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Transcript of Guerino Mazzola U & ETH Zürich [email protected] encyclospace
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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?
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• MotivationMotivation
• Local NetworksLocal Networks
• Global NetworksGlobal Networks
• Diagram LogicDiagram Logic
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• MotivationMotivation
• Local NetworksLocal Networks
• Global NetworksGlobal Networks
• Diagram LogicDiagram Logic
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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...“
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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
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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
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Are there global networks?Are there global networks?
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• MotivationMotivation
• Local NetworksLocal Networks
• Global NetworksGlobal Networks
• Diagram LogicDiagram Logic
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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)
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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
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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.
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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@@
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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
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xxii: A: Ai i FFii hhililqq//ilil
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
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
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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
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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}
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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
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2004
Example 4: Neural NetworksExample 4: Neural Networks
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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
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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
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((++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))
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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
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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
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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
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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
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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
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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.
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• MotivationMotivation
• Local NetworksLocal Networks
• Global NetworksGlobal Networks
• Diagram LogicDiagram Logic
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atlasatlas atlasatlas
rr
ss
rsrs
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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
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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.
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• 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
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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
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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|
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Dendritic transformationsDendritic transformations
Karl PribramKarl Pribram
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• MotivationMotivation
• Local NetworksLocal Networks
• Global NetworksGlobal Networks
• Diagram LogicDiagram Logic
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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
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=
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
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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
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C-major network of degreesC-major network of degrees
y =3.x + 7 y =3.x + 7
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• 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?