Guerino Mazzola U & ETH Zürich U & ETH Zürich Global Networks in Computer Science? Global Networks in Computer Science?

Motivation Motivation Local Networks Local Networks Global Networks Global Networks Diagram Logic Diagram Logic

Motivation Motivation Local Networks Local Networks Global Networks Global Networks Diagram Logic Diagram Logic

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

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

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

Are there global networks?

Motivation Motivation Local Networks Local Networks Global Networks Global Networks Diagram Logic Diagram Logic

v x w y c a bd Digraph = category of digraphs (= quivers, diagram schemes, etc.)  = A V ht x = t(a) y = h(a) a E = B W h‘t‘ uq  Digraph ( , E)

Diagram in a category C = digraph morphism D :   C i j l m a ij t a il q a jm s a li p a jl k a ll r a ll r D i = objects in C D i = objects in C D ij t = morphisms in C D ij t = morphisms in C DiDiDiDi D jD jD jD j DlDlDlDl D mD mD mD m D ij t D il q D jm s D li p D jl k D ll r D C

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

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

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

x i : A i  F i h il q /  il q h jm s /  jm s h li p /  li p h jl k /  jl k h ll r /  ll r x j : A j  F j x m : A m  F m x l : A l  F l h ij t /  ij t x i : A  F h ij t /  ij t h il q /  il q h jm s /  jm s h li p /  li p h jl k /  jl k h ll r /  ll r x j : A  F x m : A  F x l : A  F Local network in C = diagram x of C-addressed points x is flat if all addresses and spaces coincide. x :   ∫ C  D x  lim( D ) coordinate of x

Ÿ 12 Example 1: K-nets of pitch classes C = Ab abelian groups + affine maps Ÿ T 11.-1/Id T 11.5/Id T 4 /Id T 2 /Id 3 724

Example 2: K-nets of chords C = Ab 2 Ÿ {3,4,10} 2 T /Id 2 T 11.5 /Id 2 T 4 /Id 2 T 2 /Id {2,7,8} {3,4,9}{1,2,7}

Example 3: K-nets of dodecaphonic series C = Ab Ÿ 12 s Us Ks UKs T 11.-1/Id Id/T Ÿ 11 s

2004 Example 4: Neural Networks

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

p3p3p3p3 D n  D n  D DnDnDnDn p1p1p1p1 p 12 h D n  D n DD Id/ + ?  Id/ + ? Id/ + ?  Id/ + ?  ?,?  a D o DnDnDnDn p2p2p2p2DD D p1p1p1p1 pnpnpnpn pipipipi

( + w, + x, a  + w, + x  ) w p3p3p3p3 p1p1p1p1 p 12 h (w, x) ( + w, + x) +w,+x+w,+x+w,+x+w,+x a+w,+xa+w,+xa+w,+xa+w,+x Id/ + ?  Id/ + ?  ?,?  a o x x1x1x1x1 xnxnxnxn xixixixi p1p1p1p1 pnpnpnpn pipipipi p2p2p2p2 o(a  + w, + x  )

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

address A = (0, M: {0,1}    2 {0,1} ) points x: A  (e, M: S  A  2 S ) ~ states s in S local network of A-addressed points Id A = address change ~ network of states s i : A  M i h ij t /Id h il q /Id h jm s /Id h li p /Id h jl k /Id h ll r /Id s j : A  M j s m : A  M m s l : A  M l

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

Instance method in two variables: F K L (i,j): I  F, m: F M Cartesian product  multiple inheritance I i j pKpKpKpK pLpLpLpL (i,j) m(i,j) mId

Morphisms of local networks x :   ∫ C, y : E  ∫ C f: x  y xixixixi xjxjxjxj xlxlxlxl xmxmxmxm x ij t x il q x jm s x li p x jl k x ll r x = y f(i) ysysysys yryryryr y f(i)s t y f(i)r q y rr h y =y =y =y = y sr w y r f(i) p didididi f:   E for every vertex i of , there is a morphism d i : x i  y f(i) Flat morphism: x, y flat and d i = const. = h/  category L C subcategory F C

Special cases identity morphism Id x : x  x identity morphism Id x : x  x isomorphisms f: x  y there is g: y  x with g ∞ f = Id x und f ∞ g = Id y, write x  y. isomorphisms f: x  y there is g: y  x with g ∞ f = Id x und f ∞ g = Id y, write x  y. local subnetworks Local network y : E  ∫ C, f :   E subdigraph, f: y    y embedding morphism. local subnetworks Local network y : E  ∫ C, f :   E subdigraph, f: y    y embedding morphism.

Motivation Motivation Local Networks Local Networks Global Networks Global Networks Diagram Logic Diagram Logic

atlas atlas  rrrr ssss  rs

 isomorphism of local networks ij l m ij l m i j l i j l cartes. chart xixixixi xjxjxjxj xlxlxlxl xixixixi xjxjxjxj xlxlxlxl yiyiyiyi yjyjyjyj ylylylyl chart yiyiyiyi yjyjyjyj ylylylyl ymymymym  rrrr ssss  rs

Examples Local networks are global networks with one chart. Local networks are global networks with one chart. Interpretations: let y : E  ∫ C be a local network and let I = (  i ) be a covering by subdigraphs  i  E. Build the corresponding subnetworks x i = y   i. Together with the identity on the chart overlaps, this defines a global network y I, 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. Interpretations: let y : E  ∫ C be a local network and let I = (  i ) be a covering by subdigraphs  i  E. Build the corresponding subnetworks x i = y   i. Together with the identity on the chart overlaps, this defines a global network y I, 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. Locally flat global networks have flat charts and local glueing data.

Morphisms of global networks x, y over category C f: x  y = morphisms of their digraphs, which induce morphisms of local networks. Morphisms of global networks x, y over category C f: x  y = morphisms of their digraphs, which induce morphisms of local networks. Category G C of global networks over C. Category G C of global networks over C. Subcategory Lf C of locally flat networks + locally flat morphisms. Subcategory Lf C of locally flat networks + locally flat morphisms. A global network is interpretable, if it is isomorphic to an interpretation. A global network is interpretable, if it is isomorphic to an interpretation. Open problem: Under what condition are there non-interpretable global networks? Lf C  X  G C Open problem: Under what condition are there non-interpretable global networks? Lf C  X  G C

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 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 Theorem Given address A in C, we have a verification functor |?|: A Lf C red  A Glob x |x||x||x||x|~> Corollary There are non-interpretable global networks in A Lf C red

a d b c *1* |x|

Dendritic transformations Karl Pribram

Motivation Motivation Local Networks Local Networks Global Networks Global Networks Diagram Logic Diagram Logic

D  E D + ED + ED + ED + E 1 = 0 = Ø DEDEDEDE The category Digraph is a topos Alexander Grothendieck

 = T vx w y digraph In particular: The set Sub(  ) of subdigraphs of a digraph  is a Heyting algebra: have „digraph logic“. Ergo: Global networks, ANNs, Klumpenhouwer-nets, and local/global gestures, enable logical operators ( , , ,  ) digraph In particular: The set Sub(  ) of subdigraphs of a digraph  is a Heyting algebra: have „digraph logic“. Ergo: Global networks, ANNs, Klumpenhouwer-nets, and local/global gestures, enable logical operators ( , , ,  ) Subobject classifier

Heyting logic on set Sub( y ) of subnetworks of y h, k  Sub( y ) h  k := h  k h  k := h  k h  k (complicated)  h := h  Ø tertium datur: h ≤  h u: y 1  y 2 Sub(u): Sub( y 2 )  Sub( y 1 ) homomorphism of Heyting algebras = contravariant functor Sub: L C  Heyting Sub: G C  Heyting complexes Heyting logic on set Sub( y ) of subnetworks of y h, k  Sub( y ) h  k := h  k h  k := h  k h  k (complicated)  h := h  Ø tertium datur: h ≤  h u: y 1  y 2 Sub(u): Sub( y 2 )  Sub( y 1 ) homomorphism of Heyting algebras = contravariant functor Sub: L C  Heyting Sub: G C  Heyting complexes

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

V I  I VI IV =

Describe global ANNs. Describe global ANNs. Can we interpret the dendritic transformations in the theory of Karl Pribram as being glueing operations of charts for 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 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? 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? Was would global SW-engineering/programming mean? How global are VM architectures?

Problems: Investigate the possible role and semantics of network logic in concrete contexts such as local/global ANNs, automata networks, gestures, Klumpenhouwer-nets.Investigate the possible role and semantics of network logic in concrete contexts such as local/global ANNs, automata networks, gestures, Klumpenhouwer-nets. Investigate a (formal) propositional/predicate language of networks with values in Heyting algebras of digraphs.Investigate a (formal) propositional/predicate language of networks with values in Heyting algebras of digraphs.