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Optimality in Cognition and GrammarPaul Smolensky Cognitive
Science Department, Johns Hopkins University
Plan of lecturesCognitive architecture: Symbols &
optimization in neural networksOptimization in grammar: HG OT From
numerical to algebraic optimization in grammarOT and nativism The
initial state & neural/genomic encoding of UG?
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The ICS HypothesisThe Integrated Connectionist/Symbolic
Cognitive Architecture (ICS)In higher cognitive domains,
representations and fuctions are well approximated by symbolic
computationThe Connectionist Hypothesis is correctThus, cognitive
theory must supply a computational reduction of symbolic functions
to PDP computation
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Levels
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The ICS Architecture
RepresentationGrammar G Function Algorithm AConstraints:
NOCODAoptimal: opt.constr. sat.:Activation PatternConnection
WeightsHarmony Opt./ Constraint SatisfactionSpreading
Activation
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Units
Representations
tTHa(0) -- #1#2#3#4a(1) --
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dtepsilon
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A syllable: "cat"
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Tensor Product RepresentationsRepresentations:Depth 0
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Local tree realizationsRepresentations:
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tTHa(0) -- #1#2#3#4a(1) --
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H
a
Time
X
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The ICS IsomorphismTensor product representationsTensorial
networks
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Tensor Product Representations
Structuring operationSymbolic formalizationConnectionist
formalizationStructuresExampleExampleVector
operationCombiningSets{c1, c2}c1 + c2Vector sum: +Role/filler
bindingStrings, framesAB = {A/r1, B/r2} A r1 + B r2Tensor product:
Recursive embeddingTrees A B C Ar0 +[Br0+ Cr1]r1 Recursive role
vectors: rleft/right-child(x) = r0/1 rx
-
Binding by Synchrony = s= r1 [fbook + fgive-obj] + r3 [fMary +
frecipient] + r2 [fgiver + fJohn] timegive(John, book, Mary)
(Shastri & Ajjanagadde 1993)r1 [fbook + fgive-obj][Tesar &
Smolensky 1994]
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The ICS Architecture
RepresentationGrammar G Function Algorithm AConstraints:
NOCODAoptimal: opt.constr. sat.:Activation PatternConnection
WeightsHarmony Opt./ Constraint Sat.Spreading Activation
Chart6
0.031250.031250.031250.03125
-0.03125-0.03125-0.03125-0.03125
-0.03125-0.03125-0.03125-0.03125
0.031250.031250.031250.03125
-0.03125-0.03125-0.03125-0.03125
0.031250.031250.031250.03125
0.031250.031250.031250.03125
-0.03125-0.03125-0.03125-0.03125
Depth-1 Units
Depth-0 Units
W (X B)
Sheet1
X' = X+X* = XTr0*r1*u0*u1*
r01111u00.50.50.50.5
r11-11-1u10.5-0.50.5-0.5
X'*Y'*A'*B'*
X11110.250.250.250.25X'0.250.250.250.25
Y1-11-10.25-0.250.25-0.25Y'0.25-0.250.25-0.25
A11-1-10.250.25-0.25-0.25A'0.250.25-0.25-0.25
B1-1-110.25-0.25-0.250.25B'0.25-0.25-0.250.25
(X'xu0)*0.1250.1250.1250.125(X'xu1)*0.1250.1250.1250.125
0.1250.1250.1250.125-0.125-0.125-0.125-0.125
(Y'xu0)*0.125-0.1250.125-0.125(Y'xu1)*0.125-0.1250.125-0.125
0.125-0.1250.125-0.125-0.1250.125-0.1250.125
(A'xu0)*0.1250.125-0.125-0.125(A'xu1)*0.1250.125-0.125-0.125
0.1250.125-0.125-0.125-0.125-0.1250.1250.125
(B'xu0)*0.125-0.125-0.1250.125(B'xu1)*0.125-0.125-0.1250.125
0.125-0.125-0.1250.125-0.1250.1250.125-0.125
(X'xu0)*0.1250.1250.1250.1250.1250.1250.1250.125
(Y'xu0)*0.125-0.1250.125-0.1250.125-0.1250.125-0.125
(A'xu0)*0.1250.125-0.125-0.1250.1250.125-0.125-0.125
(B'xu0)*0.125-0.125-0.1250.1250.125-0.125-0.1250.125
(X'xu1)*0.1250.1250.1250.125-0.125-0.125-0.125-0.125
(Y'xu1)*0.125-0.1250.125-0.125-0.1250.125-0.1250.125
(A'xu1)*0.1250.125-0.125-0.125-0.125-0.1250.1250.125
(B'xu1)*0.125-0.125-0.1250.125-0.1250.1250.125-0.125
Axr011-1-1Axr011-1-111-1-1
11-1-1
Bxr01-1-11Bxr01-1-111-1-11
1-1-11
Axr111-1-1Axr111-1-1-1-111
-1-111
Bxr11-1-11Bxr11-1-11-111-1
-111-1
a=X+Axr0111111-1-111-1-1
Wa0.250.250.250.250.250-0.25000.250-0.25
a'Wa2
W(0,1)0.1250-0.12500000
000000.1250-0.125
0.1250-0.12500000
000000.1250-0.125
X'(A'xu0)*0.031250.03125-0.03125-0.031250.031250.03125-0.03125-0.03125X
A B0.06250-0.0625000.06250-0.0625
0.031250.03125-0.03125-0.031250.031250.03125-0.03125-0.031250.06250-0.0625000.06250-0.0625
0.031250.03125-0.03125-0.031250.031250.03125-0.03125-0.031250.06250-0.0625000.06250-0.0625
0.031250.03125-0.03125-0.031250.031250.03125-0.03125-0.031250.06250-0.0625000.06250-0.0625
X'(B'xu1)*0.03125-0.03125-0.031250.03125-0.031250.031250.03125-0.03125
0.03125-0.03125-0.031250.03125-0.031250.031250.03125-0.03125
0.03125-0.03125-0.031250.03125-0.031250.031250.03125-0.03125
0.03125-0.03125-0.031250.03125-0.031250.031250.03125-0.03125
Y'(B'xu0)*0.03125-0.03125-0.031250.031250.03125-0.03125-0.031250.03125Y
B A0.06250-0.062500-0.062500.0625
-0.031250.031250.03125-0.03125-0.031250.031250.03125-0.03125-0.062500.0625000.06250-0.0625
0.03125-0.03125-0.031250.031250.03125-0.03125-0.031250.031250.06250-0.062500-0.062500.0625
-0.031250.031250.03125-0.03125-0.031250.031250.03125-0.03125-0.062500.0625000.06250-0.0625
Y'(A'xu1)*0.031250.03125-0.03125-0.03125-0.03125-0.031250.031250.03125
-0.03125-0.031250.031250.031250.031250.03125-0.03125-0.03125
0.031250.03125-0.03125-0.03125-0.03125-0.031250.031250.03125
-0.03125-0.031250.031250.031250.031250.03125-0.03125-0.03125
11111111
22222222
33333333
44444444
55555555
66666666
77777777
88888888
99999999
1010101010101010
1111111111111111
1212121212121212
12345678
123456789101112
U1U2U3U4U10U20U30U40U11U21U31U41
X111100000000
Y1-11-100000000
A11-1-100000000
B1-1-1100000000
A 000011-1-111-1-1
B 00001-1-111-1-11
A000011-1-1-1-111
B 00001-1-11-111-1
AB000020-20020-2
BA000020-200-202
XAB111120-20020-2
YBA1-11-120-200-202
111111111111
222222222222
333333333333
444444444444
555555555555
666666666666
777777777777
888888888888
999999999999
101010101010101010101010
111111111111111111111111
121212121212121212121212
123456789101112
Sheet1
Depth-1 units
Weight
Depth-0 units
W (Total)
Sheet2
Depth-1 units
Weight
Depth-0 units
W(X --> A --)
Sheet3
Depth-1 units
Weight
Depth-0 units
W(X --> -- B)
Sheet3 (2)
Depth-1 units
Weight
Depth-0 units
W(Y --> B --)
Sheet3 (3)
Depth-1 units
Weight
Depth-0 units
W(Y --> -- A)
Depth-1 units
Weight
Depth-0 units
W(Y --> -- A)
Depth-1 units
Weight
Depth-0 units
W(Y --> B --)
Depth-1 units
Weight
Depth-0 units
W(X --> -- B)
Depth-1 units
Weight
Depth-0 units
W(X --> A --)
Depth-1 units
Weight
Depth-0 units
W (Total)
1
2
3
4
Depth-1 Units
Weight
Depth-0 Units
W (X A B, Y B A)
1
2
3
4
Depth-1 Units
Weight
Depth-0 Units
W (X A )
1
2
3
4
Depth-1 Units
Weight
Depth-0 Units
W (X B)
1
2
3
4
Depth-1 Units
Weight
Depth-0 Units
W (Y B )
1
2
3
4
Depth-1 Units
Weight
Depth-0 Units
W (Y A)
1
2
3
4
Depth-1 Units
Weight
Depth-0 Units
W (X A B)
1
2
3
4
Depth-1 Units
Weight
Depth-0 Units
W (Y B A)
X
Y
A
B
A
B
A
B
AB
BA
XAB
YBA
X
A
A
A
AB
XAB
X
Y
A
B
A
B
A
B
Depth-1 Units
Depth-0 Units
W (X A B, Y B A)
Depth-1 Units
Depth-0 Units
W (X A B)
Depth-1 Units
Depth-0 Units
W (Y B A)
Depth-1 Units
Depth-0 Units
W (X B)
Depth-1 Units
Depth-0 Units
W (Y B )
Depth-1 Units
Depth-0 Units
W (X A )
Depth-1 Units
Depth-0 Units
W (Y A)
YBA
XAB
BA
AB
B
A
B
A
B
A
Y
X
Units
Representations
tTHa(0) -- #1#2#3#4a(1) --
#5#6#7#8#9#10#11#12[Wa](0)[Wa](1)p(0)p(1)Update n
00.00110000020-20020-20.50.50.50.5000000000.51249739650.51249739650.51249739650.51249739650.50.50.50.50.50.50.50.56
10.0019.990000020-20020-20.50.50.50.5000000000.51250990120.51250990120.51250990120.51250990120.50.50.50.50.50.50.50.58
dtepsilon
0.051X1111
X'0.250.250.250.25
iterationtHa(0) -- #1#2#3#4a(1) --
#5#6#7#8#9#10#11#12[Wa](0)[Wa](1)
00000020-20020-20.50.50.50.500000000
10.0500.0250.0250.0250.02520-20020-20.50.50.50.50.006250-0.00625000.006250-0.00625
20.10.10.048750.048750.048750.0487520-20020-20.50.50.50.50.01218750-0.0121875000.01218750-0.0121875
30.150.1950.07131250.07131250.07131250.071312520-20020-20.50.50.50.50.0178281250-0.017828125000.0178281250-0.017828125
40.20.285250.0927468750.0927468750.0927468750.09274687520-20020-20.50.50.50.50.02318671880-0.0231867188000.02318671880-0.0231867188
50.250.37098750.11310953120.11310953120.11310953120.113109531220-20020-20.50.50.50.50.02827738280-0.0282773828000.02827738280-0.0282773828
60.30.4524381250.13245405470.13245405470.13245405470.132454054720-20020-20.50.50.50.50.03311351370-0.0331135137000.03311351370-0.0331135137
70.350.52981621880.1508313520.1508313520.1508313520.15083135220-20020-20.50.50.50.50.0377078380-0.037707838000.0377078380-0.037707838
80.40.60332540780.16828978440.16828978440.16828978440.168289784420-20020-20.50.50.50.50.04207244610-0.0420724461000.04207244610-0.0420724461
90.450.67315913740.18487529510.18487529510.18487529510.184875295120-20020-20.50.50.50.50.04621882380-0.0462188238000.04621882380-0.0462188238
100.50.73950118060.20063153040.20063153040.20063153040.200631530420-20020-20.50.50.50.50.05015788260-0.0501578826000.05015788260-0.0501578826
110.550.80252612150.21559995390.21559995390.21559995390.215599953920-20020-20.50.50.50.50.05389998850-0.0538999885000.05389998850-0.0538999885
120.60.86239981540.22981995620.22981995620.22981995620.229819956220-20020-20.50.50.50.50.0574549890-0.057454989000.0574549890-0.057454989
130.650.91927982470.24332895840.24332895840.24332895840.243328958420-20020-20.50.50.50.50.06083223960-0.0608322396000.06083223960-0.0608322396
140.70.97331583340.25616251040.25616251040.25616251040.256162510420-20020-20.50.50.50.50.06404062760-0.0640406276000.06404062760-0.0640406276
150.751.02465004180.26835438490.26835438490.26835438490.268354384920-20020-20.50.50.50.50.06708859620-0.0670885962000.06708859620-0.0670885962
160.81.07341753970.27993666570.27993666570.27993666570.279936665720-20020-20.50.50.50.50.06998416640-0.0699841664000.06998416640-0.0699841664
170.851.11974666270.29093983240.29093983240.29093983240.290939832420-20020-20.50.50.50.50.07273495810-0.0727349581000.07273495810-0.0727349581
180.91.16375932960.30139284080.30139284080.30139284080.301392840820-20020-20.50.50.50.50.07534821020-0.0753482102000.07534821020-0.0753482102
190.951.20557136310.31132319870.31132319870.31132319870.311323198720-20020-20.50.50.50.50.07783079970-0.0778307997000.07783079970-0.0778307997
2011.24529279490.32075703880.32075703880.32075703880.320757038820-20020-20.50.50.50.50.08018925970-0.0801892597000.08018925970-0.0801892597
211.051.28302815520.32971918690.32971918690.32971918690.329719186920-20020-20.50.50.50.50.08242979670-0.0824297967000.08242979670-0.0824297967
221.11.31887674740.33823322750.33823322750.33823322750.338233227520-20020-20.50.50.50.50.08455830690-0.0845583069000.08455830690-0.0845583069
231.151.35293291010.34632156610.34632156610.34632156610.346321566120-20020-20.50.50.50.50.08658039150-0.0865803915000.08658039150-0.0865803915
241.21.38528626450.35400548780.35400548780.35400548780.354005487820-20020-20.50.50.50.50.0885013720-0.088501372000.0885013720-0.088501372
251.251.41602195130.36130521340.36130521340.36130521340.361305213420-20020-20.50.50.50.50.09032630340-0.0903263034000.09032630340-0.0903263034
261.31.44522085380.36823995280.36823995280.36823995280.368239952820-20020-20.50.50.50.50.09205998820-0.0920599882000.09205998820-0.0920599882
271.351.47295981110.37482795510.37482795510.37482795510.374827955120-20020-20.50.50.50.50.09370698880-0.0937069888000.09370698880-0.0937069888
281.41.49931182050.38108655740.38108655740.38108655740.381086557420-20020-20.50.50.50.50.09527163930-0.0952716393000.09527163930-0.0952716393
291.451.52434622950.38703222950.38703222950.38703222950.387032229520-20020-20.50.50.50.50.09675805740-0.0967580574000.09675805740-0.0967580574
301.51.5481289180.3926806180.3926806180.3926806180.39268061820-20020-20.50.50.50.50.09817015450-0.0981701545000.09817015450-0.0981701545
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331.651.61257703110.40798704490.40798704490.40798704490.407987044920-20020-20.50.50.50.50.10199676120-0.1019967612000.10199676120-0.1019967612
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351.751.65035077060.4169583080.4169583080.4169583080.41695830820-20020-20.50.50.50.50.1042395770-0.104239577000.1042395770-0.104239577
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371.851.68444157040.4250548730.4250548730.4250548730.42505487320-20020-20.50.50.50.50.10626371820-0.1062637182000.10626371820-0.1062637182
381.91.70021949190.42880212930.42880212930.42880212930.428802129320-20020-20.50.50.50.50.10720053230-0.1072005323000.10720053230-0.1072005323
391.951.71520851730.43236202290.43236202290.43236202290.432362022920-20020-20.50.50.50.50.10809050570-0.1080905057000.10809050570-0.1080905057
4021.72944809140.43574392170.43574392170.43574392170.435743921720-20020-20.50.50.50.50.10893598040-0.1089359804000.10893598040-0.1089359804
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442.21.77963377950.44766302260.44766302260.44766302260.447663022620-20020-20.50.50.50.50.11191575570-0.1119157557000.11191575570-0.1119157557
452.251.79065209060.45027987150.45027987150.45027987150.450279871520-20020-20.50.50.50.50.11256996790-0.1125699679000.11256996790-0.1125699679
462.31.8011194860.45276587790.45276587790.45276587790.452765877920-20020-20.50.50.50.50.11319146950-0.1131914695000.11319146950-0.1131914695
472.351.81106351170.4551275840.4551275840.4551275840.45512758420-20020-20.50.50.50.50.1137818960-0.113781896000.1137818960-0.113781896
482.41.82051033610.45737120480.45737120480.45737120480.457371204820-20020-20.50.50.50.50.11434280120-0.1143428012000.11434280120-0.1143428012
492.451.82948481930.45950264460.45950264460.45950264460.459502644620-20020-20.50.50.50.50.11487566110-0.1148756611000.11487566110-0.1148756611
502.51.83801057840.46152751240.46152751240.46152751240.461527512420-20020-20.50.50.50.50.11538187810-0.1153818781000.11538187810-0.1153818781
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6031.90301094950.47696510050.47696510050.47696510050.476965100520-20020-20.50.50.50.50.11924127510-0.1192412751000.11924127510-0.1192412751
613.051.9078604020.47811684550.47811684550.47811684550.478116845520-20020-20.50.50.50.50.11952921140-0.1195292114000.11952921140-0.1195292114
623.11.91246738190.47921100320.47921100320.47921100320.479211003220-20020-20.50.50.50.50.11980275080-0.1198027508000.11980275080-0.1198027508
633.151.91684401280.4802504530.4802504530.4802504530.48025045320-20020-20.50.50.50.50.12006261330-0.1200626133000.12006261330-0.1200626133
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653.251.92495172160.48217603390.48217603390.48217603390.482176033920-20020-20.50.50.50.50.12054400850-0.1205440085000.12054400850-0.1205440085
663.31.92870413550.48306723220.48306723220.48306723220.483067232220-20020-20.50.50.50.50.1207668080-0.120766808000.1207668080-0.120766808
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683.41.93565548230.4847181770.4847181770.4847181770.48471817720-20020-20.50.50.50.50.12117954430-0.1211795443000.12117954430-0.1211795443
693.451.93887270820.48548226820.48548226820.48548226820.485482268220-20020-20.50.50.50.50.1213705670-0.121370567000.1213705670-0.121370567
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H
a
Time
X
-
Two Fundamental Questions
2.What are the constraints? Knowledge representationPrior
question:1.What are the activation patterns data structures mental
representations evaluated by these constraints? Harmony
maximization is satisfaction of parallel, violable constraints
-
Representation
Chart2
0.10000.1
0.70000.7
00.06000.06
00.21000.21
00.06000.06
00.21000.21
000.1280.5760.704
000.0480.2160.264
000.4480.2160.664
000.1680.0810.249
000.0640.0640.128
000.0240.0240.048
000.2240.0240.248
000.0840.0090.093
k/r0
/r01
t/r11
[ [k [ t]]]
Unit (Area = activation level)
Activation patterns: cat and its constituents
Sheet1
112111222333vector # (x-axis)
3-21123123123unit # (y axis)
2131321-21213activation value
xyx+y
Sheet1
00
00
00
x
x+y
Sheet2
000
000
000
x
y
x+y
Unit number
Activation level
Superposition
Sheet3
0
0
0
0
0
0
0
0
0
+
=
Vector
Unit
Activation Patterns
000
000
000
x
y
x+y
Unit
Vector
Superposition
000
000
000
x
y
x+y
Unit
Vector
Superposition
k/r0
/r01t/r11
r0r1fillerdepth 0depth 1depth 2
0.20.70.80.3
0.20.70.80.30.10.70.10.7
k/r00.20.70.80.30.30.30.060.210.060.21
/r010.20.70.80.30.80.40.1280.0480.4480.1680.0640.0240.2240.084
t/r110.20.70.80.30.90.10.5760.2160.2160.0810.0640.0240.0240.009
[ [k [
t]]]0.10.70.060.210.060.210.7040.2640.6640.2490.1280.0480.2480.093
012345678910111213
11111111111111
22222222222222
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44444444444444
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00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
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00000
00000
00000
k/r0
/r01
t/r11
[ [k [ t]]]
Unit
A syllable: "cat"
-
Two Fundamental Questions
2.What are the constraints? Knowledge representationPrior
question:1.What are the activation patterns data structures mental
representations evaluated by these constraints? Harmony
maximization is satisfaction of parallel, violable constraints
-
ConstraintsNOCODA: A syllable has no coda
[Maori/French/English]* H(a[ k [ t]]) = sNOCODA < 0
-
The ICS ArchitectureAkt[k[t]]
RepresentationGrammar G Function Algorithm AConstraints:
NOCODAoptimal: opt.constr. sat.:Activation PatternConnection
WeightsHarmony Opt./ Constraint Sat.Spreading Activation
Chart6
0.031250.031250.031250.03125
-0.03125-0.03125-0.03125-0.03125
-0.03125-0.03125-0.03125-0.03125
0.031250.031250.031250.03125
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0.031250.031250.031250.03125
0.031250.031250.031250.03125
-0.03125-0.03125-0.03125-0.03125
Depth-1 Units
Depth-0 Units
W (X B)
Sheet1
X' = X+X* = XTr0*r1*u0*u1*
r01111u00.50.50.50.5
r11-11-1u10.5-0.50.5-0.5
X'*Y'*A'*B'*
X11110.250.250.250.25X'0.250.250.250.25
Y1-11-10.25-0.250.25-0.25Y'0.25-0.250.25-0.25
A11-1-10.250.25-0.25-0.25A'0.250.25-0.25-0.25
B1-1-110.25-0.25-0.250.25B'0.25-0.25-0.250.25
(X'xu0)*0.1250.1250.1250.125(X'xu1)*0.1250.1250.1250.125
0.1250.1250.1250.125-0.125-0.125-0.125-0.125
(Y'xu0)*0.125-0.1250.125-0.125(Y'xu1)*0.125-0.1250.125-0.125
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(X'xu1)*0.1250.1250.1250.125-0.125-0.125-0.125-0.125
(Y'xu1)*0.125-0.1250.125-0.125-0.1250.125-0.1250.125
(A'xu1)*0.1250.125-0.125-0.125-0.125-0.1250.1250.125
(B'xu1)*0.125-0.125-0.1250.125-0.1250.1250.125-0.125
Axr011-1-1Axr011-1-111-1-1
11-1-1
Bxr01-1-11Bxr01-1-111-1-11
1-1-11
Axr111-1-1Axr111-1-1-1-111
-1-111
Bxr11-1-11Bxr11-1-11-111-1
-111-1
a=X+Axr0111111-1-111-1-1
Wa0.250.250.250.250.250-0.25000.250-0.25
a'Wa2
W(0,1)0.1250-0.12500000
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0.03125-0.03125-0.031250.03125-0.031250.031250.03125-0.03125
Y'(B'xu0)*0.03125-0.03125-0.031250.031250.03125-0.03125-0.031250.03125Y
B A0.06250-0.062500-0.062500.0625
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0.03125-0.03125-0.031250.031250.03125-0.03125-0.031250.031250.06250-0.062500-0.062500.0625
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-0.03125-0.031250.031250.031250.031250.03125-0.03125-0.03125
0.031250.03125-0.03125-0.03125-0.03125-0.031250.031250.03125
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11111111
22222222
33333333
44444444
55555555
66666666
77777777
88888888
99999999
1010101010101010
1111111111111111
1212121212121212
12345678
123456789101112
U1U2U3U4U10U20U30U40U11U21U31U41
X111100000000
Y1-11-100000000
A11-1-100000000
B1-1-1100000000
A 000011-1-111-1-1
B 00001-1-111-1-11
A000011-1-1-1-111
B 00001-1-11-111-1
AB000020-20020-2
BA000020-200-202
XAB111120-20020-2
YBA1-11-120-200-202
111111111111
222222222222
333333333333
444444444444
555555555555
666666666666
777777777777
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999999999999
101010101010101010101010
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Sheet1
Depth-1 units
Weight
Depth-0 units
W (Total)
Sheet2
Depth-1 units
Weight
Depth-0 units
W(X --> A --)
Sheet3
Depth-1 units
Weight
Depth-0 units
W(X --> -- B)
Sheet3 (2)
Depth-1 units
Weight
Depth-0 units
W(Y --> B --)
Sheet3 (3)
Depth-1 units
Weight
Depth-0 units
W(Y --> -- A)
Depth-1 units
Weight
Depth-0 units
W(Y --> -- A)
Depth-1 units
Weight
Depth-0 units
W(Y --> B --)
Depth-1 units
Weight
Depth-0 units
W(X --> -- B)
Depth-1 units
Weight
Depth-0 units
W(X --> A --)
Depth-1 units
Weight
Depth-0 units
W (Total)
1
2
3
4
Depth-1 Units
Weight
Depth-0 Units
W (X A B, Y B A)
1
2
3
4
Depth-1 Units
Weight
Depth-0 Units
W (X A )
1
2
3
4
Depth-1 Units
Weight
Depth-0 Units
W (X B)
1
2
3
4
Depth-1 Units
Weight
Depth-0 Units
W (Y B )
1
2
3
4
Depth-1 Units
Weight
Depth-0 Units
W (Y A)
1
2
3
4
Depth-1 Units
Weight
Depth-0 Units
W (X A B)
1
2
3
4
Depth-1 Units
Weight
Depth-0 Units
W (Y B A)
X
Y
A
B
A
B
A
B
AB
BA
XAB
YBA
X
A
A
A
AB
XAB
X
Y
A
B
A
B
A
B
Depth-1 Units
Depth-0 Units
W (X A B, Y B A)
Depth-1 Units
Depth-0 Units
W (X A B)
Depth-1 Units
Depth-0 Units
W (Y B A)
Depth-1 Units
Depth-0 Units
W (X B)
Depth-1 Units
Depth-0 Units
W (Y B )
Depth-1 Units
Depth-0 Units
W (X A )
Depth-1 Units
Depth-0 Units
W (Y A)
YBA
XAB
BA
AB
B
A
B
A
B
A
Y
X
Units
Representations
tTHa(0) -- #1#2#3#4a(1) --
#5#6#7#8#9#10#11#12[Wa](0)[Wa](1)p(0)p(1)Update n
00.00110000020-20020-20.50.50.50.5000000000.51249739650.51249739650.51249739650.51249739650.50.50.50.50.50.50.50.56
10.0019.990000020-20020-20.50.50.50.5000000000.51250990120.51250990120.51250990120.51250990120.50.50.50.50.50.50.50.58
dtepsilon
0.051X1111
X'0.250.250.250.25
iterationtHa(0) -- #1#2#3#4a(1) --
#5#6#7#8#9#10#11#12[Wa](0)[Wa](1)
00000020-20020-20.50.50.50.500000000
10.0500.0250.0250.0250.02520-20020-20.50.50.50.50.006250-0.00625000.006250-0.00625
20.10.10.048750.048750.048750.0487520-20020-20.50.50.50.50.01218750-0.0121875000.01218750-0.0121875
30.150.1950.07131250.07131250.07131250.071312520-20020-20.50.50.50.50.0178281250-0.017828125000.0178281250-0.017828125
40.20.285250.0927468750.0927468750.0927468750.09274687520-20020-20.50.50.50.50.02318671880-0.0231867188000.02318671880-0.0231867188
50.250.37098750.11310953120.11310953120.11310953120.113109531220-20020-20.50.50.50.50.02827738280-0.0282773828000.02827738280-0.0282773828
60.30.4524381250.13245405470.13245405470.13245405470.132454054720-20020-20.50.50.50.50.03311351370-0.0331135137000.03311351370-0.0331135137
70.350.52981621880.1508313520.1508313520.1508313520.15083135220-20020-20.50.50.50.50.0377078380-0.037707838000.0377078380-0.037707838
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100.50.73950118060.20063153040.20063153040.20063153040.200631530420-20020-20.50.50.50.50.05015788260-0.0501578826000.05015788260-0.0501578826
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150.751.02465004180.26835438490.26835438490.26835438490.268354384920-20020-20.50.50.50.50.06708859620-0.0670885962000.06708859620-0.0670885962
160.81.07341753970.27993666570.27993666570.27993666570.279936665720-20020-20.50.50.50.50.06998416640-0.0699841664000.06998416640-0.0699841664
170.851.11974666270.29093983240.29093983240.29093983240.290939832420-20020-20.50.50.50.50.07273495810-0.0727349581000.07273495810-0.0727349581
180.91.16375932960.30139284080.30139284080.30139284080.301392840820-20020-20.50.50.50.50.07534821020-0.0753482102000.07534821020-0.0753482102
190.951.20557136310.31132319870.31132319870.31132319870.311323198720-20020-20.50.50.50.50.07783079970-0.0778307997000.07783079970-0.0778307997
2011.24529279490.32075703880.32075703880.32075703880.320757038820-20020-20.50.50.50.50.08018925970-0.0801892597000.08018925970-0.0801892597
211.051.28302815520.32971918690.32971918690.32971918690.329719186920-20020-20.50.50.50.50.08242979670-0.0824297967000.08242979670-0.0824297967
221.11.31887674740.33823322750.33823322750.33823322750.338233227520-20020-20.50.50.50.50.08455830690-0.0845583069000.08455830690-0.0845583069
231.151.35293291010.34632156610.34632156610.34632156610.346321566120-20020-20.50.50.50.50.08658039150-0.0865803915000.08658039150-0.0865803915
241.21.38528626450.35400548780.35400548780.35400548780.354005487820-20020-20.50.50.50.50.0885013720-0.088501372000.0885013720-0.088501372
251.251.41602195130.36130521340.36130521340.36130521340.361305213420-20020-20.50.50.50.50.09032630340-0.0903263034000.09032630340-0.0903263034
261.31.44522085380.36823995280.36823995280.36823995280.368239952820-20020-20.50.50.50.50.09205998820-0.0920599882000.09205998820-0.0920599882
271.351.47295981110.37482795510.37482795510.37482795510.374827955120-20020-20.50.50.50.50.09370698880-0.0937069888000.09370698880-0.0937069888
281.41.49931182050.38108655740.38108655740.38108655740.381086557420-20020-20.50.50.50.50.09527163930-0.0952716393000.09527163930-0.0952716393
291.451.52434622950.38703222950.38703222950.38703222950.387032229520-20020-20.50.50.50.50.09675805740-0.0967580574000.09675805740-0.0967580574
301.51.5481289180.3926806180.3926806180.3926806180.39268061820-20020-20.50.50.50.50.09817015450-0.0981701545000.09817015450-0.0981701545
311.551.57072247210.39804658710.39804658710.39804658710.398046587120-20020-20.50.50.50.50.09951164680-0.0995116468000.09951164680-0.0995116468
321.61.59218634850.40314425780.40314425780.40314425780.403144257820-20020-20.50.50.50.50.10078606440-0.1007860644000.10078606440-0.1007860644
331.651.61257703110.40798704490.40798704490.40798704490.407987044920-20020-20.50.50.50.50.10199676120-0.1019967612000.10199676120-0.1019967612
341.71.63194817950.41258769260.41258769260.41258769260.412587692620-20020-20.50.50.50.50.10314692320-0.1031469232000.10314692320-0.1031469232
351.751.65035077060.4169583080.4169583080.4169583080.41695830820-20020-20.50.50.50.50.1042395770-0.104239577000.1042395770-0.104239577
361.81.6678332320.42111039260.42111039260.42111039260.421110392620-20020-20.50.50.50.50.10527759820-0.1052775982000.10527759820-0.1052775982
371.851.68444157040.4250548730.4250548730.4250548730.42505487320-20020-20.50.50.50.50.10626371820-0.1062637182000.10626371820-0.1062637182
381.91.70021949190.42880212930.42880212930.42880212930.428802129320-20020-20.50.50.50.50.10720053230-0.1072005323000.10720053230-0.1072005323
391.951.71520851730.43236202290.43236202290.43236202290.432362022920-20020-20.50.50.50.50.10809050570-0.1080905057000.10809050570-0.1080905057
4021.72944809140.43574392170.43574392170.43574392170.435743921720-20020-20.50.50.50.50.10893598040-0.1089359804000.10893598040-0.1089359804
412.051.74297568690.43895672560.43895672560.43895672560.438956725620-20020
