Guerino Mazzola U & ETH Zürich U & ETH Zürich [email protected] Global Networks in Computer...

Guerino MazzolaGuerino MazzolaU & ETH Zürich U & ETH Zürich

[email protected] www.encyclospace.org www.encyclospace.org

Global Networks in Global Networks in Computer Science?Computer Science?

• MotivationMotivation

• Local NetworksLocal Networks

• Global NetworksGlobal Networks

• Diagram LogicDiagram Logic

Course by Harald Gall:Course by Harald Gall:Soft-Summer-Seminar 31.8./1.9. 2004Soft-Summer-Seminar 31.8./1.9. 2004

SW-Architekturen/EvolutionSW-Architekturen/Evolution„„Klassifikation von Netzwerken...“ Klassifikation von Netzwerken...“

Course by Harald Gall:Course by Harald Gall:Soft-Summer-Seminar 31.8./1.9. 2004Soft-Summer-Seminar 31.8./1.9. 2004

SW-Architekturen/EvolutionSW-Architekturen/Evolution„„Klassifikation von Netzwerken...“ Klassifikation von Netzwerken...“

sets of notessets of notes

Transformational Theory, K-nets (Lewin et al.)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

TT

torus Ttorus Tcompactcompact

manifolds = global objects in differential geometrymanifolds = global objects in differential geometry

——

——

Open set UOpen set Unot compactnot compact

UU

T T U U

Are there global networks?Are there global networks?

vv

xx

ww

yy

cc

aa

bb

dd

DigraphDigraph = category of digraphs = category of digraphs(= quivers, diagram schemes, etc.) (= quivers, diagram schemes, etc.)

= A V= A Vhh

tt

x = x = t(a)t(a)

y = y = h(a)h(a)

aa

E = B WE = B Wh‘h‘

t‘t‘

u q

DigraphDigraph((,, E)E)

Diagram in a category C Diagram in a category C

= digraph morphism= digraph morphism DD: : CC

ii

jj

ll

mm

aaijijtt

aaililqq

aajmjmss

aalilipp

aajljlkk

aallllrr

• DDii = objects in = objects in CC• DDijij

tt = morphisms in = morphisms in CC

DDii

DD jj

DDll

DD mm

DDijijtt

DDililqq

DDjmjmss

DDlilipp

DDjljlkk

DDllllrr

DD

CC

Examples:Examples:

• diagram of sets diagram of sets CC = = SetSet

• diagram of topological spaces diagram of topological spaces CC = = TopTop

• diagram of real vector spaces diagram of real vector spaces CC = = LinLin——

• diagram of automatadiagram of automata C C == Automata Automata

• etc.etc.

Yoneda embeddingYoneda embedding

• Let Let CC@@ = category of contravariant functors = category of contravariant functors (= presheaves) (= presheaves) F: F: CC SetSet

• Have Yoneda embedding functor Have Yoneda embedding functor @:@: CC CC@@

@X: @X: CC SetSet: A ~> : A ~> A@X = = CC(A, X)(A, X)

(@X = representable presheaf)(@X = representable presheaf)

CC@@

CCCC @@CC@@CC@@

yyxx

Category ∫Category ∫CC of of CC-addressed points-addressed points

• Objects of ∫Objects of ∫CC

x: @A x: @A F, F = presheaf in F, F = presheaf in CC@@

~~

xx F(A), write F(A), write

x: A x: A F A = F A = addressaddress, F = , F = space space of xof x

hh

FF

AA

GG

BB

address changeaddress change

• Morphisms of ∫Morphisms of ∫CC

x: A x: A F, y: B F, y: B G G

h/h/: x : x y y

FFAA xx

xxii: A: Ai i FFii hhililqq//ilil

qq

hhjmjmss//jmjm

ss

hhlilipp//lili

pp

hhjljlkk//jljl

kk

hhllllrr//llll

rr

xxjj: A: Aj j FFjj

xxmm: A: Am m FFmm

xxll: A: Al l FFll

hhijijtt//ijij

tthhijijtt//ijij

tt

xxii: A : A FF

hhijijtt//ijij

tt

hhililqq//ilil

qq

hhjmjmss//jmjm

ss

hhlilipp//lili

pp

hhjljlkk//jljl

kk

hhllllrr//llll

rr

xxjj: A: A FF

xxmm: A: A FF

xxll: A: A FF

Local network in Local network in CC = diagram = diagram xx of of CC-addressed points-addressed points

x x is is flatflat if all addresses and spaces coincide. if all addresses and spaces coincide.

xx: : ∫ ∫CC

CC@@

DD

xx lim(lim(DD))coordinatecoordinateofof xx

ŸŸ1212

Example 1: K-nets of pitch classesExample 1: K-nets of pitch classes CC = = Ab Ab abelian groups + affine mapsabelian groups + affine maps

ŸŸ1212

ŸŸ1212

ŸŸ1212

ŸŸ1212

00 00

00 00

33

22

77

44

TT1111.-1/Id.-1/IdTT1111.5/Id.5/Id

TT44/Id/Id

TT22/Id/Id

33

77

22

44

Example 2: K-nets of chordsExample 2: K-nets of chords CC = = AbAb

22ŸŸ1212

22ŸŸ1212

22ŸŸ1212

22ŸŸ1212

00 00

00 00

{3,4,10}{3,4,10}

22TT1111.-1.-1/Id/Id22TT1111.5.5/Id/Id

22TT44/Id/Id

22TT22/Id/Id

{2,7,8}{2,7,8}

{3,4,9}{3,4,9}{1,2,7}{1,2,7}

Example 3: K-nets of dodecaphonic seriesExample 3: K-nets of dodecaphonic series CC = = AbAb

ŸŸ1212

ŸŸ1212

ŸŸ1212

ŸŸ1212

ss

UsUs

KsKs

UKsUKs

TT1111.-1/Id.-1/IdTT1111.-1/Id.-1/Id

Id/TId/T1111.-1.-1

Id/TId/T1111.-1.-1

ŸŸ1111 ŸŸ1111

ŸŸ1111 ŸŸ1111

ss

2004

Example 4: Neural NetworksExample 4: Neural Networks

Neural NetworksNeural Networks

CC = = SetSet address = address = ŸŸ

Points Points x: x: Ÿ Ÿ ——nn

at this address are time series x = (x(t))at this address are time series x = (x(t)) tt of vectors in of vectors in ——nn..

They describe input and output for neural networks. They describe input and output for neural networks.

DDnn = = Ÿ Ÿ @ @ ——nn

++??

——mm——nn

ŸŸ ŸŸ

xx yy

hh

y(t) = h(x(t-1))y(t) = h(x(t-1))h/h/++? ? : x : x y y

pp33

DDn n DDn n DDDDnn

pp11 pp1212

hh

DDn n DDnn DDn n DDnn DD DD Id/Id/++? ? Id/Id/++?? ?,??,? aa

DDoo

DDnn

pp22

DD

DD

DD

pp11

ppnn

ppii

((++w,w,++x, ax, a++w,w,++xx))ww

pp33pp11 pp1212

hh

(w, x)(w, x) ((++w,w,++x)x) ++w,w,++xx aa++w,w,++xxId/Id/++? ? Id/ Id/++? ? ?,??,? aa oo

xx

xx11

xxnn

xxii

pp11

ppnn

ppii pp22

o(ao(a++w,w,++xx))

C C = = AutomataAutomataSet S of states, alphabet Set S of states, alphabet AA

• Objects:Objects: ( (ee, , M: S M: S AA 22SS))

• Morphisms: Morphisms: h = ( h = (, , ): ): ((ee, , M: S M: S AA 22SS) ) ( (ff, , N: T N: T BB 22TT) )

S S AA 22SS

T T BB 22TT

22 (e) = f(e) = f

Example 5: Local Networks of AutomataExample 5: Local Networks of Automata

2004

addressaddress A = (0A = (0, , M: {0,1} M: {0,1} 22{0,1}{0,1} ) )

points points x: A x: A ( (ee, , M: S M: S AA 22SS) ) ~ states s in S~ states s in S

local network oflocal network ofA-addressed pointsA-addressed pointsIdIdAA = address change = address change

~ network of states~ network of states

ssii: A : A M Mii

hhijijtt/Id/Id

hhililqq/Id/Id

hhjmjmss/Id/Id

hhlilipp/Id/Id

hhjljlkk/Id/Id

hhllllrr/Id/Id

ssjj: A: A M Mjj

ssmm: A: A M Mmm

ssll: A: A M Mll

CC = = ClassClass classes and instances of a OO language classes and instances of a OO language • Objects: Objects: classes and one special address: classes and one special address:

I I = „the instance“ = „the instance“ (corresponds to final object 1)(corresponds to final object 1)

• Morphisms: Morphisms: s:s: K K LL superclass superclass v:v: K K FF field field m:m: K K MM method (without arguments) method (without arguments)i:i: I I KK instance instance

II@@KK = {instances of class = {instances of class K K }}

Example 6: Networks of OO InstancesExample 6: Networks of OO Instances

ClassClass@@

@@ClassClass@@ClassClassobjective classesobjective classes

virtual classesvirtual classes

Instance method in two variables: F = @Instance method in two variables: F = @K K @@LL(i,j):(i,j):I I F, m: F F, m: F @ @MM

Cartesian product Cartesian product multiple inheritance multiple inheritance

II

ii jj

@@LL@@KK

FFppKK ppLL

(i,j)(i,j)

@@MM

II

m(i,j)m(i,j)

mm

IdId

Morphisms of local networksMorphisms of local networks

xx: : ∫ ∫CC, , yy: E : E ∫ ∫C C

f: f: xx yy

xxii

xxjj

xxll

xxmm

xxijijtt

xxililqq

xxjmjmss

xxlilipp

xxjljlkk

xxllllrr

x x ==

yyf(i)f(i)

yyss

yyrr

yyf(i)sf(i)stt

yyf(i)rf(i)rqq yyrrrr

hh

yy ==

yysrsrww

yyrrf(i) f(i) pp

ddii

f: f: E for every vertex i of E for every vertex i of , there is a morphism d, there is a morphism dii: : xxii yyf(i)f(i)

FlatFlat morphism: morphism: x x,, y y flat and d flat and di i = const. = h/= const. = h/

category category LLCC

subcategory subcategory FFCC

Special casesSpecial cases

• identity identity morphism morphism IdIdxx: : xx xx

• isomorphismsisomorphisms f: f: xx yy there is g: there is g: yy xx with with gg∞∞f = Idf = Idxx und und ff∞∞g = Idg = Idyy, write , write xx yy..

• local local subnetworkssubnetworksLocal network Local network yy: E : E ∫ ∫C C , f, f : : E subdigraph, E subdigraph,

f: f: yy yy

embedding morphism.embedding morphism.

atlasatlas atlasatlas

rr

ss

rsrs

isomorphism of isomorphism of local networkslocal networks

ii

jj

ll

mm

ii

jj

ll

mm

ii

jj

ll

ii

jj

ll

cartes.cartes.

chartchart

xxii

xxjj

xxll

xxii

xxjj

xxll

yyii

yyjj

yyll

chartchartyyii

yyjj

yyll

yymm

rr

ss

rsrs

ExamplesExamples

• Local networksLocal networks are global networks with one chart. are global networks with one chart.

• InterpretationsInterpretations: let : let yy: E : E ∫ ∫CC be a local network and let be a local network and let

I = (I = (ii) be a covering ) be a covering by subdigraphsby subdigraphs ii E. E.

Build the corresponding subnetworks Build the corresponding subnetworks xxii = = yy ii. .

Together with the identity on the chart overlaps, Together with the identity on the chart overlaps, this defines a global network this defines a global network yyII, called , called interpretationinterpretation of of yy..

Interpretations are interesting for the classification of Interpretations are interesting for the classification of networks by coverings of a given type of charts!networks by coverings of a given type of charts!Visualization via the nerve of the covering.Visualization via the nerve of the covering.

• Locally flat global networks Locally flat global networks have flat charts and have flat charts and local glueing data.local glueing data.

• Morphisms Morphisms of global networksof global networks xx, , yy over category over category CC f: f: xx yy = morphisms of their digraphs, = morphisms of their digraphs, which induce morphisms of local networks.which induce morphisms of local networks.

• Category Category GGCC of global networks over of global networks over CC.. • SubcategorySubcategory LfLfCC of of locally flat networkslocally flat networks

+ locally flat morphisms.+ locally flat morphisms. • A global networkA global network isis interpretable interpretable, if it is , if it is

isomorphic to an interpretation.isomorphic to an interpretation.

Open problem: Under what condition are thereOpen problem: Under what condition are therenon-interpretable global networks?non-interpretable global networks?

LfLfC C X X GGCC

Open problem: Under what condition are thereOpen problem: Under what condition are therenon-interpretable global networks?non-interpretable global networks?

LfLfC C X X GGCC

COLLOQUIUM ON MATHEMATHICAL MUSIC THEORY COLLOQUIUM ON MATHEMATHICAL MUSIC THEORY H. Fripertinger, L. Reich (Eds.) H. Fripertinger, L. Reich (Eds.) Grazer Math. Ber., ISSN 1016–7692 Bericht Nr. 347 (2005), Grazer Math. Ber., ISSN 1016–7692 Bericht Nr. 347 (2005), Guerino Mazzola: Local and Global Limit Denotators and the Guerino Mazzola: Local and Global Limit Denotators and the Classification of Global CompositionsClassification of Global Compositions

COLLOQUIUM ON MATHEMATHICAL MUSIC THEORY COLLOQUIUM ON MATHEMATHICAL MUSIC THEORY H. Fripertinger, L. Reich (Eds.) H. Fripertinger, L. Reich (Eds.) Grazer Math. Ber., ISSN 1016–7692 Bericht Nr. 347 (2005), Grazer Math. Ber., ISSN 1016–7692 Bericht Nr. 347 (2005), Guerino Mazzola: Local and Global Limit Denotators and the Guerino Mazzola: Local and Global Limit Denotators and the Classification of Global CompositionsClassification of Global Compositions

TheoremTheorem Given address A in Given address A in CC, we have a verification functor , we have a verification functor

|?|: |?|: AALfLfCCredred

AAGlobGlob

22

11

33

44

66

55

xx

22

11

33

44

66

55

||xx||~>~>

CorollaryCorollary There are non-interpretable global networks in There are non-interpretable global networks in AALfLfCC

redred

aa

dd

bb

cc

11

22

33

44

2*2*

1*1*

66

55

55

66

33

44

22

11

33

44

66

55

|x||x|

Dendritic transformationsDendritic transformations

Karl PribramKarl Pribram

D D E E

D D ++ E E

1 1 = =

0 0 = Ø = Ø

DDEE

The category The category DigraphDigraph is a is a topostopos

Alexander GrothendieckAlexander Grothendieck

=

TT

vv

xx

ww

yy

In particular:The set Sub() of subdigraphsof a digraph is a Heyting algebra: have „digraphdigraph logic“.

Ergo:

Global networks,ANNs,Klumpenhouwer-nets,and local/global gestures,enable logicaloperators (, , ,)

In particular:The set Sub() of subdigraphsof a digraph is a Heyting algebra: have „digraphdigraph logic“.

Ergo:

Global networks,ANNs,Klumpenhouwer-nets,and local/global gestures,enable logicaloperators (, , ,)

Subobject classifierSubobject classifier

Heyting logic on set Sub(Heyting logic on set Sub(yy) of subnetworks of ) of subnetworks of yy

hh, , kk Sub( Sub(yy)) hh kk := := hh kk hh kk := := hh kk hh kk (complicated) (complicated) hh := := hh Ø Ø tertium datur: tertium datur: hh ≤ ≤ hh

u: u: yy11 yy22

Sub(u): Sub(Sub(u): Sub(yy22) ) Sub( Sub(yy11) )

homomorphism of Heyting algebras homomorphism of Heyting algebras = contravariant functor = contravariant functor

Sub: Sub: LLCC HeytingHeyting

Sub: Sub: GGCC Heyting complexesHeyting complexes

Heyting logic on set Sub(Heyting logic on set Sub(yy) of subnetworks of ) of subnetworks of yy

hh, , kk Sub( Sub(yy)) hh kk := := hh kk hh kk := := hh kk hh kk (complicated) (complicated) hh := := hh Ø Ø tertium datur: tertium datur: hh ≤ ≤ hh

u: u: yy11 yy22

Sub(u): Sub(Sub(u): Sub(yy22) ) Sub( Sub(yy11) )

homomorphism of Heyting algebras homomorphism of Heyting algebras = contravariant functor = contravariant functor

Sub: Sub: LLCC HeytingHeyting

Sub: Sub: GGCC Heyting complexesHeyting complexes

VIIVII

II

IIIIII

VV

IIIIVIVI

IVIV

cc

dd

ee

ffgg

aa

bb

C-major network of degreesC-major network of degrees

y =3.x + 7 y =3.x + 7

VV

II

II

VIVI

IVIV

==

• Describe Describe global ANNsglobal ANNs..

• Can we interpret the Can we interpret the dendritic transformationsdendritic transformations in the in thetheory of Karl Pribram as being theory of Karl Pribram as being glueing operations of charts for global ANNs?glueing operations of charts for global ANNs?

• What is the gain in the construction of global ANNs?What is the gain in the construction of global ANNs?Is there any proper Is there any proper „global“ thinking„global“ thinking in such a model? in such a model?

• What can be described in What can be described in OO architectures by global OO architectures by global networksnetworks, that local networks cannot?, that local networks cannot?

• Was would Was would global SW-engineering/programmingglobal SW-engineering/programmingmean? How global are VM architectures?mean? How global are VM architectures?

Guerino Mazzola U & ETH Zürich U & ETH Zürich [email protected] Global Networks in Computer...

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