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L2m`�H .2T2M/2M+v S�`bBM; Q7JQ`T?QHQ;B+�HHv@_B+? G�M;m�;2b
1`?�`/ >BM`B+?b �M/ .�MBďH /2 EQF
1`?�`/ >BM`B+?b �M/ .�MBďH /2 EQFL2m`�H .2T2M/2M+v S�`bBM; Q7 JQ`T?QHQ;B+�HHv@_B+? G�M;m�;2b
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Dependencygrammar
Morphology Word order
Transition-basedneural parsing
Word representationsRecurrent
neural networks
Informs
Models
1`?�`/ >BM`B+?b �M/ .�MBďH /2 EQFL2m`�H .2T2M/2M+v S�`bBM; Q7 JQ`T?QHQ;B+�HHv@_B+? G�M;m�;2b
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Dependencygrammar
Morphology Word order
Transition-basedneural parsing
Word representationsRecurrent
neural networks
1`?�`/ >BM`B+?b �M/ .�MBďH /2 EQFL2m`�H .2T2M/2M+v S�`bBM; Q7 JQ`T?QHQ;B+�HHv@_B+? G�M;m�;2b
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am`p2v
*Qm`b2 bm`p2v, ?iiTb,ff;QQX;Hf7Q`Kbf*/./ssH.pow�F">Bk
1`?�`/ >BM`B+?b �M/ .�MBďH /2 EQFL2m`�H .2T2M/2M+v S�`bBM; Q7 JQ`T?QHQ;B+�HHv@_B+? G�M;m�;2b
Dependency Grammar
I Modern theories of dependency grammar originate withLucien Tesniere
I Reference: Lucien Tesniere (1959). Elements de syntaxestructurale, Klincksieck, Paris. ISBN 2-252-01861-5
I Underlying ideas date back to Panini and his system of karakas
I Di↵erent contemporary frameworks of dependency grammar,including the Prague School’s Functional GenerativeDescription, Melcuk’s Meaning-Text Theory, and Hudson’sWord Grammar.
Dependency Grammar
The sentence is an organized whole, whose constituent elements arewords. [1.2] Every word that belongs to a sentence ceases by itself to beisolated as in the dictionary. Between the word and its neighbors, themind perceives connections, the totality of which forms the structure ofthe sentence. [1.3] The structural connections establish dependency
relations between the words. Each connection in principle unites asuperior term and an inferior term. [2.1] The superior term receives thename governor. The inferior term receives the name subordinate. Thus,in the sentence Alfred parle [. . . ], parle is the governor and Alfred thesubordinate.
from: Tesniere (1959)
Advantages of Dependency Grammars
I a completely word-based framework (no phrasal projections).
I most dependency grammar frameworks are non-derivationaland mono-stratal.
I allows for a surface-level syntactic account of languages withflexible word order and syntactic constructions withdiscontinuous elements. However, these syntactic phenomenaraise also challenging questions about the dependencygrammar formalism and the notion of projectivity ofdependency structures.
Parsing with Dependency Grammars
I Parsing a sentence is not a goal in itself, but ultimately needsto help provide an adequate answer to the question: ”Whodid what to whom, when, where, and why?” In other words:syntactic structure needs to be linked in a systematic fashionto semantic representation/interpretation.
I Dependency grammar o↵ers a direct interface between syntaxand semantics: dependency relations between a governor(lexical head) and its lexical dependents can link lexicalrepresentations of the main participants of an event or stateof a↵airs with lexical representations of the cicrumstancesunder which they occurred or hold.
I Parsing with dependency grammars benefits from the lexicalistcharacter of dependency relations. This is beneficial, interalia, for parsing long-distance dependencies and coordinations(see Kubler and Prokic 2006)
UD English treebank: treatment of nominal arguments
you should get a cocker spaniel .you should get a cocker spaniel .
PRON AUX VERB DET NOUN NOUN PUNCT
nsubj
aux
root
det
compound
obj
punct
UD English treebank: treatment of PP adjuncts
He announced this in January :he announce this in January :
PRON VERB PRON ADP PROPN PUNCT
nsubj
root
obj case
obl
punct
UD English treebank: treatment of clausal subjects
Great to have you on board !great to have you on board !
ADJ PART VERB PRON ADP NOUN PUNCT
root
mark
csubj
obj case
obl
punct
UD English treebank: treatment of relative clauses
Every move Google makes brings this particular future closer .every move Google make bring this particular future closer .
DET NOUN PROPN VERB VERB DET ADJ NOUN ADV PUNCT
det
nsubj
nsubj
acl:relcl
root
det
amod
obj
advmod
punct
UD English treebank: treatment of relative clauses
Malach , What you say makes sense .Malach , what you say make sense .
PROPN PUNCT PRON PRON VERB VERB NOUN PUNCT
vocative
punct
nsubj
nsubj
acl:relcl
root
obj
punct
UD English treebank: treatment of direct questions
Why were they suddenly acted on Saturday ?why be they suddenly act on Saturday ?
ADV AUX PRON ADV VERB ADP PROPN PUNCT
advmod
auxpass
nsubjpass
advmod
root
nmod
nmod:tmod
punct
Heads and Dependents
Tests for identifying a head H and a dependent D in a syntacticconstruction C :
1. H determines the syntactic category of C and can oftenreplace C.
2. H determines the semantic category of C ; D gives semanticspecification
3. H is obligatory; D may be optional.
4. H selects D and determines whether D is obligatory oroptional.
5. The form of D depends on H (agreement or government).
6. The linear position of D is specified with reference to H.
from: Kubler et. al. (2009), p.3f.
Heads and Dependents: Some Unclear Cases
I Auxiliary-main-verb constructions
I determiner-adjective-noun constructions
I prepositional phrases
I Coordination structures
The answer often depends on di↵erent purposes that thedependency structure is put to use for.
Case Study: Strong and Weak Adjectives in Dutch
(1) a. dethe
bruinebrown [weak]
beerbeer [masc]
’the brown beer’
b. eena
bruinebrown
beer[strong] beer [masc]
’a brown beer’
c. dethe
bruinebrown [weak] animal [neut]
beest
’the brown animal’
d. eena
bruinbrown [strong]
beestanimal [neut]
’a brown animal’
Universal Dependency Initiative
I objective: develop cross-linguistically consistent treebankannotation for many languages
I goal: facilitate multilingual parser development, cross-linguallearning, and parsing research from a language typologyperspective
I strategy: provide a universal inventory of categories andguidelines to facilitate consistent annotation of similarconstructions across languages, while allowinglanguage-specific extensions when necessary
UD annotations across languages – some examples
Universal Dependency Relations
Universal Tagset
Open class words Closed class words Other
ADJ ADP PUNCTADV AUX SYMINTJ CCONJ XNOUN DETPROPN NUMVERB PART
PRONSCONJ
Some Definitions: Sentence and Arc Labels
Definition 2.1. A sentence is a sequence of tokens denoted by:S = w0w1 . . .wn, where w0 = root
Definition 2.2. Let R = {r1, . . . , rm} be a finite set of possibledependency relation types that can hold between any two words ina sentence. A relation type r 2 R is additionally called an arc label.
Acknowledgement: Definitions 2.1 - 2.4; 2.16 - 2.18, and Notation 2.6 - 2.9
and are all taken from Kubler, McDonald, and Nivre (2009), chapt. 2
Dependency Structures and Dependency Trees
Definition 2.3. A dependency graph G = (V ,A) is a labeleddirected graph (digraph) in the standard graph-theoretic sense andconsists of nodes, V , and arcs, A, such that for sentenceS = w0w1 . . .wn and label set R the following holds:
1. V ✓ {w0,w1, . . . ,wn}2. A ✓ V ⇥ R ⇥ V
3. if (wi , r ,wj) 2 A then (wi , r 0,wj) /2 A for all r 0 6= r
The spanning node set VS = {w0,w0, . . . ,wn} contains all andonly the words of a sentence, including w0 = root.
Dependency Trees
Definition 2.4. A well-formed dependency graph G = (V ,A) foran input sentence S and dependency relation set R is anydependency graph that is a directed tree originating out of nodew0 and has the spanning node set V = VS . We call suchdependency graphs dependency trees.
Unique Head Property
Remark: Dependency trees rule out the following dependencyconfiguration:
head dep head
arc1 arc2
Some putative counterexample: In cases of VP coordination, asin Sandy listened and smiled, it appears at least pausible toestablish a dependency relation between each verbal head to thenominal dependent.
Some Notation
Notation 2.6. The notation wi ! wj indicates the unlabeled
dependency relation (or dependency relation for short) in a treeG = (V ,A). That is, wi ! wj if and only if (wi , r ,wj) 2 A forsome r 2 R .Notation 2.7. The notation wi !⇤
wj indicates the reflexivetransitive closure of the dependency relation in a tree G = (V ,A).That is, wi !⇤
wj if and only if i = j (reflexive) or both wi !⇤wi 0
and wi 0 ! wj hold (for some wi 0 2 V ).Notation 2.8. The notation wi $ wj indicates the undirected
dependency relation in a tree G = (V ,A). That is, wi $ wj if andonly if either wi ! wj or wj ! wi .Notation 2.9. The notation wi $⇤
wj indicates the reflexive
transitive closure of the undirected dependency relation in a treeG = (V ,A). That is, wi $⇤
wj if and only if i = j (reflexive) orboth wi $⇤
wi 0 and wi 0 $ wj hold (for some wi 0 2 V ).
Connectedness
A dependency tree G = (V ,A) satisfies the connectedness
property, which states that for all wi ,wj 2 V it is the case thatwi $⇤
wj . That is, there is a path connecting every two words ina dependency tree when the direction of the arc (dependencyrelation) is ignored.
(Non-)Projective Dependency Trees
Definition 2.16. An arc (wi , r ,wj) 2 A in a dependency treeG = (V ,A) is projective if and only if wi !⇤
wk for all i < k < j
when i < j , or j < k < i when j < i .
Definition 2.17. A dependency tree G = (V ,A) is a projective
dependency tree if (1) it is a dependency tree (definition 2.4), and(2) all (wi , r ;wj) 2 A are projective.
Definition 2.18. A dependency tree G = (V;A) is a non-projectivedependency tree if (1) it is a dependency tree (definition 2.4), and(2) it is not projective.
Converting Non-projective to Projective Dependency Trees
A hearing is scheduled on the issue today .
root
DET VC
ATT
SBJ
PC
ATT
TMP
PU
A hearing is scheduled on the issue today .
root
DET VC
SBJ:ATT
SBJ
PC
ATT
VC:TMP
PU
A dependency gammar treebank
A dependency gammar treebank consists of pairs of sentences S
and their corresponding dependency trees G : T = {(Sd ,Gd)}|T |d=0
The dependency trees G can be obtained by
I manual annotation by one or more human annotators
I automatically annotated by a parser
I derived automatically by a conversion algorithm from aconstituent grammar treebank
Tubingen Treebank of Written German (TuBa-D/Z)
I developed by my research group at the Seminar furSprachwissenschaft at the University of Tubingen since 1999.
I language data taken from the German newspaper ’dietageszeitung’ (taz).
I largest manually annotated treebank for GermanI total of 104,787 sentencesI average sentence length: 18.7 words per sentence.I total number of tokens: 1,959,474.
Tubingen Treebank of Written German (TuBa-D/Z)
I orginally annotated for constituent structure
I now also available in dependency structure format
I The annotation guidelines are published in the ’Stylebook forthe Tubingen Treebank of Written German (TuBa-D/Z)’http://www.sfs.uni-tuebingen.de/fileadmin/user_
upload/ascl/\tuebadz-stylebook-1508.pdf
I Information on how to obtain the data can be found at:http://www.sfs.uni-tuebingen.de/en/ascl/
resources/corpora/tueba-dz.html
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.�MBďH /2 EQF � 1`?�`/ >BM`B+?b
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
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.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
AMi`Q/m+iBQM UkV
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R G27i@�`+` BMi`Q/m+2b � H27ir�`/ TQBMiBM; �`+ rBi? H�#2H `Xk _B;?i@�`+` BMi`Q/m+2b � `B;?ir�`/ TQBMiBM; �`+ rBi? H�#2H `Xj a?B7i KQp2b � iQF2M 7`QK i?2 #mz2` iQ i?2 bi�+FX
q2 rBHH HQQF �i irQ i`�MbBiBQM bvbi2Kb,R h?2 bi�+F@T`QD2+iBp2 bvbi2KXk h?2 �`+@2�;2` bvbi2KX
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
h`�MbBiBQMb
� i`�MbBiBQM bvbi2K /2}M2b i`�MbBiBQMb iQ KQp2 7`QK i?2+m``2Mi T�`b2` bi�i2 iQ �MQi?2` T�`b2` bi�i2Xh?2 bBKTH2bi i`�MbBiBQM bvbi2Kb mb2 i?`22 ivT2b Q7 i`�MbBiBQMb,
R G27i@�`+` BMi`Q/m+2b � H27ir�`/ TQBMiBM; �`+ rBi? H�#2H `Xk _B;?i@�`+` BMi`Q/m+2b � `B;?ir�`/ TQBMiBM; �`+ rBi? H�#2H `Xj a?B7i KQp2b � iQF2M 7`QK i?2 #mz2` iQ i?2 bi�+FX
q2 rBHH HQQF �i irQ i`�MbBiBQM bvbi2Kb,R h?2 bi�+F@T`QD2+iBp2 bvbi2KXk h?2 �`+@2�;2` bvbi2KX
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
LQi�iBQM
[σ|rB], i?2 bi�+F rBi? rB �b i?2 iBT Q7 i?2 bi�+F �M/ σ �b i?2`2bi Q7 i?2 bi�+FX[rB|β], � #mz2` rBi? rB �b i?2 #mz2` ?2�/ �M/ β �b i?2 #mz2`i�BHX
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
ai�+F@T`QD2+iBp2 i`�MbBiBQM bvbi2K ULBp`2 kyyjV
h`�MbBiBQM S`2+QM/BiBQMbb?B7i (σ, [rB|β],�) ⇒ ([σ|rB],β,�)H27i@�`+` ([σ|rB|rD],β,�) ⇒ ([σ|rD]β,� ∪ (rD, `,rB)) B = 0`B;?i@�`+` ([σ|rB|rD],β,�) ⇒ ([σ|rB],β,� ∪ (rB, `,rD))
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
h`�MbBiBQM b2[m2M+2
� i`�MbBiBQM b2[m2M+2 7Q` � b2Mi2M+2 a Bb � b2[m2M+2 Q7 T�`b2`bi�i2b- *0,K = (+0, +1, . . . , +K) bm+? i?�i
R +0 Bb i?2 BMBiB�H T�`b2` bi�i2 +0(a) 7Q` ak +K Bb � }M�H T�`b2` bi�i2-j 7Q` 2p2`v B BM 1 . . .K- i?2`2 Bb � i`�MbBiBQM i BM i?2 i`�MbBiBQM
bvbi2K bm+? i?�i +B = i(+B−1)Xh?2 }M�H /2T2M/2M+v i`22 Bb :+K =< oa,�+K >
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
1t�KTH2
_PPh ai��ib�Mr�Hi KmĽ �qP@FQMi2M T`Ƀ72M_PPh .� ?�b@iQ �qP@�++QmMib +?2+F
_PPhal"C
�lsP"C�
_PPhal"C
�lsP"C�
PT2`�iBQM σ β �(_PPh) (ai��ib�Mr�Hi- . . .) ∅
a> (_PPh- ai��ib�Mr�Hi) (KmĽ- · · · ) ∅a> (_PPh- ai��ib�Mr�Hi- KmĽ) (�qP@EQMi2M- · · · ) ∅G�al"C (_PPh- KmĽ) (�qP@EQMi2M- · · · ) �1 = {(KmĽ, al"C, ai��ib�Mr�Hi)}a> (_PPh- KmĽ- �qP@EQMi2M) (T`Ƀ72M) �1a> (· · · - �qP@EQMi2M- T`Ƀ72M) () �1G�P"C� (_PPh- KmĽ- T`Ƀ72M) () �2 = �1 ∪ {(T`Ƀ72M, P"C�, �qP@EQMi2M)}_��ls (_PPh- KmĽ) () �3 = �2 ∪ {(KmĽ, �ls, T`Ƀ72M)}_�_PPh (_PPh) () �4 = �3 ∪ {(_PPh, _PPh, KmĽ)}
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
1t�KTH2
_PPh ai��ib�Mr�Hi KmĽ �qP@FQMi2M T`Ƀ72M_PPh .� ?�b@iQ �qP@�++QmMib +?2+F
_PPhal"C
�lsP"C�
_PPhal"C
�lsP"C�
PT2`�iBQM σ β �(_PPh) (ai��ib�Mr�Hi- . . .) ∅
a> (_PPh- ai��ib�Mr�Hi) (KmĽ- · · · ) ∅
a> (_PPh- ai��ib�Mr�Hi- KmĽ) (�qP@EQMi2M- · · · ) ∅G�al"C (_PPh- KmĽ) (�qP@EQMi2M- · · · ) �1 = {(KmĽ, al"C, ai��ib�Mr�Hi)}a> (_PPh- KmĽ- �qP@EQMi2M) (T`Ƀ72M) �1a> (· · · - �qP@EQMi2M- T`Ƀ72M) () �1G�P"C� (_PPh- KmĽ- T`Ƀ72M) () �2 = �1 ∪ {(T`Ƀ72M, P"C�, �qP@EQMi2M)}_��ls (_PPh- KmĽ) () �3 = �2 ∪ {(KmĽ, �ls, T`Ƀ72M)}_�_PPh (_PPh) () �4 = �3 ∪ {(_PPh, _PPh, KmĽ)}
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
1t�KTH2
_PPh ai��ib�Mr�Hi KmĽ �qP@FQMi2M T`Ƀ72M_PPh .� ?�b@iQ �qP@�++QmMib +?2+F
_PPhal"C
�lsP"C�
_PPhal"C
�lsP"C�
PT2`�iBQM σ β �(_PPh) (ai��ib�Mr�Hi- . . .) ∅
a> (_PPh- ai��ib�Mr�Hi) (KmĽ- · · · ) ∅a> (_PPh- ai��ib�Mr�Hi- KmĽ) (�qP@EQMi2M- · · · ) ∅
G�al"C (_PPh- KmĽ) (�qP@EQMi2M- · · · ) �1 = {(KmĽ, al"C, ai��ib�Mr�Hi)}a> (_PPh- KmĽ- �qP@EQMi2M) (T`Ƀ72M) �1a> (· · · - �qP@EQMi2M- T`Ƀ72M) () �1G�P"C� (_PPh- KmĽ- T`Ƀ72M) () �2 = �1 ∪ {(T`Ƀ72M, P"C�, �qP@EQMi2M)}_��ls (_PPh- KmĽ) () �3 = �2 ∪ {(KmĽ, �ls, T`Ƀ72M)}_�_PPh (_PPh) () �4 = �3 ∪ {(_PPh, _PPh, KmĽ)}
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
1t�KTH2
_PPh ai��ib�Mr�Hi KmĽ �qP@FQMi2M T`Ƀ72M_PPh .� ?�b@iQ �qP@�++QmMib +?2+F
_PPh
al"C
�lsP"C�
_PPh
al"C
�lsP"C�
PT2`�iBQM σ β �(_PPh) (ai��ib�Mr�Hi- . . .) ∅
a> (_PPh- ai��ib�Mr�Hi) (KmĽ- · · · ) ∅a> (_PPh- ai��ib�Mr�Hi- KmĽ) (�qP@EQMi2M- · · · ) ∅G�al"C (_PPh- KmĽ) (�qP@EQMi2M- · · · ) �1 = {(KmĽ, al"C, ai��ib�Mr�Hi)}
a> (_PPh- KmĽ- �qP@EQMi2M) (T`Ƀ72M) �1a> (· · · - �qP@EQMi2M- T`Ƀ72M) () �1G�P"C� (_PPh- KmĽ- T`Ƀ72M) () �2 = �1 ∪ {(T`Ƀ72M, P"C�, �qP@EQMi2M)}_��ls (_PPh- KmĽ) () �3 = �2 ∪ {(KmĽ, �ls, T`Ƀ72M)}_�_PPh (_PPh) () �4 = �3 ∪ {(_PPh, _PPh, KmĽ)}
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
1t�KTH2
_PPh ai��ib�Mr�Hi KmĽ �qP@FQMi2M T`Ƀ72M_PPh .� ?�b@iQ �qP@�++QmMib +?2+F
_PPh
al"C
�lsP"C�
_PPh
al"C
�lsP"C�
PT2`�iBQM σ β �(_PPh) (ai��ib�Mr�Hi- . . .) ∅
a> (_PPh- ai��ib�Mr�Hi) (KmĽ- · · · ) ∅a> (_PPh- ai��ib�Mr�Hi- KmĽ) (�qP@EQMi2M- · · · ) ∅G�al"C (_PPh- KmĽ) (�qP@EQMi2M- · · · ) �1 = {(KmĽ, al"C, ai��ib�Mr�Hi)}a> (_PPh- KmĽ- �qP@EQMi2M) (T`Ƀ72M) �1
a> (· · · - �qP@EQMi2M- T`Ƀ72M) () �1G�P"C� (_PPh- KmĽ- T`Ƀ72M) () �2 = �1 ∪ {(T`Ƀ72M, P"C�, �qP@EQMi2M)}_��ls (_PPh- KmĽ) () �3 = �2 ∪ {(KmĽ, �ls, T`Ƀ72M)}_�_PPh (_PPh) () �4 = �3 ∪ {(_PPh, _PPh, KmĽ)}
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
1t�KTH2
_PPh ai��ib�Mr�Hi KmĽ �qP@FQMi2M T`Ƀ72M_PPh .� ?�b@iQ �qP@�++QmMib +?2+F
_PPh
al"C
�lsP"C�
_PPh
al"C
�lsP"C�
PT2`�iBQM σ β �(_PPh) (ai��ib�Mr�Hi- . . .) ∅
a> (_PPh- ai��ib�Mr�Hi) (KmĽ- · · · ) ∅a> (_PPh- ai��ib�Mr�Hi- KmĽ) (�qP@EQMi2M- · · · ) ∅G�al"C (_PPh- KmĽ) (�qP@EQMi2M- · · · ) �1 = {(KmĽ, al"C, ai��ib�Mr�Hi)}a> (_PPh- KmĽ- �qP@EQMi2M) (T`Ƀ72M) �1a> (· · · - �qP@EQMi2M- T`Ƀ72M) () �1
G�P"C� (_PPh- KmĽ- T`Ƀ72M) () �2 = �1 ∪ {(T`Ƀ72M, P"C�, �qP@EQMi2M)}_��ls (_PPh- KmĽ) () �3 = �2 ∪ {(KmĽ, �ls, T`Ƀ72M)}_�_PPh (_PPh) () �4 = �3 ∪ {(_PPh, _PPh, KmĽ)}
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
1t�KTH2
_PPh ai��ib�Mr�Hi KmĽ �qP@FQMi2M T`Ƀ72M_PPh .� ?�b@iQ �qP@�++QmMib +?2+F
_PPh
al"C
�ls
P"C�_PPh
al"C
�ls
P"C�
PT2`�iBQM σ β �(_PPh) (ai��ib�Mr�Hi- . . .) ∅
a> (_PPh- ai��ib�Mr�Hi) (KmĽ- · · · ) ∅a> (_PPh- ai��ib�Mr�Hi- KmĽ) (�qP@EQMi2M- · · · ) ∅G�al"C (_PPh- KmĽ) (�qP@EQMi2M- · · · ) �1 = {(KmĽ, al"C, ai��ib�Mr�Hi)}a> (_PPh- KmĽ- �qP@EQMi2M) (T`Ƀ72M) �1a> (· · · - �qP@EQMi2M- T`Ƀ72M) () �1G�P"C� (_PPh- KmĽ- T`Ƀ72M) () �2 = �1 ∪ {(T`Ƀ72M, P"C�, �qP@EQMi2M)}
_��ls (_PPh- KmĽ) () �3 = �2 ∪ {(KmĽ, �ls, T`Ƀ72M)}_�_PPh (_PPh) () �4 = �3 ∪ {(_PPh, _PPh, KmĽ)}
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
1t�KTH2
_PPh ai��ib�Mr�Hi KmĽ �qP@FQMi2M T`Ƀ72M_PPh .� ?�b@iQ �qP@�++QmMib +?2+F
_PPh
al"C�ls
P"C�_PPh
al"C�ls
P"C�
PT2`�iBQM σ β �(_PPh) (ai��ib�Mr�Hi- . . .) ∅
a> (_PPh- ai��ib�Mr�Hi) (KmĽ- · · · ) ∅a> (_PPh- ai��ib�Mr�Hi- KmĽ) (�qP@EQMi2M- · · · ) ∅G�al"C (_PPh- KmĽ) (�qP@EQMi2M- · · · ) �1 = {(KmĽ, al"C, ai��ib�Mr�Hi)}a> (_PPh- KmĽ- �qP@EQMi2M) (T`Ƀ72M) �1a> (· · · - �qP@EQMi2M- T`Ƀ72M) () �1G�P"C� (_PPh- KmĽ- T`Ƀ72M) () �2 = �1 ∪ {(T`Ƀ72M, P"C�, �qP@EQMi2M)}_��ls (_PPh- KmĽ) () �3 = �2 ∪ {(KmĽ, �ls, T`Ƀ72M)}
_�_PPh (_PPh) () �4 = �3 ∪ {(_PPh, _PPh, KmĽ)}
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
1t�KTH2
_PPh ai��ib�Mr�Hi KmĽ �qP@FQMi2M T`Ƀ72M_PPh .� ?�b@iQ �qP@�++QmMib +?2+F
_PPhal"C
�lsP"C�
_PPhal"C
�lsP"C�
PT2`�iBQM σ β �(_PPh) (ai��ib�Mr�Hi- . . .) ∅
a> (_PPh- ai��ib�Mr�Hi) (KmĽ- · · · ) ∅a> (_PPh- ai��ib�Mr�Hi- KmĽ) (�qP@EQMi2M- · · · ) ∅G�al"C (_PPh- KmĽ) (�qP@EQMi2M- · · · ) �1 = {(KmĽ, al"C, ai��ib�Mr�Hi)}a> (_PPh- KmĽ- �qP@EQMi2M) (T`Ƀ72M) �1a> (· · · - �qP@EQMi2M- T`Ƀ72M) () �1G�P"C� (_PPh- KmĽ- T`Ƀ72M) () �2 = �1 ∪ {(T`Ƀ72M, P"C�, �qP@EQMi2M)}_��ls (_PPh- KmĽ) () �3 = �2 ∪ {(KmĽ, �ls, T`Ƀ72M)}_�_PPh (_PPh) () �4 = �3 ∪ {(_PPh, _PPh, KmĽ)}
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
S`QT2`iB2b Q7 `2bmHiBM; ;`�T?b
aBM;H2@?2�/2/, 2p2`v iQF2M ?�b � bBM;H2 ?2�/ U2t+2Ti i?2_PPh iQF2MVX
aBM+2 � iQF2M Bb `2KQp2/ 7`QK i?2 bi�+F �7i2` `2+2BpBM; � ?2�/-Bi Bb MQi TQbbB#H2 iQ �ii�+? Bi irB+2X
_QQi, MQ iQF2M ;Qp2`Mb i?2 `QQiXh?2 H27i@�`+ T`2+QM/BiBQM /Bb�HHQrb BMi`Q/m+BM; �/2T2M/2M+v r?2`2 _PPh Bb i?2 /2T2M/2MiX
aT�MMBM;, 2p2`v iQF2M Bb � p2`i2t BM i?2 ;`�T?Xq2�FHv +QMM2+i2/, i?2`2 Bb bQK2 mM/B`2+i2/ T�i? #2ir22M�Mv rQ`/ T�B` rB,rDX
aBM+2 � iQF2M +�MMQi #2 `2KQp2/ 7`QK i?2 bi�+F rBi?Qmi�ii�+?K2Mi- Bi Bb MQi TQbbB#H2 iQ T`Q/m+2 � /Bb+QMM2+i2/bm#;`�T?X
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
S`QT2`iB2b Q7 `2bmHiBM; ;`�T?b
aBM;H2@?2�/2/, 2p2`v iQF2M ?�b � bBM;H2 ?2�/ U2t+2Ti i?2_PPh iQF2MVX
aBM+2 � iQF2M Bb `2KQp2/ 7`QK i?2 bi�+F �7i2` `2+2BpBM; � ?2�/-Bi Bb MQi TQbbB#H2 iQ �ii�+? Bi irB+2X
_QQi, MQ iQF2M ;Qp2`Mb i?2 `QQiXh?2 H27i@�`+ T`2+QM/BiBQM /Bb�HHQrb BMi`Q/m+BM; �/2T2M/2M+v r?2`2 _PPh Bb i?2 /2T2M/2MiX
aT�MMBM;, 2p2`v iQF2M Bb � p2`i2t BM i?2 ;`�T?Xq2�FHv +QMM2+i2/, i?2`2 Bb bQK2 mM/B`2+i2/ T�i? #2ir22M�Mv rQ`/ T�B` rB,rDX
aBM+2 � iQF2M +�MMQi #2 `2KQp2/ 7`QK i?2 bi�+F rBi?Qmi�ii�+?K2Mi- Bi Bb MQi TQbbB#H2 iQ T`Q/m+2 � /Bb+QMM2+i2/bm#;`�T?X
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
S`QT2`iB2b Q7 `2bmHiBM; ;`�T?b
aBM;H2@?2�/2/, 2p2`v iQF2M ?�b � bBM;H2 ?2�/ U2t+2Ti i?2_PPh iQF2MVX
aBM+2 � iQF2M Bb `2KQp2/ 7`QK i?2 bi�+F �7i2` `2+2BpBM; � ?2�/-Bi Bb MQi TQbbB#H2 iQ �ii�+? Bi irB+2X
_QQi, MQ iQF2M ;Qp2`Mb i?2 `QQiXh?2 H27i@�`+ T`2+QM/BiBQM /Bb�HHQrb BMi`Q/m+BM; �/2T2M/2M+v r?2`2 _PPh Bb i?2 /2T2M/2MiX
aT�MMBM;, 2p2`v iQF2M Bb � p2`i2t BM i?2 ;`�T?X
q2�FHv +QMM2+i2/, i?2`2 Bb bQK2 mM/B`2+i2/ T�i? #2ir22M�Mv rQ`/ T�B` rB,rDX
aBM+2 � iQF2M +�MMQi #2 `2KQp2/ 7`QK i?2 bi�+F rBi?Qmi�ii�+?K2Mi- Bi Bb MQi TQbbB#H2 iQ T`Q/m+2 � /Bb+QMM2+i2/bm#;`�T?X
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
S`QT2`iB2b Q7 `2bmHiBM; ;`�T?b
aBM;H2@?2�/2/, 2p2`v iQF2M ?�b � bBM;H2 ?2�/ U2t+2Ti i?2_PPh iQF2MVX
aBM+2 � iQF2M Bb `2KQp2/ 7`QK i?2 bi�+F �7i2` `2+2BpBM; � ?2�/-Bi Bb MQi TQbbB#H2 iQ �ii�+? Bi irB+2X
_QQi, MQ iQF2M ;Qp2`Mb i?2 `QQiXh?2 H27i@�`+ T`2+QM/BiBQM /Bb�HHQrb BMi`Q/m+BM; �/2T2M/2M+v r?2`2 _PPh Bb i?2 /2T2M/2MiX
aT�MMBM;, 2p2`v iQF2M Bb � p2`i2t BM i?2 ;`�T?Xq2�FHv +QMM2+i2/, i?2`2 Bb bQK2 mM/B`2+i2/ T�i? #2ir22M�Mv rQ`/ T�B` rB,rDX
aBM+2 � iQF2M +�MMQi #2 `2KQp2/ 7`QK i?2 bi�+F rBi?Qmi�ii�+?K2Mi- Bi Bb MQi TQbbB#H2 iQ T`Q/m+2 � /Bb+QMM2+i2/bm#;`�T?X
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
S`QT2`iB2b Q7 `2bmHiBM; ;`�T?b UkV
S`QD2+iBp2, i?2 bi�+F@T`QD2+iBp2 bvbi2K +�MMQi T`Q/m+2MQM@T`QD2+iBp2 �`+bX
bR bk bj b9
hQ �ii�+? bj �b � /2T2M/2Mi iQ bR- r2 M22/ bR �M/ bj QM iQTQ7 i?2 bi�+FX
AM Q`/2` iQ /Q bQ, `2KQp2 bk }`biXhQ �ii�+? bk �b � /2T2M/2Mi iQ b9- r M22/ bk �M/ b9 QM iQTQ7 i?2 bi�+FX
AM Q`/2` iQ /Q bQ, `2KQp2 bj }`biX.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
Sb2m/Q@T`QD2+iBp2 T�`bBM; ULBp`2 �M/ LBHbbQM kyy8V
r?BH2 � Bb MQM@T`QD2+iBp2 /Q�← aK�HH2bi@LQMS@�`+(�)�← (�− {�}) ∪ GB7i(�)
2M/ r?BH2
GB7i(rD → rF) =
!rB → rF B7 rB → rDmM/27BM2/ Qi?2`rBb2
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
Sb2m/Q@T`QD2+iBp2 T�`bBM; ULBp`2 �M/ LBHbbQM kyy8V
r?BH2 � Bb MQM@T`QD2+iBp2 /Q�← aK�HH2bi@LQMS@�`+(�)�← (�− {�}) ∪ GB7i(�)
2M/ r?BH2
GB7i(rD → rF) =
!rB → rF B7 rB → rDmM/27BM2/ Qi?2`rBb2
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
ai�+F@br�T bvbi2K ULBp`2 kyyNV
h?2 bi�+F@T`QD2+iBp2 bvbi2K +�M T`Q/m+2 MQM@T`QD2+iBp2 i`22b #v�//BM; � br�T i`�MbBiBQM,
h`�MbBiBQM S`2+QM/BiBQMbb?B7i (σ, [rB|β],�) ⇒ ([σ|rB],β,�)H27i@�`+ ([σ|rB|rD],β,�) ⇒ ([σ|rD]β,� ∪ (rD, `,rB)) B = 0`B;?i@�`+ ([σ|rB|rD],β,�) ⇒ ([σ|rB],β,� ∪ (rB, `,rD))br�T ([σ|rB|rD],β,�) ⇒ ([σ|rD|rB],β,�) y I B I D
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
"QiiQK@mT T�`bBM;
h?2 bi�+F@T`QD2+iBp2 �H;Q`Bi?K Bb � #QiiQK mT T�`bBM;�H;Q`Bi?K,
G27i@�`+ �M/ _B;?i@�`+ `2KQp2 i?2 /2T2M/2Mi 7`QK i?2bi�+FX*QMb2[m2MiHv- QM+2 � iQF2M Bb �ii�+?2/ iQ Bib ?2�/- Bi +�MMQi`2+2Bp2 M2r /2T2M/2MibX
h?2 /2T2M/2Mib Q7 � iQF2M rB Kmbi #2 �ii�+?2/ #27Q`2 rB Bb�ii�+?2/ iQ Bib ?2�/X
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
1t�KTH2 U#QiiQK@mT T�`bBM;V
_PPh ai��ib�Mr�Hi T`Ƀ7i �qP@FQMi2M pQM >�K#m`;_PPh .� +?2+Fb �qP@�++QmMib Q7 >�K#m`;
al"C_PPh
SLSSP"C�
al"C_PPh
PT2`�iBQM σ β �(_PPh) (ai��ib�Mr�Hi- . . .) ∅
a> (_PPh- ai��ib�Mr�Hi) (T`Ƀ7i- · · · ) ∅a> (_PPh- ai��ib�Mr�Hi- T`Ƀ7i) (�qP@EQMi2M- · · · ) ∅G�al"C (_PPh- T`Ƀ7i) (�qP@EQMi2M- · · · ) �1 = {(T`Ƀ7i, al"C, ai��ib�Mr�Hi)}_�_PPh (_PPh) (�qP@EQMi2M- · · · ) �2 = �1 ∪ {(_PPh, _PPh, T`Ƀ7i)}
S`Q#H2K, _B;?i@�`+ i`�MbBiBQMb QM [σ|r1|r2] Kmbi #2TQbiTQM2/ mMiBH �HH /2T2M/2Mib �`2 �ii+?2/ iQ r2X
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
1t�KTH2 U#QiiQK@mT T�`bBM;V
_PPh ai��ib�Mr�Hi T`Ƀ7i �qP@FQMi2M pQM >�K#m`;_PPh .� +?2+Fb �qP@�++QmMib Q7 >�K#m`;
al"C_PPh
SLSSP"C�
al"C_PPh
PT2`�iBQM σ β �(_PPh) (ai��ib�Mr�Hi- . . .) ∅
a> (_PPh- ai��ib�Mr�Hi) (T`Ƀ7i- · · · ) ∅
a> (_PPh- ai��ib�Mr�Hi- T`Ƀ7i) (�qP@EQMi2M- · · · ) ∅G�al"C (_PPh- T`Ƀ7i) (�qP@EQMi2M- · · · ) �1 = {(T`Ƀ7i, al"C, ai��ib�Mr�Hi)}_�_PPh (_PPh) (�qP@EQMi2M- · · · ) �2 = �1 ∪ {(_PPh, _PPh, T`Ƀ7i)}
S`Q#H2K, _B;?i@�`+ i`�MbBiBQMb QM [σ|r1|r2] Kmbi #2TQbiTQM2/ mMiBH �HH /2T2M/2Mib �`2 �ii+?2/ iQ r2X
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
1t�KTH2 U#QiiQK@mT T�`bBM;V
_PPh ai��ib�Mr�Hi T`Ƀ7i �qP@FQMi2M pQM >�K#m`;_PPh .� +?2+Fb �qP@�++QmMib Q7 >�K#m`;
al"C_PPh
SLSSP"C�
al"C_PPh
PT2`�iBQM σ β �(_PPh) (ai��ib�Mr�Hi- . . .) ∅
a> (_PPh- ai��ib�Mr�Hi) (T`Ƀ7i- · · · ) ∅a> (_PPh- ai��ib�Mr�Hi- T`Ƀ7i) (�qP@EQMi2M- · · · ) ∅
G�al"C (_PPh- T`Ƀ7i) (�qP@EQMi2M- · · · ) �1 = {(T`Ƀ7i, al"C, ai��ib�Mr�Hi)}_�_PPh (_PPh) (�qP@EQMi2M- · · · ) �2 = �1 ∪ {(_PPh, _PPh, T`Ƀ7i)}
S`Q#H2K, _B;?i@�`+ i`�MbBiBQMb QM [σ|r1|r2] Kmbi #2TQbiTQM2/ mMiBH �HH /2T2M/2Mib �`2 �ii+?2/ iQ r2X
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
1t�KTH2 U#QiiQK@mT T�`bBM;V
_PPh ai��ib�Mr�Hi T`Ƀ7i �qP@FQMi2M pQM >�K#m`;_PPh .� +?2+Fb �qP@�++QmMib Q7 >�K#m`;
al"C
_PPhSLSSP"C�al"C
_PPh
PT2`�iBQM σ β �(_PPh) (ai��ib�Mr�Hi- . . .) ∅
a> (_PPh- ai��ib�Mr�Hi) (T`Ƀ7i- · · · ) ∅a> (_PPh- ai��ib�Mr�Hi- T`Ƀ7i) (�qP@EQMi2M- · · · ) ∅G�al"C (_PPh- T`Ƀ7i) (�qP@EQMi2M- · · · ) �1 = {(T`Ƀ7i, al"C, ai��ib�Mr�Hi)}
_�_PPh (_PPh) (�qP@EQMi2M- · · · ) �2 = �1 ∪ {(_PPh, _PPh, T`Ƀ7i)}
S`Q#H2K, _B;?i@�`+ i`�MbBiBQMb QM [σ|r1|r2] Kmbi #2TQbiTQM2/ mMiBH �HH /2T2M/2Mib �`2 �ii+?2/ iQ r2X
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
1t�KTH2 U#QiiQK@mT T�`bBM;V
_PPh ai��ib�Mr�Hi T`Ƀ7i �qP@FQMi2M pQM >�K#m`;_PPh .� +?2+Fb �qP@�++QmMib Q7 >�K#m`;
al"C_PPh
SLSSP"C�al"C_PPh
PT2`�iBQM σ β �(_PPh) (ai��ib�Mr�Hi- . . .) ∅
a> (_PPh- ai��ib�Mr�Hi) (T`Ƀ7i- · · · ) ∅a> (_PPh- ai��ib�Mr�Hi- T`Ƀ7i) (�qP@EQMi2M- · · · ) ∅G�al"C (_PPh- T`Ƀ7i) (�qP@EQMi2M- · · · ) �1 = {(T`Ƀ7i, al"C, ai��ib�Mr�Hi)}_�_PPh (_PPh) (�qP@EQMi2M- · · · ) �2 = �1 ∪ {(_PPh, _PPh, T`Ƀ7i)}
S`Q#H2K, _B;?i@�`+ i`�MbBiBQMb QM [σ|r1|r2] Kmbi #2TQbiTQM2/ mMiBH �HH /2T2M/2Mib �`2 �ii+?2/ iQ r2X
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
1t�KTH2 U#QiiQK@mT T�`bBM;V
_PPh ai��ib�Mr�Hi T`Ƀ7i �qP@FQMi2M pQM >�K#m`;_PPh .� +?2+Fb �qP@�++QmMib Q7 >�K#m`;
al"C_PPh
SLSSP"C�al"C_PPh
PT2`�iBQM σ β �(_PPh) (ai��ib�Mr�Hi- . . .) ∅
a> (_PPh- ai��ib�Mr�Hi) (T`Ƀ7i- · · · ) ∅a> (_PPh- ai��ib�Mr�Hi- T`Ƀ7i) (�qP@EQMi2M- · · · ) ∅G�al"C (_PPh- T`Ƀ7i) (�qP@EQMi2M- · · · ) �1 = {(T`Ƀ7i, al"C, ai��ib�Mr�Hi)}_�_PPh (_PPh) (�qP@EQMi2M- · · · ) �2 = �1 ∪ {(_PPh, _PPh, T`Ƀ7i)}
S`Q#H2K, _B;?i@�`+ i`�MbBiBQMb QM [σ|r1|r2] Kmbi #2TQbiTQM2/ mMiBH �HH /2T2M/2Mib �`2 �ii+?2/ iQ r2X
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
1t�KTH2 U#QiiQK@mT T�`bBM;V
_PPh ai��ib�Mr�Hi T`Ƀ7i �qP@FQMi2M pQM >�K#m`;_PPh .� +?2+Fb �qP@�++QmMib Q7 >�K#m`;
al"C SLSSP"C�_PPh
al"C SLSSP"C�_PPh
PT2`�iBQM σ β �(_PPh) (ai��ib�Mr�Hi- . . .) ∅
a> (_PPh- ai��ib�Mr�Hi) (T`Ƀ7i- · · · ) ∅a> (_PPh- ai��ib�Mr�Hi- T`Ƀ7i) (�qP@EQMi2M- · · · ) ∅G�al"C (_PPh- T`Ƀ7i) (�qP@EQMi2M- · · · ) �1 = {(T`Ƀ7i, al"C, ai��ib�Mr�Hi)}a> (· · · - T`Ƀ7i- �qP@EQMi2M) (pQM- · · · ) �1a> (· · · - �qP@EQMi2M- pQM) (>�K#m`;) �1a> (· · · - pQM- >�K#m`;) () �1_�SL (· · · - �qP@EQMi2M- pQM) () �2 = �1 ∪ {(pQM, SL, >�K#m`;)}_�SS (· · · - T`Ƀ7i- �qP@EQMi2M) () �3 = �2 ∪ {(�qP@EQMi2M, SS, pQM)}_�P"C� (_PPh- T`Ƀ7i) () �4 = �3 ∪ {(T`Ƀ7i, P"C�, �qP@EQMi2M)}_�_PPh (_PPh) () �5 = �4 ∪ {(_PPh, _PPh, S`Ƀ7i)}
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
1t�KTH2 U#QiiQK@mT T�`bBM;V
_PPh ai��ib�Mr�Hi T`Ƀ7i �qP@FQMi2M pQM >�K#m`;_PPh .� +?2+Fb �qP@�++QmMib Q7 >�K#m`;
al"C SLSSP"C�_PPh
al"C SLSSP"C�_PPh
PT2`�iBQM σ β �(_PPh) (ai��ib�Mr�Hi- . . .) ∅
a> (_PPh- ai��ib�Mr�Hi) (T`Ƀ7i- · · · ) ∅
a> (_PPh- ai��ib�Mr�Hi- T`Ƀ7i) (�qP@EQMi2M- · · · ) ∅G�al"C (_PPh- T`Ƀ7i) (�qP@EQMi2M- · · · ) �1 = {(T`Ƀ7i, al"C, ai��ib�Mr�Hi)}a> (· · · - T`Ƀ7i- �qP@EQMi2M) (pQM- · · · ) �1a> (· · · - �qP@EQMi2M- pQM) (>�K#m`;) �1a> (· · · - pQM- >�K#m`;) () �1_�SL (· · · - �qP@EQMi2M- pQM) () �2 = �1 ∪ {(pQM, SL, >�K#m`;)}_�SS (· · · - T`Ƀ7i- �qP@EQMi2M) () �3 = �2 ∪ {(�qP@EQMi2M, SS, pQM)}_�P"C� (_PPh- T`Ƀ7i) () �4 = �3 ∪ {(T`Ƀ7i, P"C�, �qP@EQMi2M)}_�_PPh (_PPh) () �5 = �4 ∪ {(_PPh, _PPh, S`Ƀ7i)}
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
1t�KTH2 U#QiiQK@mT T�`bBM;V
_PPh ai��ib�Mr�Hi T`Ƀ7i �qP@FQMi2M pQM >�K#m`;_PPh .� +?2+Fb �qP@�++QmMib Q7 >�K#m`;
al"C SLSSP"C�_PPh
al"C SLSSP"C�_PPh
PT2`�iBQM σ β �(_PPh) (ai��ib�Mr�Hi- . . .) ∅
a> (_PPh- ai��ib�Mr�Hi) (T`Ƀ7i- · · · ) ∅a> (_PPh- ai��ib�Mr�Hi- T`Ƀ7i) (�qP@EQMi2M- · · · ) ∅
G�al"C (_PPh- T`Ƀ7i) (�qP@EQMi2M- · · · ) �1 = {(T`Ƀ7i, al"C, ai��ib�Mr�Hi)}a> (· · · - T`Ƀ7i- �qP@EQMi2M) (pQM- · · · ) �1a> (· · · - �qP@EQMi2M- pQM) (>�K#m`;) �1a> (· · · - pQM- >�K#m`;) () �1_�SL (· · · - �qP@EQMi2M- pQM) () �2 = �1 ∪ {(pQM, SL, >�K#m`;)}_�SS (· · · - T`Ƀ7i- �qP@EQMi2M) () �3 = �2 ∪ {(�qP@EQMi2M, SS, pQM)}_�P"C� (_PPh- T`Ƀ7i) () �4 = �3 ∪ {(T`Ƀ7i, P"C�, �qP@EQMi2M)}_�_PPh (_PPh) () �5 = �4 ∪ {(_PPh, _PPh, S`Ƀ7i)}
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
1t�KTH2 U#QiiQK@mT T�`bBM;V
_PPh ai��ib�Mr�Hi T`Ƀ7i �qP@FQMi2M pQM >�K#m`;_PPh .� +?2+Fb �qP@�++QmMib Q7 >�K#m`;
al"C
SLSSP"C�_PPh
al"C
SLSSP"C�_PPh
PT2`�iBQM σ β �(_PPh) (ai��ib�Mr�Hi- . . .) ∅
a> (_PPh- ai��ib�Mr�Hi) (T`Ƀ7i- · · · ) ∅a> (_PPh- ai��ib�Mr�Hi- T`Ƀ7i) (�qP@EQMi2M- · · · ) ∅G�al"C (_PPh- T`Ƀ7i) (�qP@EQMi2M- · · · ) �1 = {(T`Ƀ7i, al"C, ai��ib�Mr�Hi)}
a> (· · · - T`Ƀ7i- �qP@EQMi2M) (pQM- · · · ) �1a> (· · · - �qP@EQMi2M- pQM) (>�K#m`;) �1a> (· · · - pQM- >�K#m`;) () �1_�SL (· · · - �qP@EQMi2M- pQM) () �2 = �1 ∪ {(pQM, SL, >�K#m`;)}_�SS (· · · - T`Ƀ7i- �qP@EQMi2M) () �3 = �2 ∪ {(�qP@EQMi2M, SS, pQM)}_�P"C� (_PPh- T`Ƀ7i) () �4 = �3 ∪ {(T`Ƀ7i, P"C�, �qP@EQMi2M)}_�_PPh (_PPh) () �5 = �4 ∪ {(_PPh, _PPh, S`Ƀ7i)}
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
1t�KTH2 U#QiiQK@mT T�`bBM;V
_PPh ai��ib�Mr�Hi T`Ƀ7i �qP@FQMi2M pQM >�K#m`;_PPh .� +?2+Fb �qP@�++QmMib Q7 >�K#m`;
al"C
SLSSP"C�_PPh
al"C
SLSSP"C�_PPh
PT2`�iBQM σ β �(_PPh) (ai��ib�Mr�Hi- . . .) ∅
a> (_PPh- ai��ib�Mr�Hi) (T`Ƀ7i- · · · ) ∅a> (_PPh- ai��ib�Mr�Hi- T`Ƀ7i) (�qP@EQMi2M- · · · ) ∅G�al"C (_PPh- T`Ƀ7i) (�qP@EQMi2M- · · · ) �1 = {(T`Ƀ7i, al"C, ai��ib�Mr�Hi)}a> (· · · - T`Ƀ7i- �qP@EQMi2M) (pQM- · · · ) �1
a> (· · · - �qP@EQMi2M- pQM) (>�K#m`;) �1a> (· · · - pQM- >�K#m`;) () �1_�SL (· · · - �qP@EQMi2M- pQM) () �2 = �1 ∪ {(pQM, SL, >�K#m`;)}_�SS (· · · - T`Ƀ7i- �qP@EQMi2M) () �3 = �2 ∪ {(�qP@EQMi2M, SS, pQM)}_�P"C� (_PPh- T`Ƀ7i) () �4 = �3 ∪ {(T`Ƀ7i, P"C�, �qP@EQMi2M)}_�_PPh (_PPh) () �5 = �4 ∪ {(_PPh, _PPh, S`Ƀ7i)}
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
1t�KTH2 U#QiiQK@mT T�`bBM;V
_PPh ai��ib�Mr�Hi T`Ƀ7i �qP@FQMi2M pQM >�K#m`;_PPh .� +?2+Fb �qP@�++QmMib Q7 >�K#m`;
al"C
SLSSP"C�_PPh
al"C
SLSSP"C�_PPh
PT2`�iBQM σ β �(_PPh) (ai��ib�Mr�Hi- . . .) ∅
a> (_PPh- ai��ib�Mr�Hi) (T`Ƀ7i- · · · ) ∅a> (_PPh- ai��ib�Mr�Hi- T`Ƀ7i) (�qP@EQMi2M- · · · ) ∅G�al"C (_PPh- T`Ƀ7i) (�qP@EQMi2M- · · · ) �1 = {(T`Ƀ7i, al"C, ai��ib�Mr�Hi)}a> (· · · - T`Ƀ7i- �qP@EQMi2M) (pQM- · · · ) �1a> (· · · - �qP@EQMi2M- pQM) (>�K#m`;) �1
a> (· · · - pQM- >�K#m`;) () �1_�SL (· · · - �qP@EQMi2M- pQM) () �2 = �1 ∪ {(pQM, SL, >�K#m`;)}_�SS (· · · - T`Ƀ7i- �qP@EQMi2M) () �3 = �2 ∪ {(�qP@EQMi2M, SS, pQM)}_�P"C� (_PPh- T`Ƀ7i) () �4 = �3 ∪ {(T`Ƀ7i, P"C�, �qP@EQMi2M)}_�_PPh (_PPh) () �5 = �4 ∪ {(_PPh, _PPh, S`Ƀ7i)}
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
1t�KTH2 U#QiiQK@mT T�`bBM;V
_PPh ai��ib�Mr�Hi T`Ƀ7i �qP@FQMi2M pQM >�K#m`;_PPh .� +?2+Fb �qP@�++QmMib Q7 >�K#m`;
al"C
SLSSP"C�_PPh
al"C
SLSSP"C�_PPh
PT2`�iBQM σ β �(_PPh) (ai��ib�Mr�Hi- . . .) ∅
a> (_PPh- ai��ib�Mr�Hi) (T`Ƀ7i- · · · ) ∅a> (_PPh- ai��ib�Mr�Hi- T`Ƀ7i) (�qP@EQMi2M- · · · ) ∅G�al"C (_PPh- T`Ƀ7i) (�qP@EQMi2M- · · · ) �1 = {(T`Ƀ7i, al"C, ai��ib�Mr�Hi)}a> (· · · - T`Ƀ7i- �qP@EQMi2M) (pQM- · · · ) �1a> (· · · - �qP@EQMi2M- pQM) (>�K#m`;) �1a> (· · · - pQM- >�K#m`;) () �1
_�SL (· · · - �qP@EQMi2M- pQM) () �2 = �1 ∪ {(pQM, SL, >�K#m`;)}_�SS (· · · - T`Ƀ7i- �qP@EQMi2M) () �3 = �2 ∪ {(�qP@EQMi2M, SS, pQM)}_�P"C� (_PPh- T`Ƀ7i) () �4 = �3 ∪ {(T`Ƀ7i, P"C�, �qP@EQMi2M)}_�_PPh (_PPh) () �5 = �4 ∪ {(_PPh, _PPh, S`Ƀ7i)}
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
1t�KTH2 U#QiiQK@mT T�`bBM;V
_PPh ai��ib�Mr�Hi T`Ƀ7i �qP@FQMi2M pQM >�K#m`;_PPh .� +?2+Fb �qP@�++QmMib Q7 >�K#m`;
al"C SL
SSP"C�_PPh
al"C SL
SSP"C�_PPh
PT2`�iBQM σ β �(_PPh) (ai��ib�Mr�Hi- . . .) ∅
a> (_PPh- ai��ib�Mr�Hi) (T`Ƀ7i- · · · ) ∅a> (_PPh- ai��ib�Mr�Hi- T`Ƀ7i) (�qP@EQMi2M- · · · ) ∅G�al"C (_PPh- T`Ƀ7i) (�qP@EQMi2M- · · · ) �1 = {(T`Ƀ7i, al"C, ai��ib�Mr�Hi)}a> (· · · - T`Ƀ7i- �qP@EQMi2M) (pQM- · · · ) �1a> (· · · - �qP@EQMi2M- pQM) (>�K#m`;) �1a> (· · · - pQM- >�K#m`;) () �1_�SL (· · · - �qP@EQMi2M- pQM) () �2 = �1 ∪ {(pQM, SL, >�K#m`;)}
_�SS (· · · - T`Ƀ7i- �qP@EQMi2M) () �3 = �2 ∪ {(�qP@EQMi2M, SS, pQM)}_�P"C� (_PPh- T`Ƀ7i) () �4 = �3 ∪ {(T`Ƀ7i, P"C�, �qP@EQMi2M)}_�_PPh (_PPh) () �5 = �4 ∪ {(_PPh, _PPh, S`Ƀ7i)}
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
1t�KTH2 U#QiiQK@mT T�`bBM;V
_PPh ai��ib�Mr�Hi T`Ƀ7i �qP@FQMi2M pQM >�K#m`;_PPh .� +?2+Fb �qP@�++QmMib Q7 >�K#m`;
al"C SLSS
P"C�_PPh
al"C SLSS
P"C�_PPh
PT2`�iBQM σ β �(_PPh) (ai��ib�Mr�Hi- . . .) ∅
a> (_PPh- ai��ib�Mr�Hi) (T`Ƀ7i- · · · ) ∅a> (_PPh- ai��ib�Mr�Hi- T`Ƀ7i) (�qP@EQMi2M- · · · ) ∅G�al"C (_PPh- T`Ƀ7i) (�qP@EQMi2M- · · · ) �1 = {(T`Ƀ7i, al"C, ai��ib�Mr�Hi)}a> (· · · - T`Ƀ7i- �qP@EQMi2M) (pQM- · · · ) �1a> (· · · - �qP@EQMi2M- pQM) (>�K#m`;) �1a> (· · · - pQM- >�K#m`;) () �1_�SL (· · · - �qP@EQMi2M- pQM) () �2 = �1 ∪ {(pQM, SL, >�K#m`;)}_�SS (· · · - T`Ƀ7i- �qP@EQMi2M) () �3 = �2 ∪ {(�qP@EQMi2M, SS, pQM)}
_�P"C� (_PPh- T`Ƀ7i) () �4 = �3 ∪ {(T`Ƀ7i, P"C�, �qP@EQMi2M)}_�_PPh (_PPh) () �5 = �4 ∪ {(_PPh, _PPh, S`Ƀ7i)}
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
1t�KTH2 U#QiiQK@mT T�`bBM;V
_PPh ai��ib�Mr�Hi T`Ƀ7i �qP@FQMi2M pQM >�K#m`;_PPh .� +?2+Fb �qP@�++QmMib Q7 >�K#m`;
al"C SLSSP"C�
_PPhal"C SLSSP"C�
_PPh
PT2`�iBQM σ β �(_PPh) (ai��ib�Mr�Hi- . . .) ∅
a> (_PPh- ai��ib�Mr�Hi) (T`Ƀ7i- · · · ) ∅a> (_PPh- ai��ib�Mr�Hi- T`Ƀ7i) (�qP@EQMi2M- · · · ) ∅G�al"C (_PPh- T`Ƀ7i) (�qP@EQMi2M- · · · ) �1 = {(T`Ƀ7i, al"C, ai��ib�Mr�Hi)}a> (· · · - T`Ƀ7i- �qP@EQMi2M) (pQM- · · · ) �1a> (· · · - �qP@EQMi2M- pQM) (>�K#m`;) �1a> (· · · - pQM- >�K#m`;) () �1_�SL (· · · - �qP@EQMi2M- pQM) () �2 = �1 ∪ {(pQM, SL, >�K#m`;)}_�SS (· · · - T`Ƀ7i- �qP@EQMi2M) () �3 = �2 ∪ {(�qP@EQMi2M, SS, pQM)}_�P"C� (_PPh- T`Ƀ7i) () �4 = �3 ∪ {(T`Ƀ7i, P"C�, �qP@EQMi2M)}
_�_PPh (_PPh) () �5 = �4 ∪ {(_PPh, _PPh, S`Ƀ7i)}
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
1t�KTH2 U#QiiQK@mT T�`bBM;V
_PPh ai��ib�Mr�Hi T`Ƀ7i �qP@FQMi2M pQM >�K#m`;_PPh .� +?2+Fb �qP@�++QmMib Q7 >�K#m`;
al"C SLSSP"C�_PPh
al"C SLSSP"C�_PPh
PT2`�iBQM σ β �(_PPh) (ai��ib�Mr�Hi- . . .) ∅
a> (_PPh- ai��ib�Mr�Hi) (T`Ƀ7i- · · · ) ∅a> (_PPh- ai��ib�Mr�Hi- T`Ƀ7i) (�qP@EQMi2M- · · · ) ∅G�al"C (_PPh- T`Ƀ7i) (�qP@EQMi2M- · · · ) �1 = {(T`Ƀ7i, al"C, ai��ib�Mr�Hi)}a> (· · · - T`Ƀ7i- �qP@EQMi2M) (pQM- · · · ) �1a> (· · · - �qP@EQMi2M- pQM) (>�K#m`;) �1a> (· · · - pQM- >�K#m`;) () �1_�SL (· · · - �qP@EQMi2M- pQM) () �2 = �1 ∪ {(pQM, SL, >�K#m`;)}_�SS (· · · - T`Ƀ7i- �qP@EQMi2M) () �3 = �2 ∪ {(�qP@EQMi2M, SS, pQM)}_�P"C� (_PPh- T`Ƀ7i) () �4 = �3 ∪ {(T`Ƀ7i, P"C�, �qP@EQMi2M)}_�_PPh (_PPh) () �5 = �4 ∪ {(_PPh, _PPh, S`Ƀ7i)}
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
�`+@1�;2` bvbi2K ULBp`2 kyy9V
h`�MbBiBQM S`2+QM/BiBQMbb?B7i (σ, [rB|β],�) ⇒ ([σ|rB],β,�)`B;?i@�`+` ([σ|rB], [rD|β],�) ⇒ ([σ|rB|rD],β,� ∪ (rB, `,rD))H27i@�`+` ([σ|rB], [rD|β],�) ⇒ (σ, [rD|β],� ∪ (rD, `,rB)) B = 0 ∧
(rF, `′,rB) ∈ �`2/m+2 ([σ|rB],β,�) ⇒ (σ,β,�) (rF, `′,rB) ∈ �
E2v TQBMi, _B;?i@�`+ /Q2b MQi `2KQp2 i?2 /2T2M/2Mi 7`QK i?2bi�+FX
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
�`+@1�;2` bvbi2K ULBp`2 kyy9V
h`�MbBiBQM S`2+QM/BiBQMbb?B7i (σ, [rB|β],�) ⇒ ([σ|rB],β,�)`B;?i@�`+` ([σ|rB], [rD|β],�) ⇒ ([σ|rB|rD],β,� ∪ (rB, `,rD))H27i@�`+` ([σ|rB], [rD|β],�) ⇒ (σ, [rD|β],� ∪ (rD, `,rB)) B = 0 ∧
(rF, `′,rB) ∈ �`2/m+2 ([σ|rB],β,�) ⇒ (σ,β,�) (rF, `′,rB) ∈ �
E2v TQBMi, _B;?i@�`+ /Q2b MQi `2KQp2 i?2 /2T2M/2Mi 7`QK i?2bi�+FX
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
1t�KTH2
_PPh ai��ib�Mr�Hi T`Ƀ7i �qP@FQMi2M pQM >�K#m`;_PPh .� +?2+Fb �qP@�++QmMib Q7 >�K#m`;
al"C_PPh
P"C� SS SL
al"C_PPh
P"C� SS SL
PT2`�iBQM σ β �(_PPh) (ai��ib�Mr�Hi- . . .) ∅
a> (_PPh- ai��ib�Mr�Hi) (T`Ƀ7i- · · · ) ∅G�al"C (_PPh) (T`Ƀ7i- · · · ) �1 = {(T`Ƀ7i, al"C, ai��ib�Mr�Hi)}_�_PPh (_PPh- T`Ƀ7i) (�qP@EQMi2M- · · · ) �2 = �1 ∪ {(_PPh, _PPh, T`Ƀ7i)}_�P"C� (· · · - �qP@EQMi2M) (pQM- >�K#m`;) �3 = �2 ∪ {(T`Ƀ7i, P"C�, �qP@EQMi2M)}_�SS (· · · - pQM) (>�K#m`;) �4 = �3 ∪ {(�qP@EQMi2M, SS, pQM)}_�SL (· · · - >�K#m`;) () �5 = �4 ∪ {(pQM, SS, >�K#m`;)}4× _. (_PPh) () �5
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
1t�KTH2
_PPh ai��ib�Mr�Hi T`Ƀ7i �qP@FQMi2M pQM >�K#m`;_PPh .� +?2+Fb �qP@�++QmMib Q7 >�K#m`;
al"C_PPh
P"C� SS SL
al"C_PPh
P"C� SS SL
PT2`�iBQM σ β �(_PPh) (ai��ib�Mr�Hi- . . .) ∅
a> (_PPh- ai��ib�Mr�Hi) (T`Ƀ7i- · · · ) ∅
G�al"C (_PPh) (T`Ƀ7i- · · · ) �1 = {(T`Ƀ7i, al"C, ai��ib�Mr�Hi)}_�_PPh (_PPh- T`Ƀ7i) (�qP@EQMi2M- · · · ) �2 = �1 ∪ {(_PPh, _PPh, T`Ƀ7i)}_�P"C� (· · · - �qP@EQMi2M) (pQM- >�K#m`;) �3 = �2 ∪ {(T`Ƀ7i, P"C�, �qP@EQMi2M)}_�SS (· · · - pQM) (>�K#m`;) �4 = �3 ∪ {(�qP@EQMi2M, SS, pQM)}_�SL (· · · - >�K#m`;) () �5 = �4 ∪ {(pQM, SS, >�K#m`;)}4× _. (_PPh) () �5
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
1t�KTH2
_PPh ai��ib�Mr�Hi T`Ƀ7i �qP@FQMi2M pQM >�K#m`;_PPh .� +?2+Fb �qP@�++QmMib Q7 >�K#m`;
al"C
_PPhP"C� SS SLal"C
_PPhP"C� SS SL
PT2`�iBQM σ β �(_PPh) (ai��ib�Mr�Hi- . . .) ∅
a> (_PPh- ai��ib�Mr�Hi) (T`Ƀ7i- · · · ) ∅G�al"C (_PPh) (T`Ƀ7i- · · · ) �1 = {(T`Ƀ7i, al"C, ai��ib�Mr�Hi)}
_�_PPh (_PPh- T`Ƀ7i) (�qP@EQMi2M- · · · ) �2 = �1 ∪ {(_PPh, _PPh, T`Ƀ7i)}_�P"C� (· · · - �qP@EQMi2M) (pQM- >�K#m`;) �3 = �2 ∪ {(T`Ƀ7i, P"C�, �qP@EQMi2M)}_�SS (· · · - pQM) (>�K#m`;) �4 = �3 ∪ {(�qP@EQMi2M, SS, pQM)}_�SL (· · · - >�K#m`;) () �5 = �4 ∪ {(pQM, SS, >�K#m`;)}4× _. (_PPh) () �5
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
1t�KTH2
_PPh ai��ib�Mr�Hi T`Ƀ7i �qP@FQMi2M pQM >�K#m`;_PPh .� +?2+Fb �qP@�++QmMib Q7 >�K#m`;
al"C_PPh
P"C� SS SLal"C_PPh
P"C� SS SL
PT2`�iBQM σ β �(_PPh) (ai��ib�Mr�Hi- . . .) ∅
a> (_PPh- ai��ib�Mr�Hi) (T`Ƀ7i- · · · ) ∅G�al"C (_PPh) (T`Ƀ7i- · · · ) �1 = {(T`Ƀ7i, al"C, ai��ib�Mr�Hi)}_�_PPh (_PPh- T`Ƀ7i) (�qP@EQMi2M- · · · ) �2 = �1 ∪ {(_PPh, _PPh, T`Ƀ7i)}
_�P"C� (· · · - �qP@EQMi2M) (pQM- >�K#m`;) �3 = �2 ∪ {(T`Ƀ7i, P"C�, �qP@EQMi2M)}_�SS (· · · - pQM) (>�K#m`;) �4 = �3 ∪ {(�qP@EQMi2M, SS, pQM)}_�SL (· · · - >�K#m`;) () �5 = �4 ∪ {(pQM, SS, >�K#m`;)}4× _. (_PPh) () �5
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
1t�KTH2
_PPh ai��ib�Mr�Hi T`Ƀ7i �qP@FQMi2M pQM >�K#m`;_PPh .� +?2+Fb �qP@�++QmMib Q7 >�K#m`;
al"C_PPh
P"C�
SS SLal"C_PPh
P"C�
SS SL
PT2`�iBQM σ β �(_PPh) (ai��ib�Mr�Hi- . . .) ∅
a> (_PPh- ai��ib�Mr�Hi) (T`Ƀ7i- · · · ) ∅G�al"C (_PPh) (T`Ƀ7i- · · · ) �1 = {(T`Ƀ7i, al"C, ai��ib�Mr�Hi)}_�_PPh (_PPh- T`Ƀ7i) (�qP@EQMi2M- · · · ) �2 = �1 ∪ {(_PPh, _PPh, T`Ƀ7i)}_�P"C� (· · · - �qP@EQMi2M) (pQM- >�K#m`;) �3 = �2 ∪ {(T`Ƀ7i, P"C�, �qP@EQMi2M)}
_�SS (· · · - pQM) (>�K#m`;) �4 = �3 ∪ {(�qP@EQMi2M, SS, pQM)}_�SL (· · · - >�K#m`;) () �5 = �4 ∪ {(pQM, SS, >�K#m`;)}4× _. (_PPh) () �5
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
1t�KTH2
_PPh ai��ib�Mr�Hi T`Ƀ7i �qP@FQMi2M pQM >�K#m`;_PPh .� +?2+Fb �qP@�++QmMib Q7 >�K#m`;
al"C_PPh
P"C� SS
SLal"C_PPh
P"C� SS
SL
PT2`�iBQM σ β �(_PPh) (ai��ib�Mr�Hi- . . .) ∅
a> (_PPh- ai��ib�Mr�Hi) (T`Ƀ7i- · · · ) ∅G�al"C (_PPh) (T`Ƀ7i- · · · ) �1 = {(T`Ƀ7i, al"C, ai��ib�Mr�Hi)}_�_PPh (_PPh- T`Ƀ7i) (�qP@EQMi2M- · · · ) �2 = �1 ∪ {(_PPh, _PPh, T`Ƀ7i)}_�P"C� (· · · - �qP@EQMi2M) (pQM- >�K#m`;) �3 = �2 ∪ {(T`Ƀ7i, P"C�, �qP@EQMi2M)}_�SS (· · · - pQM) (>�K#m`;) �4 = �3 ∪ {(�qP@EQMi2M, SS, pQM)}
_�SL (· · · - >�K#m`;) () �5 = �4 ∪ {(pQM, SS, >�K#m`;)}4× _. (_PPh) () �5
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
1t�KTH2
_PPh ai��ib�Mr�Hi T`Ƀ7i �qP@FQMi2M pQM >�K#m`;_PPh .� +?2+Fb �qP@�++QmMib Q7 >�K#m`;
al"C_PPh
P"C� SS SL
al"C_PPh
P"C� SS SL
PT2`�iBQM σ β �(_PPh) (ai��ib�Mr�Hi- . . .) ∅
a> (_PPh- ai��ib�Mr�Hi) (T`Ƀ7i- · · · ) ∅G�al"C (_PPh) (T`Ƀ7i- · · · ) �1 = {(T`Ƀ7i, al"C, ai��ib�Mr�Hi)}_�_PPh (_PPh- T`Ƀ7i) (�qP@EQMi2M- · · · ) �2 = �1 ∪ {(_PPh, _PPh, T`Ƀ7i)}_�P"C� (· · · - �qP@EQMi2M) (pQM- >�K#m`;) �3 = �2 ∪ {(T`Ƀ7i, P"C�, �qP@EQMi2M)}_�SS (· · · - pQM) (>�K#m`;) �4 = �3 ∪ {(�qP@EQMi2M, SS, pQM)}_�SL (· · · - >�K#m`;) () �5 = �4 ∪ {(pQM, SS, >�K#m`;)}
4× _. (_PPh) () �5
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
1t�KTH2
_PPh ai��ib�Mr�Hi T`Ƀ7i �qP@FQMi2M pQM >�K#m`;_PPh .� +?2+Fb �qP@�++QmMib Q7 >�K#m`;
al"C_PPh
P"C� SS SL
al"C_PPh
P"C� SS SL
PT2`�iBQM σ β �(_PPh) (ai��ib�Mr�Hi- . . .) ∅
a> (_PPh- ai��ib�Mr�Hi) (T`Ƀ7i- · · · ) ∅G�al"C (_PPh) (T`Ƀ7i- · · · ) �1 = {(T`Ƀ7i, al"C, ai��ib�Mr�Hi)}_�_PPh (_PPh- T`Ƀ7i) (�qP@EQMi2M- · · · ) �2 = �1 ∪ {(_PPh, _PPh, T`Ƀ7i)}_�P"C� (· · · - �qP@EQMi2M) (pQM- >�K#m`;) �3 = �2 ∪ {(T`Ƀ7i, P"C�, �qP@EQMi2M)}_�SS (· · · - pQM) (>�K#m`;) �4 = �3 ∪ {(�qP@EQMi2M, SS, pQM)}_�SL (· · · - >�K#m`;) () �5 = �4 ∪ {(pQM, SS, >�K#m`;)}4× _. (_PPh) () �5
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
:mB/2
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
AMi`Q/m+iBQM
LmK#2` Q7 TQbbB#H2 i`�MbBiBQM b2[m2M+2b BM i?2 Q`/2` Q72|_||a|- r?2`2,
|_|, i?2 bBx2 Q7 i?2 /2T2M/2M+v H�#2H b2iX|a|, i?2 b2Mi2M+2 H2M;i?X
PMHv � bK�HH MmK#2` Q7 b2[m2M+2b T`Q/m+2b i?2 +Q``2+i/2T2M/2M+v ;`�T?Xh?2 ;mB/2 Bb � 7mM+iBQM ; : *→ h i?�i b2H2+ib i?2 #2bii`�MbBiBQM 7Q` � ;Bp2M T�`b2` bi�i2X
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
hvT2b Q7 ;mB/2b
h?2`2 �`2 irQ ivT2b Q7 ;mB/2b,R P`�+H2, `2im`Mb 7Q` � ;Bp2M +QM};m`�iBQM + i?2 +Q``2+i
i`�MbBiBQM i mbBM; i?2 ;QH/ bi�M/�`/ /2T2M/2M+v i`22Xk *H�bbB}2`, T`2/B+ib 7Q` � ;Bp2M +QM};m`�iBQM + i?2 #2bi
i`�MbBiBQM i mbBM; � /�i�@/`Bp2M KQ/2HX
P`�+H2b �`2 mb2/ /m`BM; i`�BMBM; iQ ;2M2`�i2 i`�BMBM; 2t�KTH2bX*H�bbB}2`b �`2 mb2/ BM T�`bBM; Q7 mMb22M b2Mi2M+2bX
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
hvT2b Q7 ;mB/2b
h?2`2 �`2 irQ ivT2b Q7 ;mB/2b,R P`�+H2, `2im`Mb 7Q` � ;Bp2M +QM};m`�iBQM + i?2 +Q``2+i
i`�MbBiBQM i mbBM; i?2 ;QH/ bi�M/�`/ /2T2M/2M+v i`22Xk *H�bbB}2`, T`2/B+ib 7Q` � ;Bp2M +QM};m`�iBQM + i?2 #2bi
i`�MbBiBQM i mbBM; � /�i�@/`Bp2M KQ/2HXP`�+H2b �`2 mb2/ /m`BM; i`�BMBM; iQ ;2M2`�i2 i`�BMBM; 2t�KTH2bX*H�bbB}2`b �`2 mb2/ BM T�`bBM; Q7 mMb22M b2Mi2M+2bX
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
S�`bBM; �H;Q`Bi?K
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
AMi`Q/m+iBQM
6Q` � ;Bp2M T�`b2` bi�i2, KmHiBTH2 TQbbB#H2 i`�MbBiBQMbX.2i2`KBMBbiB+ T�`b2`,
�Hr�vb +?QQb2 i?2 #2bi i`�MbBiBQM `2im`M2/ #v i?2 ;mB/2X"2�K@b2�`+? T�`b2`,
�i 2�+? bi2T- `2i�BM i?2 F ?B;?2bi@T`Q#�#BHBiv i`�MbBiBQMb2[m2M+2bX
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
.2i2`KBMBbiB+ T�`b2`
+← +0(a)r?BH2 ¬Bb6BM�H(+) /Q
i← ;(+)+← i(+)
2M/ r?BH2
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
S�`bBM; +QKTH2tBiv Ubi�+F@T`QD2+iBp2 bvbi2KV
O(M)- r?2`2 M Bb |a|- T`QpB/2/ i?�i i?2 ;mB/2 �M/ i`�MbBiBQM7mM+iBQMb �`2 +QMbi�Mi iBK2X
h?2 #mz2` bBx2 Bb #QmM/ #v M,h?2 K�tBKmK bBx2 Q7 i?2 #mz2` Bb |a| UBM i?2 BMBiB�H+QM};m`�iBQMVXLQM2 Q7 i?2 i`�MbBiBQMb BM+`2�b2b i?2 bBx2 Q7 i?2 #mz2`X
h?2 bi�+F bBx2 Bb �HbQ #QmM/ #v M,1�+? a?B7i BM+`2�b2b i?2 bi�+F bBx2 #v 1XJ�tBKmK bi�+F bBx2, a?B7i mMiBH �HH #mz2` iQF2Mb �`2 QM i?2bi�+FXJ�tBKmK TQbbB#H2 MmK#2` Q7 a?B7ib, |a|
1�+? i`�MbBiBQM /2+`2�b2b i?2 bi�+F Q` #mz2` bBx2 #v 1X
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
S�`bBM; +QKTH2tBiv Ubi�+F@T`QD2+iBp2 bvbi2KV
O(M)- r?2`2 M Bb |a|- T`QpB/2/ i?�i i?2 ;mB/2 �M/ i`�MbBiBQM7mM+iBQMb �`2 +QMbi�Mi iBK2Xh?2 #mz2` bBx2 Bb #QmM/ #v M,
h?2 K�tBKmK bBx2 Q7 i?2 #mz2` Bb |a| UBM i?2 BMBiB�H+QM};m`�iBQMVXLQM2 Q7 i?2 i`�MbBiBQMb BM+`2�b2b i?2 bBx2 Q7 i?2 #mz2`X
h?2 bi�+F bBx2 Bb �HbQ #QmM/ #v M,1�+? a?B7i BM+`2�b2b i?2 bi�+F bBx2 #v 1XJ�tBKmK bi�+F bBx2, a?B7i mMiBH �HH #mz2` iQF2Mb �`2 QM i?2bi�+FXJ�tBKmK TQbbB#H2 MmK#2` Q7 a?B7ib, |a|
1�+? i`�MbBiBQM /2+`2�b2b i?2 bi�+F Q` #mz2` bBx2 #v 1X
.�MBďH /2 EQF � 1`?�`/ >BM`B+?bh`�MbBiBQM@#�b2/ /2T2M/2M+v T�`bBM;
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S�`b2` bi�i2 h`�MbBiBQM bvbi2Kb :mB/2 S�`bBM; �H;Q`Bi?K h?2 2M/
S�`bBM; +QKTH2tBiv Ubi�+F@T`QD2+iBp2 bvbi2KV
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