Coherence Spaces for Computable Analysis II K e i M a t s u - - PowerPoint PPT Presentation

SLIDE 1

1

Coherence Spaces for Computable Analysis II

K e i M a t s u mo t

• a

n d K a z u s h i g e T e r u i

( R I M S , K y

• t
• U

n i v e r s i t y )

Supported by JSPS Core-to-Core Program

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 e 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 ]

, w h e r e

t w

• mo

r p h i s ms coexist: s t a b l e / l i n e a r ma p 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 .

SLIDE 4

4

Main Result from [MT '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

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 .

SLIDE 5

5

Main Result of This Talk

Conjecture

( b y a L I C S r e v i e w e r ) .

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

• n

s b a s e d

• n

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

• n

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

• n

s b a s e d

• n

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

• n

s

I would very much like to know whether the author just reinvented John Longley's sequentially realizable functions. If they did, then their notion of computability for coherence spaces will presumably coincide with that for sequentially realizable functionals. (…) I conjecture that your realizability model is equivalent to the category

• f representations over the sequential functionals.

CohRep ≃ SeqRep

’’

SLIDE 6

6

Main Result of This Talk

Conjecture

( b y a L I C S r e v i e w e r ) .

T h e c

• n

j e c t u r e i s F A L S E .

I would very much like to know whether the author just reinvented John Longley's sequentially realizable functions. If they did, then their notion of computability for coherence spaces will presumably coincide with that for sequentially realizable functionals. (…) I conjecture that your realizability model is equivalent to the category

• f representations over the sequential functionals.

CohRep ≃ SeqRep

’’

SLIDE 7

7

Ⅰ. Review: Coherent Representations Ⅱ. Sequential Representations Ⅲ. Inequivalence Ⅳ. Conclusion and Future Works

SLIDE 8

8

Coherence Spaces

Def. A c

• h

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

• 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 x y i f f

x ≠ y a

n d

x y (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 . T h e c l i q u e s p a c e Clq(X) f

• r

ms a S c

• t

t d

• ma

i n w . r . t . t h e s e t i n c l u s i

• n

.

• 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 .

X=(|X|, )

( )

|X|

( () ()

|X|

SLIDE 9

9

Examples

Def. D e f i n e a d i s c r e t e c

• h

e r e n c e s p a c e N b y :

• |N| = ℕ
• n

m i f f n=m Clq(N) = {{n}: n ∈ ℕ} ∪ { } ≈

ℕ⊥

T h e s e t

• f

d y a d i c r a t i

• n

a l s

D =

× ℤ ℕ, i

d e n t i f i e d a s

(m,n) ～ m/2n

Def. D e f i n e a c

• h

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

• l

l

• ws

:

• F
• r

e a c h

d=(m,n) and e=(m',n'), d e i f f n≠n' and |d-e| ≤ 2－n + 2－n'

Ma x i ma l c l i q u e s

• f

R ≈

( 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|=D (

( )

∅ ∅ {0} {1} {2}

…

SLIDE 10

10

Summary: Coherence Spaces

• 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 e S c

• t

t

• d
• ma

i n s .

• T

wo k i n d s

• f

mo r p h i s ms c

• e

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

– S

t a b i l i t y i s a n a n a l

• g

u e

• f

p r

• g

r a ms f

• r

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

• f

i n p u t n e e d e d t

• p

r

• d

u c e a n

• u

t p u t i s u n i q u e l y d e t e r mi n e d ,

– L

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

• g

u e

• f

p r

• g

r a ms wh i c h ma k e s e x a c t l y

• n

e q u e r y d u r i n g c

• mp

u t a t i

• n

.

• T

wo c a t e g

• r

i 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 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 .

• L

i n e a r d e c

• mp
• s

i t i

• n

:

– T

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

• l

i n e a r l

• g

i c [ G i r a r d ' 8 6 ] .

Stbl(X ,Y )≃Lin(! X ,Y )

SLIDE 11

11

Coh-Representations

Def. A ( C

• h
• )

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

• n

i s a t u p l e X → X

• f

a c

• h

e r e n c e s p a c e X, a s e t X, a n d a s u r j e c t i v e ma p ρX:⊆Clq(X)→X . Def. A f u n c t i

• n

f: X→Y i s s t a b l y r e a l i z a b l e i f t h e r e e x i s t s a s t a b l e ma p F: X→Y s . t . : Ex. T h e C

• h
• r

e p r e s e n t a t i

• n

N → ℕ⊥ ma p s t h e s i n g l e t

• n

{n} t

• t

h e n u mb e r n a n d t h e e mp t y s e t t

• t

h e b

• t

t

• m.

Ex. T h e C

• h
• r

e p r e s e n t a t i

• n

R →

ℝ s

e n d s a C a u c h y s e q u e n c e t

• i

t s l i mi t . Prop. ( [ MT ' 1 5 ] ) ( i ) f :ℕ⊥→ℕ⊥ i s s t a b l y r e a l i z a b l e ⇔ i t i s mo n

• t
• n

e . ( i i ) f :ℝ→ℝ i s 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 . ( i i i ) f :ℝ→ℝ i 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 .

ρX ρY f

Y X Y

F

X

ρX

1 2

…

⊥

ρ0 ρR

SLIDE 12

12

The Category CohRep

Def. CohRep i s t h e c a t e g

• r

y

• f

c

• h

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

• n

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

• n

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

• r

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

• n

s c

• r

r e s p

• n

d t

• mo

d e s t s e t s

• v

e r s

• me

P C A ( p a r t i a l c

• mb

i n a t

• r

y a l g e b r a ) .

• L

e t Ucoh b e t h e P C A

• b

t a 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 .

• CohRep Mod(Ucoh

) S i n c e c a t e g

• r

i e s

• f

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

• s

e d , s

• i

s CohRep.

≃

SLIDE 13

13

Ⅰ. Review: Coherent Representations Ⅱ. Sequential Representations Ⅲ. Inequivalence Ⅳ. Conclusion and Future Works

SLIDE 14

14

Sequential functionals

S e q u e n t i a l i t y : a p r

• p

e r t y

• f

p r

• g

r a ms wh i c h ma k e q u e r i e s t

• t

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

• d

e t e r mi n e d s e q u e n t i a l wa y . Def. ( v a n O

• s

t e n ' s s e q u e n t i a l g a me ) L e t [ ℕ ℕ ] b e t h e s e t

• f

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

• n

s . A f u n c t i

• n

a l F: [ ℕ ℕ ] → ℕ

⊥

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

• mp

u t e d b y a s t r a t e g y α: ℕ ℕ s u c h t h a t g i v e n β: ℕ ℕ , F(β) = n i f f

• α[] = 2m0, β(m0) =u1
• α[u1] = 2m1, β(m1) =u2
• α[u1;u2 ] = 2m2, β(m2) =u3

。。。 。。。

• α[u1...uk-1] = 2mk-1, β(mk-1) =uk
• α[u1...uk] = 2n+1 f
• r

s

• me

{ui}, {mi},

SLIDE 15

15

Decision Tree Strategy

T h e f

• r

me r p l a y e r α i s a s t r a t e g y d e s c r i b e d b y a d e c i s i

• n

t r e e l i k e t h i s : I t s a c t i

• n

i s e i t h e r

• f

:

• ?n :

q u e r y t h e v a l u e β(n)

• !m :
• u

t p u t t h e v a l u e m a n d h a l t . T h e c h

• i

c e i s c

• mp

l e t e l y d e t e r mi n e d b y t h e l a t t e r p l a y e r ' s h i s t

• r

y . ?5 ?3 ?1 !4 !7 ?0

2 3 1

... ...

SLIDE 16

16

Decision Tree Strategy

T h e f

• r

me r p l a y e r α i s a s t r a t e g y d e s c r i b e d b y a d e c i s i

• n

t r e e l i k e t h i s : I t s a c t i

• n

i s e i t h e r

• f

:

• ?n :

q u e r y t h e v a l u e β(n)

• !m :
• u

t p u t t h e v a l u e m a n d h a l t . T h e c h

• i

c e i s c

• mp

l e t e l y d e t e r mi n e d b y t h e l a t t e r p l a y e r ' s h i s t

• r

y . ?5 ?3 ?1 !4 !7 ?0

2 3 1

... ...

G i v e n β: ℕ ℕ

SLIDE 17

17

Decision Tree Strategy

T h e f

• r

me r p l a y e r α i s a s t r a t e g y d e s c r i b e d b y a d e c i s i

• n

t r e e l i k e t h i s : I t s a c t i

• n

i s e i t h e r

• f

:

• ?n :

q u e r y t h e v a l u e β(n)

• !m :
• u

t p u t t h e v a l u e m a n d h a l t . T h e c h

• i

c e i s c

• mp

l e t e l y d e t e r mi n e d b y t h e l a t t e r p l a y e r ' s h i s t

• r

y . ?5 ?3 ?1 !4 !7 ?0

2 3 1

... ...

q u e r y β(5)

SLIDE 18

18

Decision Tree Strategy

T h e f

• r

me r p l a y e r α i s a s t r a t e g y d e s c r i b e d b y a d e c i s i

• n

t r e e l i k e t h i s : I t s a c t i

• n

i s e i t h e r

• f

:

• ?n :

q u e r y t h e v a l u e β(n)

• !m :
• u

t p u t t h e v a l u e m a n d h a l t . T h e c h

• i

c e i s c

• mp

l e t e l y d e t e r mi n e d b y t h e l a t t e r p l a y e r ' s h i s t

• r

y . ?5 ?3 ?1 !4 !7 ?0

2 3 1

... ...

I f β(5)=2,

SLIDE 19

19

Decision Tree Strategy

T h e f

• r

me r p l a y e r α i s a s t r a t e g y d e s c r i b e d b y a d e c i s i

• n

t r e e l i k e t h i s : I t s a c t i

• n

i s e i t h e r

• f

:

• ?n :

q u e r y t h e v a l u e β(n)

• !m :
• u

t p u t t h e v a l u e m a n d h a l t . T h e c h

• i

c e i s c

• mp

l e t e l y d e t e r mi n e d b y t h e l a t t e r p l a y e r ' s h i s t

• r

y . ?5 ?3 ?1 !4 !7 ?0

2 3 1

... ...

q u e r y β(3)

SLIDE 20

20

Decision Tree Strategy

T h e f

• r

me r p l a y e r α i s a s t r a t e g y d e s c r i b e d b y a d e c i s i

• n

t r e e l i k e t h i s : I t s a c t i

• n

i s e i t h e r

• f

:

• ?n :

q u e r y t h e v a l u e β(n)

• !m :
• u

t p u t t h e v a l u e m a n d h a l t . T h e c h

• i

c e i s c

• mp

l e t e l y d e t e r mi n e d b y t h e l a t t e r p l a y e r ' s h i s t

• r

y . ?5 ?1 !4 !7 ?0

2 3 1

... ...

i f β(3)=0,

?3

SLIDE 21

21

Decision Tree Strategy

T h e f

• r

me r p l a y e r α i s a s t r a t e g y d e s c r i b e d b y a d e c i s i

• n

t r e e l i k e t h i s : I t s a c t i

• n

i s e i t h e r

• f

:

• ?n :

q u e r y t h e v a l u e β(n)

• !m :
• u

t p u t t h e v a l u e m a n d h a l t . T h e c h

• i

c e i s c

• mp

l e t e l y d e t e r mi n e d b y t h e l a t t e r p l a y e r ' s h i s t

• r

y .

... ...

Output 7

?5 ?1 !4 !7 ?0

2 3 1

?3

SLIDE 22

22

Decision Tree Strategy

T h e f

• r

me r p l a y e r α i s a s t r a t e g y d e s c r i b e d b y a d e c i s i

• n

t r e e l i k e t h i s : I t s a c t i

• n

i s e i t h e r

• f

:

• ?n :

q u e r y t h e v a l u e β(n)

• !m :
• u

t p u t t h e v a l u e m a n d h a l t . T h e c h

• i

c e i s c

• mp

l e t e l y d e t e r mi n e d b y t h e l a t t e r p l a y e r ' s h i s t

• r

y . ?5 ?3 ?1 !4 !7 ?0

2 3 1

... ...

q u e r y β(5)

SLIDE 23

23

Decision Tree Strategy

T h e f

• r

me r p l a y e r α i s a s t r a t e g y d e s c r i b e d b y a d e c i s i

• n

t r e e l i k e t h i s : I t s a c t i

• n

i s e i t h e r

• f

:

• ?n :

q u e r y t h e v a l u e β(n)

• !m :
• u

t p u t t h e v a l u e m a n d h a l t . T h e c h

• i

c e i s c

• mp

l e t e l y d e t e r mi n e d b y t h e l a t t e r p l a y e r ' s h i s t

• r

y . ?5 ?3 ?1 !4 !7 ?0

2 3 1

... ...

i f β(5)=3,

SLIDE 24

24

Decision Tree Strategy

T h e f

• r

me r p l a y e r α i s a s t r a t e g y d e s c r i b e d b y a d e c i s i

• n

t r e e l i k e t h i s : I t s a c t i

• n

i s e i t h e r

• f

:

• ?n :

q u e r y t h e v a l u e β(n)

• !m :
• u

t p u t t h e v a l u e m a n d h a l t . T h e c h

• i

c e i s c

• mp

l e t e l y d e t e r mi n e d b y t h e l a t t e r p l a y e r ' s h i s t

• r

y . ?5 ?3 ?1 !4 !7 ?0

2 3 1

... ...

No Arrow! =diverge

SLIDE 25

25

The Category SeqRep Mod(B)

L e t B b e a P C A f

• r

s e q u e n t i a l r e a l i z a b i l i t y p r

• p
• s

e d b y [ v a n O

• s

t e n ' 9 7 ] . Def. SeqRep i s t h e c a t e g

• r

y wh

• s

e

• o

b j e c t s a r e s u r j e c t i v e p a r t i a l ma p s f r

• m

[ ℕ ℕ ]

• mo

r p h i s ms a r e s e q u e n t i a l l y r e a l i z a b l e f u n c t i

• n

s SeqRep Mod(B) i s a l s

• a

r e g u l a r c . c . c . ≃

≃

σX σY f

Y

[ ℕ ℕ] [ ℕ ℕ]

F

X

s e q u e n t i a l

SLIDE 26

26

Relationship with Hypercoherence Model

• [

v a n O

• s

t e n ' 9 7 ] [ L

• n

g l e y ' 2 ] : F

• r

f u n c t i

• n

a l s

• f

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

• n

g l y s t a b l e

• T

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

• n

i s d

• n

e i n t h e c a t e g

• r

y

• f

h y p e r c

• h

e r e n c e a n d s t r

• n

g l y s t a b l e ma p s i n t r

• d

u c e d b y E h r h a r d .

• C
• h

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

• f

h y p e r c

• h

e r e n c e .

• S
• w

e a r e mo t i v a t e d t

• c
• mp

a r e s e q u e n t i a l r e a l i z a b i l i t y w i t h

• u

r s t a b l e r e a l i z a b i l i t y .

SLIDE 27

27

Ⅰ. Review: Coherence Representations Ⅱ. Sequential Representations Ⅲ. Inequivalence Ⅳ. Conclusion and Future Works

SLIDE 28

28

Main Theorem

Th. T h e c a t e g

• r

i e s CohRep a n d SeqRep a r e n

• t

e q u i v a l e n t . I t i s i n t u i t i v e l y c l e a r : we h a v e a c

• u

n t e r e x a mp l e a s k n

• wn

a s G u s t a v f u n c t i

• n

, t h e t r i p l e v e r s i

• n
• f

t h e p a r a l l e l

• r

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

• t

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

• r

s p e c i f i c “ n a t u r a l ” r e p r e s e n t a t i

• n

s

• f

ℕ

⊥.

Continuous Stable Sequential Parallel-or Gustav

SLIDE 29

29

Gustav Function (1)

A G u s t a v f u n c t i

• n

G: [ℕ

⊥ → ℕ ⊥] → ℕ ⊥ i

s d e f i n e d b y

• G(f):= 0 i

f

• n

e

• f

t h e m i s s a t i s f i e d :

– (I) f(0)=0, f(1)=1, – (II)f(0)=1,

f(2)=0,

– (III) f(1)=0, f(2)=1

• G(f):=⊥ o

t h e r wi s e .

• Prop. G

i s n

• t

s e q u e n t i a l l y r e a l i z a b l e w . r . t . [ ] ℕ ℕ → ℕ

⊥ ,

wh e r e σ0(α):=n ∈ ℕ

⊥ i

f f α(0)= n a n d α(m+1)= ⊥ f

• r

a n y m.

σ0

?0 ?1 1 ?2

Strategy must make a query first.

SLIDE 30

30

Gustav Function (1)

A G u s t a v f u n c t i

• n

G: [ℕ

⊥ → ℕ ⊥] → ℕ ⊥ i

s d e f i n e d b y

• G(f):= 0 i

f

• n

e

• f

t h e m i s s a t i s f i e d :

– (I) f(0)=0, f(1)=1, – (II)f(0)=1,

f(2)=0,

– (III) f(1)=0, f(2)=1

• G(f):=⊥ o

t h e r wi s e .

• Prop. G

i s n

• t

s e q u e n t i a l l y r e a l i z a b l e w . r . t . [ ] ℕ ℕ → ℕ

⊥ ,

wh e r e σ0(α):=n ∈ ℕ

⊥ i

f f α(0)= n a n d α(m+1)= ⊥ f

• r

a n y m.

σ0

?0 ?1 1 ?2

assume that you query f(0) first

SLIDE 31

31

Gustav Function (1)

A G u s t a v f u n c t i

• n

G: [ℕ

⊥ → ℕ ⊥] → ℕ ⊥ i

s d e f i n e d b y

• G(f):= 0 i

f

• n

e

• f

t h e m i s s a t i s f i e d :

– (I) f(0)=0, f(1)=1, – (II)f(0)=1,

f(2)=0,

– (III) f(1)=0, f(2)=1

• G(f):=⊥ o

t h e r wi s e .

• Prop. G

i s n

• t

s e q u e n t i a l l y r e a l i z a b l e w . r . t . [ ] ℕ ℕ → ℕ

⊥ ,

wh e r e σ0(α):=n ∈ ℕ

⊥ i

f f α(0)= n a n d α(m+1)= ⊥ f

• r

a n y m.

σ0

?0 ?1 1 ?2

Take an opponent

f(1)=0 f(2)=1 f(0)↑

SLIDE 32

32

Gustav Function (1)

A G u s t a v f u n c t i

• n

G: [ℕ

⊥ → ℕ ⊥] → ℕ ⊥ i

s d e f i n e d b y

• G(f):= 0 i

f

• n

e

• f

t h e m i s s a t i s f i e d :

– (I) f(0)=0, f(1)=1, – (II)f(0)=1,

f(2)=0,

– (III) f(1)=0, f(2)=1

• G(f):=⊥ o

t h e r wi s e .

• Prop. G

i s n

• t

s e q u e n t i a l l y r e a l i z a b l e w . r . t . [ ] ℕ ℕ → ℕ

⊥ ,

wh e r e σ0(α):=n ∈ ℕ

⊥ i

f f α(0)= n a n d α(m+1)= ⊥ f

• r

a n y m.

σ0

?0 ?1 1 ?2 f(0) i

s u n d e f i n e d ,

SLIDE 33

33

Gustav Function (1)

A G u s t a v f u n c t i

• n

G: [ℕ

⊥ → ℕ ⊥] → ℕ ⊥ i

s d e f i n e d b y

• G(f):= 0 i

f

• n

e

• f

t h e m i s s a t i s f i e d :

– (I) f(0)=0, f(1)=1, – (II)f(0)=1,

f(2)=0,

– (III) f(1)=0, f(2)=1

• G(f):=⊥ o

t h e r wi s e .

• Prop. G

i s n

• t

s e q u e n t i a l l y r e a l i z a b l e w . r . t . [ ] ℕ ℕ → ℕ

⊥ ,

wh e r e σ0(α):=n ∈ ℕ

⊥ i

f f α(0)= n a n d α(m+1)= ⊥ f

• r

a n y m.

σ0

?0 ?1 1 ?2

Case(III) if f(1)=0, f(2)=1, but f(0)↑

while G(f) i s d e f i n e d .

SLIDE 34

34

Gustav Function (1)

A G u s t a v f u n c t i

• n

G: [ℕ

⊥ → ℕ ⊥] → ℕ ⊥ i

s d e f i n e d b y

• G(f):= 0 i

f

• n

e

• f

t h e m i s s a t i s f i e d :

– (I) f(0)=0, f(1)=1, – (II)f(0)=1,

f(2)=0,

– (III) f(1)=0, f(2)=1

• G(f):=⊥ o

t h e r wi s e .

• Prop. G

i s n

• t

s e q u e n t i a l l y r e a l i z a b l e w . r . t . [ ] ℕ ℕ → ℕ

⊥ ,

wh e r e σ0(α):=n ∈ ℕ

⊥ i

f f α(0)= n a n d α(m+1)= ⊥ f

• r

a n y m.

σ0

?0 ?1 1 ?2

×

So, you fail the game

Case(III) if f(1)=0, f(2)=1, but f(0)↑

SLIDE 35

35

Gustav Function (1)

A G u s t a v f u n c t i

• n

G: [ℕ

⊥ → ℕ ⊥] → ℕ ⊥ i

s d e f i n e d b y

• G(f):= 0 i

f

• n

e

• f

t h e m i s s a t i s f i e d :

– (I) f(0)=0, f(1)=1, – (II)f(0)=1,

f(2)=0,

– (III) f(1)=0, f(2)=1

• G(f):=⊥ o

t h e r wi s e .

• Prop. G

i s n

• t

s e q u e n t i a l l y r e a l i z a b l e w . r . t . [ ] ℕ ℕ → ℕ

⊥ ,

wh e r e σ0(α):=n ∈ ℕ

⊥ i

f f α(0)= n a n d α(m+1)= ⊥ f

• r

a n y m.

σ0

?1 ?2 1 ?0

Case(II) if f(2)=0, f(0)=1, but f(1)↓

×

Whatever the first query is...

SLIDE 36

36

Gustav Function (1)

A G u s t a v f u n c t i

• n

G: [ℕ

⊥ → ℕ ⊥] → ℕ ⊥ i

s d e f i n e d b y

• G(f):= 0 i

f

• n

e

• f

t h e m i s s a t i s f i e d :

– (I) f(0)=0, f(1)=1, – (II)f(0)=1,

f(2)=0,

– (III) f(1)=0, f(2)=1

• G(f):=⊥ o

t h e r wi s e .

• Prop. G

i s n

• t

s e q u e n t i a l l y r e a l i z a b l e w . r . t . [ ] ℕ ℕ → ℕ

⊥ ,

wh e r e σ0(α):=n ∈ ℕ

⊥ i

f f α(0)= n a n d α(m+1)= ⊥ f

• r

a n y m.

σ0

?2 ?0 1 ?1

Case(I) if f(0)=0, f(1)=1, but f(2)↓

×

Whatever the first query is...

SLIDE 37

37

Gustav Function (1)

A G u s t a v f u n c t i

• n

G: [ℕ

⊥ → ℕ ⊥] → ℕ ⊥ i

s d e f i n e d b y

• G(f):= 0 i

f

• n

e

• f

t h e m i s s a t i s f i e d :

– (I) f(0)=0, f(1)=1, – (II)f(0)=1,

f(2)=0,

– (III) f(1)=0, f(2)=1

• G(f):=⊥ o

t h e r wi s e .

• Prop. G

i s n

• t

s e q u e n t i a l l y r e a l i z a b l e w . r . t . [ ] ℕ ℕ → ℕ

⊥ ,

wh e r e σ0(α):=n ∈ ℕ

⊥ i

f f α(0)= n a n d α(m+1)= ⊥ f

• r

a n y m.

σ0

?0 ?1 1 ?2

Case(III) if f(1)=0, f(2)=1, but f(0)↓

×

You must fail for some opp.

Contradiction.

SLIDE 38

38

Gustav Function (2)

A G u s t a v f u n c t i

• n

G: [ℕ⊥ → ℕ⊥] → ℕ

⊥ i

s d e f i n e d b y

• G(f):= 0 i

f

• n

e

• f

t h e m i s s a t i s f i e d :

– (I) f(0)=0, f(1)=1, – (II)f(0)=1,

f(2)=0,

– (III) f(1)=0, f(2)=1

• G(f):=⊥ o

t h e r wi s e .

• Prop. O

n t h e

• t

h e r h a n d , G i s s t a b l y r e a l i z a b l e w . r . t . N → ℕ

⊥ :

Tr(G):= {(0,0)(1,1) → 0; (0,1)(2,0) → 0; (1,0)(2,1) → 0} S

• a

f u n c t

• r

F: SeqRep → CohRep c a n n

• t

b e p a r t

• f

e q u i v a l e n c e i f i t ma p s σ0 t

• ρ0

ρ0

SLIDE 39

39

Proof Idea (1)

F

• r

a r i g

• r
• u

s p r

• f

, we h a v e t

• s

h

• w

t h a t n

• f

u n c t

• r

F: SeqRep → CohRep i

s p a r t

• f

e q u i v a l e n c e .

B y c

• n

t r a d i c t i

• n

. A s s u me t h a t F: SeqRep → CohRep i s p a r t

• f

e q u i v a l e n c e . B y t h e l e mma s b e l

• w

, we c a n s h

• w

t h a t i f t h e c

• h

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

• n

ρX= F(σ0), i

t h a s a s i mi l a r s t r u c t u r e t

• ρ0 ,

S

• ,

( mo d i f i e d ) G u s t a v f u n c t i

• n

i s a l s

• s

t a b l y r e a l i z a b l e w . r . t .

ρX,

wh i l e i t i s n

• t

s e q u e n t i a l l y r e a l i z a b l e w . r . t .

σ0,

wh i c h c

• n

t r a d i c t s t h e a s s u mp t i

• n

.

SLIDE 40

40

Proof Idea (2)

T h e c

• h
• r

e p r e s e n t a t i

• n

ρX:= F(σ0) s a t i s f i e s t h e f

• l

l

• wi

n g p r

• p

e r t i e s : Lem. ρX r e p r e s e n t s

ℕ ⊥

a n d e a c h n ∈ℕ

⊥

h a s e x a c t l y

• n

e r e a l i z e r d e n

• t

e d b y an Clq(X) ∈ . Lem. ∀n≠m ∈

ℕ an, am Clq(X)

∈ a r e n

• t

c

• h

e r e n t . Lem. ∀n∈

ℕ an ⊃ a⊥

N

• w

we h a v e : ∅ {0} {1} {2}

…

N= a0 a1 a2

…

a⊥ X=

ℕ⊥

ρX ρ0

incoherent incoherent

≒

SLIDE 41

41

Gustav Function is Stably Realizable

SLIDE 42

42

Ⅰ. Review: Coherence Representations Ⅱ. Sequential Representations Ⅲ. Inequivalence Ⅳ. Conclusion and Future Works

SLIDE 43

43

• C
• mp

a r i s

• n

w i t h s

• me

r e a l i z a b i l i t y mo d e l s

– We

h a v e s h

• w

n ( u s i n g G u s t a v f u n c t i

• n

) :

• S

e q R e p = M

• d

( B ) ≠ M

• d

( U

c

• h

) = C

• h

R e p ,

– We

c a n s i mi l a r l y s e p a r a t e ( u s i n g p a r a l l e l

• r

) :

• M
• d

( P

ω)

≠ M

• d

( U

c

• h

)

• M
• d

( K

2

) ≠ M

• d

( U

c

• h

)

– c

a t e g

• r

y

• f

T T E

• r

e p r e s e n t a t i

• n

s [ B a u e r ' 2 ] .

• C
• h

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

• n

s a r e n

• t

a r e i n v e n t i

• n
• f
• t

h e r r e a l i z a b i l i t y .

SLIDE 44

44

• Th. f :ℝ→

ℝ i 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 .

• We

a l s

• h

a v e a l i n e a r c

• mb

i n a t

• r

y a l g e b r a U

l i n

f

• r

l i n e a r r e a l i z a b i l i t y

• M
• d

( U

l i n

) i s a mo d e l

• f

l i n e a r l

• g

i c .

• C

a n C

• mp

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

• mp
• s

e d i n t

• “

L i n e a r A n a l y s i s ” ?

– A

n a l

• g

y f r

• m

t h e d i s c

• v

e r y

• f

l i n e a r l

• g

i c :

• I

n t u i t i

• n

i s t i c l

• g

i c i s d e c

• mp
• s

e d i n t

• l

i n e a r l

• g

i c .