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Dependencygrammar

Morphology Word order

Transition-basedneural parsing

Word representationsRecurrent

neural networks

Informs

Models

Dependencygrammar

Morphology Word order

Transition-basedneural parsing

Word representationsRecurrent

neural networks

am`p2v

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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

compound

UD English treebank: treatment of PP adjuncts

He announced this in January :he announce this in January :

PRON VERB PRON ADP PROPN PUNCT

obj case

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

obj case

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

acl:relcl

advmod

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

acl:relcl

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

nmod:tmod

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]

’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 .

DET VC

A hearing is scheduled on the issue today .

DET VC

SBJ:ATT

VC:TMP

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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1t�KTH2

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_PPhal"C

�lsP"C�

_PPhal"C

�lsP"C�

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1t�KTH2

_PPhal"C

�lsP"C�

_PPhal"C

�lsP"C�

a> (_PPh- ai��ib�Mr�Hi) (KmĽ- · · · ) ∅

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1t�KTH2

_PPhal"C

�lsP"C�

_PPhal"C

�lsP"C�

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1t�KTH2

�lsP"C�

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1t�KTH2

�lsP"C�

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)}

1t�KTH2

�lsP"C�

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)}

1t�KTH2

P"C�_PPh

P"C�

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)}

1t�KTH2

al"C�ls

P"C�_PPh

al"C�ls

P"C�

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)}

1t�KTH2

_PPhal"C

�lsP"C�

_PPhal"C

�lsP"C�

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)}

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

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

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;

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

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

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,

"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

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

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

al"C_PPh

SLSSP"C�

al"C_PPh

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)}

al"C_PPh

SLSSP"C�

al"C_PPh

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)}

_PPhSLSSP"C�al"C

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)}

al"C_PPh

SLSSP"C�al"C_PPh

al"C_PPh

SLSSP"C�al"C_PPh

al"C SLSSP"C�_PPh

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)}

al"C SLSSP"C�_PPh

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)}

al"C SLSSP"C�_PPh

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)}

SLSSP"C�_PPh

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)}

SLSSP"C�_PPh

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)}

SLSSP"C�_PPh

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)}

SLSSP"C�_PPh

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)}

al"C SL

SSP"C�_PPh

al"C SL

SSP"C�_PPh

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)}

al"C SLSS

P"C�_PPh

al"C SLSS

P"C�_PPh

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)}

al"C SLSSP"C�

_PPhal"C SLSSP"C�

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)}

al"C SLSSP"C�_PPh

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)}

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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

1t�KTH2

al"C_PPh

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al"C_PPh

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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

1t�KTH2

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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

1t�KTH2

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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

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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

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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

1t�KTH2

al"C_PPh

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al"C_PPh

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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

1t�KTH2

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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

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