b l e wi t h a t

p
l
g

i c a l s p a c e .

We ’ v e s e e n t h a t s i t u a t i

n

!

A

u n i f

r

m s p a c e i s a s e t wi t h a u n i f

r

mi t y : a c

l

l e c t i

n
f

c

v

e r i n g s

f

t h e s e t .

–

E a c h u n i f

r

m c

v

e r i s c

n

s i d e r e d t

b

e c

n

s i s t i n g

f

b a l l s

f

t h e “ s a me s i z e ”

–

T h e y a r e p a r t i a l l y

r

d e r e d b y t h e r e f i n e me n t r e l a t i

n

a n d f

r

m a f i l t e r .

T

h e y a l s

i

n d u c e a t

p
l
g

y i n t h e “ v e r t i c a l ” w a y .

–

T h e b a l l s s u r r

u

n d i n g e a c h p

i

n t f

r

m a n e i g h b

r

h

d

f i l t e r , wh i c h g e n e r a t e s t h e u n i f

r

m t

p
l
g

y .

E

v e r y u n i f

r

mi z a b l e s p a c e h a s t h e f i n e s t u n i f

r

mi t y : t h e f i n e u n i f

r

m s p a c e .

SLIDE 33

33

Girard’s Formula

T wo c l

s

e d s t r u c t u r e s

f

c

h

e r e n c e s p a c e s :

T

h e c a t e g

r

y Stbl

f

c

h

. s p a c e s a n d s t a b l e ma p s i s c a r t e s i a n c l

s

e d

–

: t h e c

h

e r e n c e s p a c e f

r

s t a b l e ma p s .

T

h e c a t e g

r

y Lin

f

c

h

. s p a c e s a n d l i n e a r ma p s i s mo n

i

d a l c l

s

e d

–

: t h e c

h

e r e n c e s p a c e f

r

l i n e a r ma p s . T h e y a r e c

mb

i n e d b y i n t r

d

u c i n g t h e “

f

c

u

r s e ” mo d a l i t y : n a t u r a l l y d e f i n e d b y .

M

d

e l

f

I n t u i t . L

g

i c

Th.

L i n e a r D e c

mp
s

t i

n
f

C a r t e s i a n c l

s

e d S t r u c t u r e .

SLIDE 34

34

Coherence as Uniformity

R e c a l l : I n c

h

e r e n c e i mp l i e s d i s j

i

n t n e s s : A p a r t i t i

n
f

i s a n a n t i c l i q u e wh i c h i n d u c e s t h e d i s j

i

n t c

v

e r i n g . C

n

d i t i

n

: a c

h

e r e n c e s p a c e i s d i s j

i

n t l y c

v

e r a b l e i f e v e r y t

k

e n c a n b e e x t e n d e d t

a

p a r t i t i

n

:

A n t i c l i q u e s i n d u c e U n i f

r

mi t y

Th. 1

) P a r t i t i

n

s

f

f

r

m a s u b b a s i s f

r

a u n i f

r

mi t y . 2 ) P a r t i t i

n

s

f

f

r

m a b a s i s f

r

a u n i f

r

mi t y . 3 ) B

t

h u n i f

r

mi t i e s i n d u c e t h e S c

t

t t

p
l
g

y a s t h e u n i f

r

m t

p
l
g

i e s . 4 ) T h e i n d u c e d u n i f

r

mi t y

n

i s f i n e .

Prop.

(homeomorphic.)

C

h

. S p . f

r

a n t i c l i q u e s

SLIDE 35

35

Cauchy Sequences Again

Ex. D e f i n e a c

h

e r e n c e s p a c e f

r

d y a d i c C a u c h y s e q u e n c e s a s :

I

n a n y p a r t i t i

n

, s p

t

l i g h t s

f

d i f f e r e n t c

l
r

s mu s t p r

j

e c t d i s j

i

n t s e t s b y t h e s e c

n

d c

n

d i t i

n
f

t h e i n c

h

e r e n c e .

T

h i s i s i mp

s

s i b l e e s s e n t i a l l y d u e t

S

i e r p i ń s k i ' s t h e

r

e m.

T

h e p a r t i t i

n

s t h e n g e n e r a t e t h e u n i f

r

mi t y c

mp

a t i b l e wi t h t h e r e a l l i n e , t h e r e p r e s e n t a t i

n

i s a u n i f

r

ml y

p

e n ma p .

E a c h i s a p a r t i t i

n
f

, b e c a u s e a ma x i ma l c l i q u e mu s t c

n

t a i n “ a l l c

l
r

s ” . T h e r e a r e n

t

h e r p a r t i t i

n

s c

n

s i s t i n g

f

“ s e v e r a l c

l
r

s ” .

( i n c

h

e r e n c e )

SLIDE 36

36

Linear ⇒ Uniform Continuous

R e c a l l t h a t

Th. E

v e r y l i n e a r ma p i s u n i f

r

ml y c

n

t i n u

u

s wi t h r e s p e c t t

t

h e u n i f

r

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

n

s .

Cor. E

v e r y s t a b l e ma p i s c

n

t i n u

u

s .

( a l t h

u

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

n
f

t h e wh e e l . . . )

Th. Th.

We t h e n c

mb

i n e t h e s e r e s u l t s . A s s u me t h a t p r e s e r v e s ma x i ma l i t y

f

c l i q u e s .

SLIDE 37

37

Ⅰ. Review: : Coherent Spa paces Ⅱ. Coherence e as Uniformity Ⅲ. Linear Admissibility Ⅳ. Concluding Comments

SLIDE 38

38

Standard Representations

L e t b e a H a u s d

r

f f u n i f

r

m s p a c e wi t h a c

u

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

n

s i s t i n g

f

c

u

n t a b l e c

v

e r i n g s .

Fact.

S u c h a s p a c e i s k n

wn

t

b

e s e p a r a b l e me t r i z a b l e .

Def.

T h e s t a n d a r d r e p r e s e n t a t i

n
f

i s g i v e n b y t h e c

h

e r e n c e s p a c e d e f i n e d b y a n d

E a c h Ma x i ma l c l i q u e

f

s p e c i f i e s a t mo s t

n

e p

i

n t .

SLIDE 39

39

Linear Admissibility

S t a n d a r d r e p r e s e n t a t i

n

d

e

s n

t

d e p e n d

n

t h e c h

i

c e

f

b a s i s : T h i s g e n e r a l i z e s t

t

h e n

t

i

n
f

a d mi s s i b i l i t y . An Idea. A r e p r e s e n t a t i

n

i s l i n e a r a d mi s s i b l e i f 1 . f

r

e v e r y u n i f

r

m c

v

e r

f

t h e r e e x i s t s a u n i f

r

m c

v

e r

f

wh i c h r e f i n e s , a n d 2 . f

r

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

n

s a t i s f y i n g ( 1 ) t h e i n c l u s i

n

ma p i s t r a c k e d b y s

me

l i n e a r ma p .

u n i f

r

m c

n

t .

A n a i v e i d e a i s t

mi

mi c A d mi s s i b i l i t y i n T T E , b u t i t d

e

s n ’ t wo r k ! !

SLIDE 40

40

Linear Admissibility

S t a n d a r d r e p r e s e n t a t i

n

d

e

s n

t

d e p e n d

n

t h e c h

i

c e

f

b a s i s : T h i s g e n e r a l i z e s t

t

h e n

t

i

n
f

a d mi s s i b i l i t y . An Idea. A r e p r e s e n t a t i

n

i s l i n e a r a d mi s s i b l e i f 1 . f

r

e v e r y u n i f

r

m c

v

e r

f

t h e r e e x i s t s a u n i f

r

m c

v

e r

f

wh i c h r e f i n e s , a n d 2 . f

r

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

n

s a t i s f y i n g ( 1 ) t h e i n c l u s i

n

ma p i s t r a c k e d b y s

me

l i n e a r ma p .

u n i f

r

m c

n

t .

A n a i v e i d e a i s t

mi

mi c A d mi s s i b i l i t y i n T T E , b u t i t d

e

s n ’ t wo r k ! !

SLIDE 41

41

Linear Admissibility

S t a n d a r d r e p r e s e n t a t i

n

d

e

s n

t

d e p e n d

n

t h e c h

i

c e

f

b a s i s : T h i s g e n e r a l i z e s t

t

h e n

t

i

n
f

a d mi s s i b i l i t y .

Def. A

r e p r e s e n t a t i

n

i s l i n e a r a d mi s s i b l e i f 1 . f

r

e v e r y u n i f

r

m c

v

e r

f

t h e r e e x i s t s a p a r t i t i

n
f

wh i c h r e f i n e s , a n d 2 . f

r

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

n

s a t i s f y i n g ( 1 ) t h e i n c l u s i

n

ma p i s t r a c k e d b y s

me

l i n e a r ma p .

Th. (

1 ) E v e r y u n i f

r

ml y c

n

t i n u

u

s ma p s i s t h e n l i n e a r l y r e a l i z a b l e w . r . t . l i n e a r a d mi s s i b l e r e p r e s e n t a t i

n

s . ( 2 ) E v e r y s t a n d a r d r e p r e s e n t a t i

n

i s l i n e a r a d mi s s i b l e .

s t r

n

g l y u n i f

r

m c

n

t .

SLIDE 42

42

Chain-Connectedness

Def.

A u n i f

r

m s p a c e i s c h a i n

c
n

n e c t e d i f A t y p i c a l e x a mp l e : ( t h

u

g h i t i s t

t

a l l y d i s c

n

n e c t e d ) I n p a r t i c u l a r , e v e r y t w

me

mb e r s

f

a u n i f

r

m c

v

e r i s “ c h a i n l y c

n

n e c t e d ”

SLIDE 43

43

Chain-Connectedness

Def.

A u n i f

r

m s p a c e i s c h a i n

c
n

n e c t e d i f A t y p i c a l e x a mp l e : ( t h

u

g h i t i s t

t

a l l y d i s c

n

n e c t e d ) I n p a r t i c u l a r , e v e r y t w

me

mb e r s

f

a u n i f

r

m c

v

e r i s “ c h a i n l y c

n

n e c t e d ”

SLIDE 44

44

Chain-Connectedness

Def.

A u n i f

r

m s p a c e i s c h a i n

c
n

n e c t e d i f A t y p i c a l e x a mp l e : ( t h

u

g h i t i s t

t

a l l y d i s c

n

n e c t e d ) I n p a r t i c u l a r , e v e r y t w

me

mb e r s

f

a u n i f

r

m c

v

e r i s “ c h a i n l y c

n

n e c t e d ”

SLIDE 45

45

Chain-Connectedness

Def.

A u n i f

r

m s p a c e i s c h a i n

c
n

n e c t e d i f A t y p i c a l e x a mp l e : ( t h

u

g h i t i s t

t

a l l y d i s c

n

n e c t e d ) I n p a r t i c u l a r , e v e r y t w

me

mb e r s

f

a u n i f

r

m c

v

e r i s “ c h a i n l y c

n

n e c t e d ”

SLIDE 46

46

Principal Lemma

S u p p

s

e t h a t i s a c h a i n

c
n

n e c t e d s e p a r a b l e me t r i z a b l e s p a c e a n d t h a t i s a u n i f

r

m b a s i s c

n

s i t i n g

f
p

e n c

v

e r i n g s . N.B. E v e r y u n i f

r

m s p a c e h a s a b a s i s

f
p

e n c

v

e r i n g s .

Lem. T h e s t a n d a r d r e p r e s e n t a t i

n

i s a t

p
l
g

i c a l l y a n d u n i f

r

ml y

p

e n ma p .

T
p
l
g

i c a l

p

e n n e s s i s i mme d i a t e f r

m

t h e a s s u mp t i

n

.

U

n i f

r

m

p

e n n e s s i s e s s e n t i a l l y d u e t

“
n

e

c
l
r

e d n e s s ”

f

p a r t i t i

n

s .

SLIDE 47

47

Principal Lemma

S u p p

s

e t h a t i s a c h a i n

c
n

n e c t e d s e p a r a b l e me t r i z a b l e s p a c e a n d t h a t i s a u n i f

r

m b a s i s c

n

s i t i n g

f
p

e n c

v

e r i n g s . N.B. E v e r y u n i f

r

m s p a c e h a s a b a s i s

f
p

e n c

v

e r i n g s .

Lem. T h e s t a n d a r d r e p r e s e n t a t i

n

i s a t

p
l
g

i c a l l y a n d u n i f

r

ml y

p

e n ma p .

T
p
l
g

i c a l

p

e n n e s s i s i mme d i a t e f r

m

t h e a s s u mp t i

n

.

U

n i f

r

m

p

e n n e s s i s e s s e n t i a l l y d u e t

“
n

e

c
l
r

e d n e s s ”

f

p a r t i t i

n

s .

SLIDE 48

48

Principal Lemma

S u p p

s

e t h a t i s a c h a i n

c
n

n e c t e d s e p a r a b l e me t r i z a b l e s p a c e a n d t h a t i s a u n i f

r

m b a s i s c

n

s i t i n g

f
p

e n c

v

e r i n g s . N.B. E v e r y u n i f

r

m s p a c e h a s a b a s i s

f
p

e n c

v

e r i n g s .

Lem. T h e s t a n d a r d r e p r e s e n t a t i

n

i s a t

p
l
g

i c a l l y a n d u n i f

r

ml y

p

e n ma p .

T
p
l
g

i c a l

p

e n n e s s i s i mme d i a t e f r

m

t h e a s s u mp t i

n

.

U

n i f

r

m

p

e n n e s s i s e s s e n t i a l l y d u e t

“
n

e

c
l
r

e d n e s s ”

f

p a r t i t i

n

s .

SLIDE 49

49

Principal Lemma

S u p p

s

e t h a t i s a c h a i n

c
n

n e c t e d s e p a r a b l e me t r i z a b l e s p a c e a n d t h a t i s a u n i f

r

m b a s i s c

n

s i t i n g

f
p

e n c

v

e r i n g s . N.B. E v e r y u n i f

r

m s p a c e h a s a b a s i s

f
p

e n c

v

e r i n g s .

Lem. T h e s t a n d a r d r e p r e s e n t a t i

n

i s a t

p
l
g

i c a l l y a n d u n i f

r

ml y

p

e n ma p .

T
p
l
g

i c a l

p

e n n e s s i s i mme d i a t e f r

m

t h e a s s u mp t i

n

.

U

n i f

r

m

p

e n n e s s i s e s s e n t i a l l y d u e t

“
n

e

c
l
r

e d n e s s ”

f

p a r t i t i

n

s .

SLIDE 50

50

Principal Lemma

S u p p

s

e t h a t i s a c h a i n

c
n

n e c t e d s e p a r a b l e me t r i z a b l e s p a c e a n d t h a t i s a u n i f

r

m b a s i s c

n

s i t i n g

f
p

e n c

v

e r i n g s . N.B. E v e r y u n i f

r

m s p a c e h a s a b a s i s

f
p

e n c

v

e r i n g s .

Lem. T h e s t a n d a r d r e p r e s e n t a t i

n

i s a t

p
l
g

i c a l l y a n d u n i f

r

ml y

p

e n ma p .

T
p
l
g

i c a l

p

e n n e s s i s i mme d i a t e f r

m

t h e a s s u mp t i

n

.

U

n i f

r

m

p

e n n e s s i s e s s e n t i a l l y d u e t

“
n

e

c
l
r

e d n e s s ”

f

p a r t i t i

n

s .

SLIDE 51

51

Main Result

Th. I f i s c h a i n

c
n

n e c t e d ,

:

s t a b l y r e a l i z a b l e ⇔ i t i s c

n

t i n u

u

s .

:

l i n e a r l y r e a l i z a b l e ⇔ i t i s u n i f

r

ml y c

n

t i n u

u

s . Cor. I f i s l i n e a r l y r e a l i z a b l e t h e n i t i s u n i f

r

ml y c

n

t i n u

u

s

n

e a c h c h a i n

c
n

n e c t e d c

mp
n

e n t .

L e t a n d b e s e p a r a b l e me t r i z a b l e u n i f

r

m s p a c e s wi t h l i n e a r a d mi s s i b l e r e p r e s e n t a t i

n

s , a n d .

C

n

v e r s e l y , e v e r y u n i f

r

ml y c

n

t i n u

u

s f u n c t i

n

i s l i n e a r l y r e a l i z a b l e . T

c
mp

l e t e t h e c

r
l

l a r y , w e n e e d t

r

e d u c e t h e c

mp
n

e n t s b y i d e n t i f y i n g s

me
f

t h e m. R e c a l l : e v e r y s e p a r a b l e me t r i z a b l e s p a c e h a s t h e s t a n d a r d r e p r e s e n t a t i

n

, h e n c e h a s l i n e a r a d mi s s i b l e r e p r e s e n t a t i

n

s .

SLIDE 52

52

Ⅰ. Review: : Coherent Spa paces Ⅱ. Coherence e as Uniformity Ⅲ. Linear Admissibility Ⅳ. Concluding Comments

SLIDE 53

53

T

w

a r d s L i n e a r R e a l i z a b i l i t y

A

l i n e a r c

mb

i n a t

r

y a l g e b r a ( L C A ) U

l i n

i s d e f i n e d f r

m

t h e “ u n i v e r s a l c

h

e r e n c e s p a c e ” .

C
h

e r e n t R e p r e s e n t a t i

n

s = M

d

e s t S e t s

v

e r U

l i n

Mo

d

( U

l i n

) i s a mo d e l

f

l i n e a r l

g

i c .

I

' m e s s e n t i a l l y a r e a l i s t . . . b u t s t i l l a b i t a d r e a me r

s
I

i ma g i n e t h e r e ' s a k i n d

f

“ L i n e a r A n a l y s i s ” a s t h e d e c

mp
s

i t i

n
f

C

mp

u t a b l e A n a l y s i s .

– A

n a n a l

g

y

f

t h e d i s c

v

e r y

f

L i n e a r L

g

i c .

– E

v e r y ma t h e ma t i c a l s p a c e h a s “ a d mi s s i b l e ” r e p r e s e n t a t i

n

s i n s

me

s e n s e , a n d f u n c t i

n

s a r e a l l l i n e a r l y c

mp

u t a b l e . . .

SLIDE 54

54

T

w

a r d s C

mp

l e x i t y i n A n a l y s i s

I

t s e e ms h

p

e l e s s t h a t t h e c a t e g

r

y

f

l i n e a r a d mi s s i b l e r e p r e s e n t a t i

n

s i s mo n

i

d a l c l

s

e d b e c a u s e f u n c t i

n

s p a c e s a r e n

t

s e p a r a b l e me t r i z a b l e .

N
n

e t h e l e s s , I s t i l l b e l i e v e t h a t l i n e a r i t y i s s t r

n

g l y r e l a t e d t

u

n i f

r

m s t r u c t u r e s b e c a u s e : Th (Férée-Gomaa-Hoyrup' 13). F

r

a n y r e a l f u n c t i

n

a l

l i n e a r i n

u

r t e r mi n

l
g

y

F i s “ u n i f

r

ml y c

n

t i n u

u

s ” w . r . t . a k i n d

f

u n i f

r

mi t y

n

C [ , 1 ] : We c a n e x p l a i n t h i s p h e n

me

n

n

i n

u

r mo d e l :

U n i f

r

mi t y

n

C [ , 1 ] a p

i

n t i n [ , 1 ] & a u n i f

r

mi t y

f

R

SLIDE 55

55

A P

s

s i b l e W a y : Q u a s i

U

n i f

r

mi t y

I

n s

me

s e n s e , s e p a r a b l e me t r i s a b i l i t y

f

l i n e a r a d mi s s i b l e r e p r e s e n t a t i

n

s s e e ms i n e v i t a b l e :

– B

e c a u s e

f

c

u

n t a b i l i t y

f

c

h

e r e n c e s p a c e s .

We

mu s t h a v e a c

u

n t a b l e b a s i s

f

c

u

n t a b l e c

v

e r i n g s .

B

u t . . . t h e H a u s d

r

f f p r

p

e r t y s e e ms n

t

n e c e s s a r y .

j

u s t u s e d f

r

w e l l

d

e f i n e d n e s s

f

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

n

.

T

h e u s e

f

n

n
H

a u s d

r

f f me t r i c s e e ms a p

s

s i b l e a n s w e r .

E

v e r y s e c

n

d

c
u

n t a b l e T

1

s

p a c e i s s e p a r a b l e q u a s i

me

t r i z a b l e .

– I

t i s v e r y l i k e l y t h a t t h e y h a v e t h e s t a n d a r d r e p r e s e n t a t i

n

s f

r

q u a s i

u

n i f

r

m s p a c e s .