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Discover the intricacies of global network structures in computer science through a detailed course by Harald Gall. Learn about transformational theory, K-nets, and diagram logic. Delve into neural networks, local networks, and the categorization of networks. Understand the concept of diagram embeddings and the mapping of network points. Join us in unraveling the complexities of global networks in this comprehensive study.
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Global Networks in Computer Science? Guerino Mazzola U & ETH Zürich guerino@mazzola.ch www.encyclospace.org
Motivation • Local Networks • Global Networks • Diagram Logic
Motivation • Local Networks • Global Networks • Diagram Logic
Course by Harald Gall: Soft-Summer-Seminar 31.8./1.9. 2004 SW-Architekturen/Evolution „Klassifikation von Netzwerken...“
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
— U T — torus T compact Open set U not compact T U manifolds = global objects in differential geometry
Motivation • Local Networks • Global Networks • Diagram Logic
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)
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
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.
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
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@~xF(A), write x: A F A = address, F = space of x • Morphisms of ∫C • x: A F, y: B G h/: x y
hllr/llr hllr/llr xi: Ai Fi xi: A F hilq/ilq hilq/ilq xl: Al Fl xl: AF hlip/lip hlip/lip hijt/ijt hijt/ijt hijt/ijt hjlk/jlk hjlk/jlk D xj: Aj Fj xj: AF C@ hjms/jms hjms/jms xm: Am Fm xm: AF coordinateofx Local network in C= diagram x of C-addressed points x: ∫C xlim(D) x is flat if all addresses and spaces coincide.
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
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
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
2004 Example 4: Neural Networks
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))
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
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)
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
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
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
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
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/
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.
Motivation • Local Networks • Global Networks • Diagram Logic
s r rs atlas
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
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.
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
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 |?|: ALfCredAGlob 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
|x| 4 6 4 1* 5 6 2 4 c d 3 b a 5 2 1 3 2* 6 5 3 1
Dendritic transformations Karl Pribram
Motivation • Local Networks • Global Networks • Diagram Logic
1 = Alexander Grothendieck The category Digraph is a topos D E D+E DE 0 = Ø
= 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
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
c b IV V d a III II VI VII I e g f C-major network of degrees y =3.x + 7
V IV I VI I =
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?