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Coherence Spaces for Resource-Sensitive Computation in Analysis K e i M a t s u mo t o ( 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

  1. Coherence Spaces for Resource-Sensitive Computation in Analysis K e i M a t s u mo t o ( R I M S , K y o t o U n i v e r s i t y ) 1

  2. 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

  3. 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 i n g 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 ] . coexists ● B t w o mo r p h i s ms i n c o h e r e n c e s p a c e s : T W, 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 o n t h e n a r r i s e s : 3 Wh a t a r e L i n e a r C o mp u t a t i o n s i n T o p o l o g y ?

  4. Girard's Linear Logic O n e o 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 0 ' s i n b o t h l o g i c a n d c o mp u t e r s c i e n c e . A t t r a c t i v e I d e a s : ● R e s t r u c t u r i n g b o t h C l a s s i c a l & I n t u i t i o n i s t i c L o g i c ● P r o o f N e t s ● R e s o u r c e - C o n c i o u s n e s s

  5. Girard's Linear Logic O n e o 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 0 ' s i n b o t h l o g i c a n d c o mp u t e r s c i e n c e . A t t r a c t i v e I d e a s : ● R e s t r u c t u r i n g b o t h C l a s s i c a l & I n t u i t i o n i s t i c L o g i c ● P r o o f N e t s ● R e s o u r c e - C o n c i o u s n e s s

  6. Girard's Linear Logic O n e o 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 0 ' s i n b o t h l o g i c a n d c o mp u t e r s c i e n c e . A t t r a c t i v e I d e a s : ● R e s t r u c t u r i n g b o t h C l a s s i c a l & I n t u i t i o n i s t i c L o g i c ● P r o o f N e t s ● R e s o u r c e - C o n c i o u s n e s s

  7. Girard's Linear Logic O n e o 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 0 ' s i n b o t h l o g i c a n d c o mp u t e r s c i e n c e . A t t r a c t i v e I d e a s : ● R e s t r u c t u r i n g b o t h C l a s s i c a l & I n t u i t i o n i s t i c L o g i c ● P r o o f N e t s ● R e s o u r c e - C o n c i o u s n e s s

  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 ... 8

  9. Resource-Sensitivity Ex. I n I n t u i t i o n i s t i c L o g i c , i s t r u e . S u b s t i t u t e : ● A : = “ t o p a y ￥ 4 0 0 ” ● B : = “ t o g e t a p a c k o f c i g a r e t t e s ” ● C : = “ t o g e t a c u p o f c a k e ” a p e r s o n i n t h e I n t u i t i o n i s t i c L o g i c wo r l d 9

  10. Resource-Sensitivity Ex. I n I n t u i t i o n i s t i c L o g i c , i s t r u e . S u b s t i t u t e : ● A : = “ t o p a y ￥ 4 0 0 ” ● B : = “ t o g e t a p a c k o f c i g a r e t t e s ” ● C : = “ t o g e t a c u p o f c a k e ” 10

  11. Resource-Sensitivity Ex. I n I n t u i t i o n i s t i c L o g i c , i s t r u e . S u b s t i t u t e : ● A : = “ t o p a y ￥ 4 0 0 ” ● B : = “ t o g e t a p a c k o f c i g a r e t t e s ” ● C : = “ t o g e t a c u p o f c a k e ” 11

  12. Resource-Sensitivity Ex. I n I n t u i t i o n i s t i c L o g i c , i s t r u e . S u b s t i t u t e : ● A : = “ t o p a y ￥ 4 0 0 ” ● B : = “ t o g e t a p a c k o f c i g a r e t t e s ” ● C : = “ t o g e t a c u p o f c a k e ” P a r a d o x 12

  13. Resource-Sensitivity Ex. I n I n t u i t i o n i s t i c L o g i c , i s t r u e . S u b s t i t u t e : ● A : = “ t o p a y ￥ 4 0 0 ” ● B : = “ t o g e t a p a c k o f c i g a r e t t e s ” ● C : = “ t o g e t a c u p o f c a k e ” A is used twice. L a c k o f c o n c i o u s n e s s t o c o ms u me a s s u mp t i o n s ! 13

  14. Resource-Sensitivity Ex. I n L i n e a r L o g i c , i s f a l s e . N e w c o n j u n c t i o n / i mp l i c a t i o n 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 o n e x a c t l y o n c e i n t h e p r o o f . C o h e r e n c e S p a c e s a r e p r o p o s e d a s a d e n o t a t i o 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 o p e r t y . V i a t h e C u r r y - H o w a r d i s o mo r p h i s m, resource-sensitive computations t h e y a r e a l s o a mo d e l o f o f l i n e a r f u n c t i o n p r o g r a ms . 14

  15. Main Result from CCA’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 e v e r y 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 . L e t u s e mp h a s i z e t h a t t h e s e c o r r e s p o n d e n c e s h o l d f o r r e a l f u n c t i o 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 o a wi d e r c l a s s . 15

  16. Ⅰ. Review: : Coherent Spa paces Ⅱ. Coherence e as Uniformity Ⅲ. Linear Admissibility Ⅳ. Concluding Comments 16

  17. Coherence Spaces Def. A c o 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 o u n t a b l e s e t o f t o 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 . o n ● Write i f f a n d ( s 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 . A n a n t i c l i q u e i s a s e t o f t o k e n s i n wh i c h e v e r y p a i r i s n o t c o h e r e n t . ● : t h e s e t o f a l l c l i u q e s . ● : t h e s e t o f a l l f i n i t e c l i q u e s . ● : t h e s e t o f a l l ma x i ma l c l i q u e s . 17

  18. Example: Cauchy Sequences L e t . E a c h me mb e r o 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 o n a l a s . F o r e a c h , d e f i n e ● ● ● Ex. D e f i n e a c o h e r e n c e s p a c e f 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 : Ma x i ma l c l i q u e s ≈ ( 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 18 R e a l i z a t i o n o f R e a l N u mb e r s

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