Exploring Global Networks in Computer Science

1. Global Networks in Computer Science? Guerino Mazzola U & ETH Zürich     guerino@mazzola.ch      www.encyclospace.org        

2. Motivation • Local Networks • Global Networks • Diagram Logic

3. Motivation • Local Networks • Global Networks • Diagram Logic

4. Course by Harald Gall: Soft-Summer-Seminar 31.8./1.9. 2004 SW-Architekturen/Evolution „Klassifikation von Netzwerken...“

5. Transformational Theory, K-nets (Lewin et al.) sets of notes Perspectives of New Music (2006) Guerino Mazzola & Moreno Andreatta: From a Categorical Point of View: K-nets as Limit Denotators

6. — U T — torus T compact Open set U not compact T  U  manifolds = global objects in differential geometry

7. Are there global networks?

8. Motivation • Local Networks • Global Networks • Diagram Logic

9. h h‘  = A V E = B W d v t t‘ w u q x = t(a) c a b x y = h(a) a y Digraph = category of digraphs (= quivers, diagram schemes, etc.)  Digraph(,E)

10. Dllr Di allr i Dilq ailq Dl l Dlip alip Dijt aijt  D Djlk ajlk j Dj Djms ajms C m Dm Diagram in a category C = digraph morphismD:   C • Di = objects in C • Dijt = morphisms in C

11. Examples: • diagram of sets C =Set • diagram of topological spaces C =Top • diagram of real vector spaces C =Lin— • diagram of automata C = Automata • etc.

12. C@ @ @C • Yoneda embedding • Let C@ = category of contravariant functors (= presheaves) F: C Set • Have Yoneda embedding functor @:C C@ • @X: C Set: A ~> A@X = C(A, X) (@X = representable presheaf) C

13. F x A h F G    A B x y address change • Category ∫C of C-addressed points • Objects of ∫C • x: @A  F, F = presheaf in C@~xF(A), write x: A  F A = address, F = space of x • Morphisms of ∫C • x: A  F, y: B  G h/: x  y

14. hllr/llr hllr/llr xi: Ai Fi xi: A F hilq/ilq hilq/ilq xl: Al Fl xl: AF hlip/lip hlip/lip hijt/ijt hijt/ijt hijt/ijt  hjlk/jlk hjlk/jlk D xj: Aj Fj xj: AF C@ hjms/jms hjms/jms xm: Am Fm xm: AF coordinateofx Local network in C= diagram x of C-addressed points x:   ∫C xlim(D) x is flat if all addresses and spaces coincide.

15. Ÿ12 Ÿ12 Ÿ12 2 3 T4/Id 7 3 Ÿ12 Ÿ12 0 0 4 7 T11.5/Id T11.-1/Id 0 0 2 4 T2/Id Example 1: K-nets of pitch classesC = Ab abelian groups + affine maps

16. 2Ÿ12 2Ÿ12 2T4/Id {2,7,8} {3,4,10} 0 0 2T11.5/Id 2T11.-1/Id 0 0 {1,2,7} {3,4,9} 2T2/Id 2Ÿ12 2Ÿ12 Example 2: K-nets of chordsC = Ab

17. Ÿ12 Ÿ12 Id/T11.-1 Ks s Ÿ12 Ÿ12 Ÿ11 Ÿ11 T11.-1/Id T11.-1/Id Ÿ11 Ÿ11 s Us UKs Id/T11.-1 Example 3: K-nets of dodecaphonic seriesC = Ab

18. 2004 Example 4: Neural Networks

19. h —n —m x y Ÿ Ÿ +? Neural Networks C = Set address = Ÿ Points x: Ÿ —nat this address are time series x = (x(t))t of vectors in —n.They describe input and output for neural networks. Dn = Ÿ @ —n h/+? : x  y y(t) = h(x(t-1))

20. D h Dn Dn D Dn p1 p1 p12 p3 Dn Dn Dn Dn D D D D Dn pi Id/+? Id/+? p2 o a ?,? pn D

21. x1 h p1 p1 p12 p3 xi pi Id/+?  Id/+? ?,? p2 a o pn xn (+w,+x, a+w,+x) w o(a+w,+x) (w, x) +w,+x a+w,+x x (+w,+x)

22. 2004    (e) = f 2 Example 5: Local Networks of Automata • C = AutomataSet S of states, alphabet A • Objects: (e, M: S  A 2S) • Morphisms: h = (, ): (e, M: S  A 2S)  (f, N: T  B 2T) S  A 2S T  B 2T

23. hllr/Id si: A  Mi hilq/Id sl: A Ml hlip/Id hijt/Id hjlk/Id sj: A Mj hjms/Id sm: A Mm addressA = (0, M: {0,1}   2{0,1}) points x: A (e, M: S  A 2S) ~ states s in S local network ofA-addressed pointsIdA = address change~ network of states

24. Class@ virtual classes objective classes Example 6: Networks of OO Instances • C = Class classes and instances of a OO language • Objects: classes and one special address: I = „the instance“ (corresponds to final object 1) • Morphisms: s: K L superclass v: K F field m: K M method (without arguments) i: I K instance • I@K = {instances of class K } @Class

25. m Id @M F m(i,j) pK pL @K @L (i,j) I I i j Cartesian product  multiple inheritance Instance method in two variables: F = @K  @L (i,j):I  F, m: F  @M

26. xllr xi xilq xl xlip xijt di x = xjlk xj yf(i) yrrh yf(i)rq xjms yrf(i) p yr xm yf(i)st ysrw ys y= Morphisms of local networks x:   ∫C, y: E  ∫C f: x y category LC f:   E for every vertex i of , there is a morphism di: xi yf(i) subcategory FC Flat morphism: x, y flat and di = const. = h/

27. Special cases • identity morphism Idx: x  x • isomorphisms f: x  y there is g: y  x with g∞f = Idxund f∞g = Idy, write x y. • local subnetworksLocal network y: E  ∫C , f:   E subdigraph, f: y  y embedding morphism.

28. Motivation • Local Networks • Global Networks • Diagram Logic

29.  s r rs atlas

30. chart yi i i l  yl l r j yj j ym m m cartes. rs xi i yi l xl yl xj j s  yj i isomorphism of local networks l xi xl j xj chart

31. Examples • Local networks are global networks with one chart. • Interpretations: let y: E  ∫C be a local network and letI = (i) be a covering by subdigraphs i  E. Build the corresponding subnetworks xi = y i. Together with the identity on the chart overlaps, this defines a global network yI, called interpretation of y.Interpretations are interesting for the classification of networks by coverings of a given type of charts!Visualization via the nerve of the covering. • Locally flat global networks have flat charts and local glueing data.

32. Morphisms of global networksx, y over category C f: x  y= morphisms of their digraphs, which induce morphisms of local networks. • Category GC of global networks over C. • SubcategoryLfCof locally flat networks + locally flat morphisms. • A global networkis interpretable, if it is isomorphic to an interpretation. Open problem: Under what condition are therenon-interpretable global networks? LfC  X  GC

33. 4 6 x |x| ~> 4 3 6 5 2 3 5 1 2 1 Theorem Given address A in C, we have a verification functor |?|: ALfCredAGlob Corollary There are non-interpretable global networks in ALfCred COLLOQUIUM ON MATHEMATHICAL MUSIC THEORY H. Fripertinger, L. Reich (Eds.) Grazer Math. Ber., ISSN 1016–7692 Bericht Nr. 347 (2005), Guerino Mazzola: Local and Global Limit Denotators and the Classification of Global Compositions

34. |x| 4 6 4 1* 5 6 2 4 c d 3 b a 5 2 1 3 2* 6 5 3 1

35. Dendritic transformations Karl Pribram

36. Motivation • Local Networks • Global Networks • Diagram Logic

37. 1 = Alexander Grothendieck The category Digraph is a topos D  E D+E DE 0 = Ø

38.  = v w x y T In particular:The set Sub(D) of subdigraphsof a digraph Dis a Heyting algebra: have „digraph logic“. Ergo: Global networks, ANNs, Klumpenhouwer-nets, and local/global gestures, enable logicaloperators (, , ,) Subobject classifier

39. Heyting logic on set Sub(y) of subnetworks of y h, k  Sub(y)h  k := h  kh  k := h  kh  k (complicated)  h := h  Ø tertium datur: h ≤  h u: y1  y2Sub(u): Sub(y2)  Sub(y1) homomorphism of Heyting algebras = contravariant functor Sub: LC Heyting Sub: GC Heyting complexes

40. c b IV V d a III II VI VII I e g f C-major network of degrees y =3.x + 7

41. V IV I VI I  =

42. Describe global ANNs. • Can we interpret the dendritic transformations in the theory of Karl Pribram as being glueing operations of charts for global ANNs? • What is the gain in the construction of global ANNs? Is there any proper „global“ thinking in such a model? • What can be described in OO architectures by global networks, that local networks cannot? • Was would global SW-engineering/programming mean? How global are VM architectures?