Guerino Mazzola

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Global Networks in Computer Science?
Guerino Mazzola U ETH Zürich     guerino_at_mazzola
.ch      www.encyclospace.org        
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• Motivation
• Local Networks
• Global Networks
• Diagram Logic

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• Motivation
• Local Networks
• Global Networks
• Diagram Logic

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Course by Harald Gall Soft-Summer-Seminar
31.8./1.9. 2004 SW-Architekturen/Evolution Klass
ifikation von Netzwerken...
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Perspectives of New Music (2006) Guerino Mazzola
Moreno Andreatta From a Categorical Point of
View K-nets as Limit Denotators
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manifolds global objects in differential
geometry
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Are there global networks?
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• Motivation
• Local Networks
• Global Networks
• Diagram Logic

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Digraph category of digraphs ( quivers,
diagram schemes, etc.)
?
Digraph(?, E)
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Diagram in a category C digraph morphism
D ? ? C

• Di objects in C
• Dijt morphisms in C

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• 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.

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• Yoneda embedding
• Let C_at_ category of contravariant functors
  ( presheaves) F C ? Set
• Have Yoneda embedding functor _at_ C ? C_at_
• _at_X C ? Set A gt A_at_X C(A, X) (_at_X
  representable presheaf)

C
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• Category ?C of C-addressed points
• Objects of ?C
• x _at_A ? F, F presheaf in C_at_ 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

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Local network in C diagram x of C-addressed
points
x ? ? ?C
x ?lim(D)
x is flat if all addresses and spaces coincide.
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Example 1 K-nets of pitch classes C Ab
abelian groups affine maps
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Example 2 K-nets of chords C Ab
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Example 3 K-nets of dodecaphonic series C Ab
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Example 4 Neural Networks
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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 _at_ n
h/? x ? y
y(t) h(x(t-1))
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(w,x, a?w,x?)
w
(w, x)
?w,x?
o(a?w,x?)
a?w,x?
(w,x)
x
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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
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address A (0, M 0,1 ? ? ? 20,1 )
points x A ? (e, M S ? A ? 2S)
states s in S local network
ofA-addressed pointsIdA address change
network of states
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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_at_K instances of class K

_at_Class
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Instance method in two variables F _at_K ?
_at_L (i,j)I ? F, m F ? _at_M
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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/?
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• Special cases
• identity morphism Idx x ? x
• isomorphisms f x ? y there is g y ? x
  with g8f Idx und f8g Idy, write x ? y.
• local subnetworks Local network y E ? ?C , f
  ? ? E subdigraph, f y?? ? y embedding
  morphism.

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• Motivation
• Local Networks
• Global Networks
• Diagram Logic

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atlas
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• Examples
• 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 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.

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• Morphisms of global networks x, y over category
  C f x ? y morphisms of their digraphs,
  which induce morphisms of local networks.
• Category GC of global networks over C.
• Subcategory LfC of locally flat networks
  locally flat morphisms.
• A global network is interpretable, if it is
  isomorphic to an interpretation.

Open problem Under what condition are
therenon-interpretable global networks? LfC
? X ? GC
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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 10167692 Bericht Nr. 347 (2005), Guerino
Mazzola Local and Global Limit Denotators and
the Classification of Global Compositions
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Karl Pribram
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• Motivation
• Local Networks
• Global Networks
• Diagram Logic

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The category Digraph is a topos
D ? E
D E
DE
0 Ø
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In particularThe set Sub(D) of subdigraphsof a
digraph D is a Heyting algebra have digraph
logic. Ergo Global networks, ANNs, Klumpenhouw
er-nets, and local/global gestures, enable
logicaloperators (?, ?, ?,?)
Subobject classifier
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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 y1 ? y2 Sub(u) Sub(y2) ?
Sub(y1) homomorphism of Heyting algebras
contravariant functor Sub LC ?
Heyting Sub GC ? Heyting complexes
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C-major network of degrees
y 3.x 7
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?

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• 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 mea
  n? How global are VM architectures?

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• Problems
• 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.