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

Supported by JSPS Core-to-Core Program Coherence Spaces for Computable Analysis II K e i M a t s u mo t o a n d K a z u s h i g e T e r u i ( R I M S , K y o t o U n i v e r s i t y ) 1

Background ● C o mp u t a b l e a n a l y s i s s t u d i e s c o mp u t a t i o n o v e r t o p o l o 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 o n s . – T y p e t wo t h e o r y o f E f f e c t i v i t y – D o ma i n r e p r e s e n t a t i o n s ● T h e i r a p p r o a c h e s a r e t o t r a c k c o mp u t a t i o n b y c o n t i n u o u s ma p s o v e r “ s y mb o l i c ” s p a c e s . B a i r e s p . , S c o t t d o ma i n s , . . . T h e p r i n c i p l e : C o mp u t a b l e ⇒ c o n t i n u o u s 2

Our Proposal T r a c k e d b y s t a b l e ma p . F X Y C o h e r e n c e s p . T o p o l o g i c a l s p . X Y f O u r p r i n c i p l e : C o 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 o f S c o t t - d o ma i n s c o h e r e n c e s p a c e s , [ G i r a r d ' 8 6 ] coexist : t w o mo r p h i s ms s t a b l e / l i n e a r ma p s w h e r e 3

Main Result from [MT '15] R e p r e s e n t a t i o n s b a s e d o n c o 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 o r r e a l f u n c i t o n s , w e h a v e s h o w n t h a t ● s t a b l y r e a l i z a b l e ⇔ c o n t i n u o u s ● l i n e a r l y r e a l i z a b l e ⇔ u n i f o r ml y c o n t i n u o u s . 4

Main Result of This Talk Conjecture ( b y a L I C S r e v i e w e r ) . ’’ 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 of representations over the sequential functionals. CohRep ≃ SeqRep R e p r e s e n t a t i o n s R e p r e s e n t a t i o n s b a s e d o n b a s e d o n s t a b l e f u n c t i o n s s e q u e n t i a l f u n c t i o n s 5

Main Result of This Talk Conjecture ( b y a L I C S r e v i e w e r ) . ’’ 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 of representations over the sequential functionals. CohRep ≃ SeqRep T h e c o n j e c t u r e i s F A L S E . 6

Ⅰ. Review: Coherent Representations Ⅱ. Sequential Representations Ⅲ. Inequivalence Ⅳ. Conclusion and Future Works 7

Coherence Spaces X =(| X | , ) Def. A c o h e r e n c e s p a c e i s ( ) | X | ● a c o u n t a b l e s e t o f t o k e n s wi t h | X | ● 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 . o n () Write x y i x ≠ y a x y ( s ) ( f f n d t r i c t c o h e r e n c e () A c l i q u e i s a s e t o f t o k e n s wh i c h a r e p a i r wi s e c o h e r e n t . T h e c l i q u e s p a c e Clq(X) f o r ms a S c o t t d o ma i n w . r . t . t h e s e t i n c l u s i o n . ● C o mp a c t e l e me n t s = f i n i t e c l i q u e s . 8

Examples Def. D e f i n e a d i s c r e t e c o h e r e n c e s p a c e N b y : ● |N| = ℕ … {0} {1} {2} ● n m i f f n=m ( ) ∅ ∈ ℕ } ∪ { } ≈ ℕ ⊥ ∅ Clq(N) = {{n}: n D = ℤ × ℕ , i (m,n) ～ m/2 n T h e s e t o f d y a d i c r a t i o n a l s d e n t i f i e d a s Def. R f D e f i n e a c o h e r e n c e s p a c e o 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 o l l o ws : | R |= D ● ● F d=(m,n) and e=(m',n'), o r e a c h n ≠n' and |d-e| ≤ 2 － n + 2 － n' d e i ( f f R ≈ Ma x i ma l c l i q u e s o f ( r a p i d l y c o n v e r g i n g ) C a u c h y s e q u e n c e s 9

Summary: Coherence Spaces ● C o 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 o t t - d o ma i n s . ● T wo k i n d s o f mo r p h i s ms c o 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 o g u e o f p r o g r a ms f o r wh i c h t h e a mo u n t o f i n p u t n e e d e d t o p r o d u c e a n o 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 o g u e o f p r o g r a ms wh i c h ma k e s e x a c t l y o n e q u e r y d u r i n g c o mp u t a t i o n . ● T wo c a t e g o r i e s o f c o h e r e n c e s p a c e s – T Stbl h e c a t e g o r y o f c o 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 o s e d . Lin – T h e c a t e g o r y o f c o h . s p a c e s a n d l i n e a r ma p s i s mo n o i d a l c l o s e d . ● L Stbl ( X ,Y )≃ Lin ( ! X ,Y ) i n e a r d e c o mp o s i t i o n : – T h i s g a v e b i r t h t o l i n e a r l o g i c [ G i r a r d ' 8 6 ] . 10

Coh-Representations ρ X Def. A ( C o h - ) r e p r e s e n t a t i o n i s a t u p l e X → X o f a c o h e r e n c e s p a c e X , ρ X : ⊆ Clq(X)→ X . a s e t X , a n d a s u r j e c t i v e ma p F Def. A f u n c t i o n f: X → Y i s s t a b l y r e a l i z a b l e i f Y X F: X→Y t h e r e e x i s t s a s t a b l e ma p s . t . : ρ X ρ Y ρ 0 Ex. N → ℕ ⊥ T h e C o h - r e p r e s e n t a t i o n f ma p s t h e s i n g l e t o n {n} t o t h e n u mb e r n X Y a n d t h e e mp t y s e t t o t h e b o t t o m. ρ R Ex. R → T h e C o h - r e p r e s e n t a t i o n ℝ s e n d s a C a u c h y s e q u e n c e t o i t s l i mi t . … 0 1 2 Prop. ( [ MT ' 1 5 ] ) f : ℕ ⊥ → ℕ ⊥ i ( i ) s s t a b l y r e a l i z a b l e ⇔ i t i s mo n o t o 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 o n t i n u o u s . 11 ( 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 o r ml y c o n t i n u o u s .

The Category CohRep Def. CohRep i s t h e c a t e g o r y o f c o h e r e n t r e p r e s e n t a t i o 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 o n s . I n r e a l i z a b i l i t y t h e o r y , r e p r e s e n t a t i o n s c o r r e s p o n d t o mo d e s t s e t s o v e r s o me P C A ( p a r t i a l c o mb i n a t o r y a l g e b r a ) . ● L e t U coh b e t h e P C A o b t a i n e d f r o m t h e u n i v e r s a l c o h e r e n c e s p a c e . ● CohRep Mod (U coh ≃ ) S i n c e c a t e g o r i e s o 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 CohRep. c l o s e d , s o i s 12

Ⅰ. Review: Coherent Representations Ⅱ. Sequential Representations Ⅲ. Inequivalence Ⅳ. Conclusion and Future Works 13