Drawings as Models of Syntactic Structure: Theory and Algorithms

1
Drawings as Models of Syntactic Structure
Theory and Algorithms

• by Mathias Möhl
• supervised by Marco Kuhlmann
• final talk of diploma thesis
• at Programming Systems Lab. Saarland University,
  Prof. Smolka

2
Dependency analysis

• The depedency analysis of a sentence consists of
  two relations
• a tree (? dependencies among words)
• a total order (? word order)

is
a
This
sentence
formal model drawings
Definition
A drawing is a relational structure (VS, ),
where (VS) forms a tree and (V ) is a total
order.
3
The task
two relaxations of projectivity gap
degree well-nestedness constraint language for
well-nested drawings saturation algorithm for
enumeration definition of TAG drawing TAGness
well-nestedness gap 1
structural properties of drawings description
language Tree Adjoining Grammar (TAG)
4

• Part I
• Structural properties of drawings

5
Some terminology

• for any node v of a drawing (VS, ) we define
• yield(v) Sv
• cover(v) Convex-Hull(Sv)

Example
yield(5)2,4,5
cover(5)2,3,4,5
1
3
4
5
2
Definition
A drawing is projective, iff the yield of each
node equals its cover.
6
Gaps in drawings

• A gap of a node v is a maximal convex set in
  cover(v)-yield(v).
• The number of gaps of a node is called its gap
  degree.

Example
node 1 has two gaps 3 and 5,6 its gap degree
is two
1
3
4
5
2
6
7

• The gap degree of a drawing is the maximum of the
  gap degrees of the nodes
• gap degree 0 ? projective
• The gap degree is a measure for the
  non-projectivity of a drawing

7
Contributions related to gaps

• Algorithm to compute the gap degree O(ng)
• (n number of nodes ggap degree gltn)
• A drawing has at most O(n²) gaps and at most
  O(nlog(n)) different gaps

worst case examples
O(nlog(n))
O(n²)
1
2
4
7
3
5
6
1
2
4
7
3
5
6
8
Well-nestedness

• Well-nestedness is a second kind of relaxation of
  projectivity
• Orthogonal to the gap degree

Definition
A drawing is well-nested if is satisfies the
constraint All disjoint subtrees are
non-interleaving. Two subtrees T1, T2 interleave,
if there are nodes l1, r1 ? T1 and l2, r2 ?
T2 such that l1 l2 r1 r2 .
interleaving subtrees
non-interleaving subtrees
l1
r1
l2
l1
l2
r1
l1
l2
r1
l1
l2
r1
r2
r2
r2
r2
projective
well-nested
9
Contributions related to well-nesterness

• A well-nested drawing has at most O(n) different
  gaps

worst case examples
O(nlog(n))
O(n²)
1
2
4
7
3
5
6
1
2
4
7
3
5
6
not well-nested
bound for well-nested drawings O(n)
10
Contributions related to well-nestedness

• two algorithms to test well-nestedness both
  O(n²)
• first algorithm tests for sibling nodes, if
  their subtrees interleave
• second algorithm reduction to a cycle test (?
  details later)
• projective drawings ? planar drawings ?
  well-nested drawings

4
1
2
3
1
2
3
1
2
3
1
2
3
4
1
2
3
1
2
3
planar
well-nested
projective
11
Contributions related to well-nestedness

• In well-nested drawings each node has a gap
  forest

The gap forest of a node v describes the relative
position among the subtrees rooted at its children
?
If the drawing is not well-nested, some nodes
have no gap forest
12
Contributions related to well-nestedness
Example
gap forest of node b
a
c
e
f
a
b
c
d
e
f
g
h
b
e
a
c
f
13

• Part II
• A description language for drawings

14
Description language

• projective drawings are describable as tree
  structure local order

is
sentence obj
subj
obj
det
This subj
a det
is
a
This
sentence

• for well-nested drawings local order is not
  sufficient
• ? goal find local description for order in
  well-nested drawings

This
is
a
a
a
sentence
15
Description language

• description of order in our approach extended
  form of gap forests
• (contains some additional nodes and edge labels)

Example
c
1
2
a
self
b
b
c
d
e
self
self

• each well-nested drawing has a unique description
• order is described locally for each node

16
Description language

• underspecified description of gap forests with
  constraint language

Example
constraints self is in the first gap of c b
before c ...
a
b
c
...
...
...
d
e

• describes sets of drawings (with the same
  tree structure)
• saturation algorithm to enumerate all described
  gap forests (NP)
• (related to saturation algorithms for dominance
  constraints)

17

• Part III
• Tree Adjoining Grammar

18
Tree Adjoining Grammar (TAG)

• TAG derivation combines elementary trees into
  derived tree

derived tree
elementary trees

VB
NP1
NP2
like

does
?
NP2
NP1
Dan
what

• derivation tree records the combining operations
• lexicalised TAG each elementary tree corresponds
  to one word

derivation tree
does
like
Dan
what
19
TAG drawings

• consist of derivation tree leaf-order of
  derived tree

derived tree
derivation tree
drawing
does

?
like
Dan
what
what
does
Dan
like

• structural characterisation

Theorem
TAG drawings wellnested drawings with gap
degree 1
20

• finally a technical detail
• Reducing well-nestedness to a cycle-test

21
The gap graph of a drawing
Two types of edges tree edges and gap edges
tree edges gap edges
c
?
d
b
b
c
a
d
a
Theorem A drawing is well-nested if and only if
its gap graph is acyclic
tree edges gap edges
c
?
d
b
b
c
a
d
x
a
x
22
The gap graph of a drawing
Proof I. If a drawing is not well-nested, its
gap graph contains a cycle. II. If the gap graph
contains a cycle, the drawing is not well-nested.
Part I.
If the drawing is not well-nested, there exist
two disjoint subtrees with interleaving nodes
?
l1
r1
l2
r2
l1
l2
r1
r2
drawing with interleaving subtrees
gap graph with cycle
23
The gap graph of a drawing
Proof II. If the gap graph contains a cycle,
the drawing is not well-nested.
If the gap graph contains a cycle, it contains a
cycle in which all nodes reached by a gap edge
are pairwise disjoint
x1
x2
xn
...
y1
y2
yn
If x1 and x2 are not disjoint
x1
x2
xn
x1
x2
xn
...
...
y1
y2
yn
y1
y2
yn
either x1 dominates x2
or x2 dominates x1
24
The gap graph of a drawing
Proof II. If the gap graph contains a cycle,
the drawing is not well-nested.
Assume that the drawing is well-nested. Then the
path implies C(x1) ? C(x2)
x1
x2
xn
...
y1
y2
yn
y1
x2
x1
C(x1) ? C(x2) ? ... ? C(xn) ? C(x1) ? C(x1) ?
C(x1)
x1
y1
x2
25
Main contributions

• Formalisation of drawings
• Measures for non-projectivity of drawings
• gap-degree
• well-nestedness
• Description language for well-nested drawings.
• Characterisation of TAG drawings (well-nested
  gap 1)
• Future work
• tree bank evaluations
• grammar formalism based on drawings
• structural properties of other formalisms

26
References

• Manuel Bodirsky, Marco Kuhlmann, and Mathias
  Möhl. Well-nested drawings as
• models of syntactic structure. In 10th Conference
  on Formal Grammar and 9th
• Meeting on Mathematics of Language, Edinburgh,
  Scotland, UK, 2005.
• Manuel Bodirsky and Martin Kutz. Pure dominance
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• of the 19th Annual Symposium on Theoretical
  Aspects of Computer Science (STACS
• 2002), 2002.
• Mike Daniels and W. Detmar Meurers. Improving the
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• constituents. In Shuly Wintner, editor,
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• International Workshop on Natural Language
  Understanding and Logic Program-
• ming, number 92 in Datalogiske Skrifter, pages
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• Universitetscenter.
• Denys Duchier and Joachim Niehren. Dominance
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• Proceedings of the First International Conference
  on Computational Logic (CL2000),
• volume 1861 of Lecture Notes in Computer Science,
  pages 326341. Springer, July
• 2000.

27
References

• Alexander Koller. Constraint-based and
  graph-based resolution of ambiguities in
• natural language. PhD thesis, Universität des
  Saarlandes, 2004.
• Martin Plátek, Tomáš Holan, and Vladislav Kubo?n.
  On relax-ability of word-order
• by d-grammars. In Cristian Calude, Michael
  Dinneen, and Smaranda Sburlan, editors,
  Combinatorics, Computability and Logic, Discrete
  Mathematics and Theoretical Computer Science,
  pages 159174. Springer, Berlin, 2001.
• Anssi Yli-Jyrä. Multiplanarity a model for
  dependency structures in treebanks. In
• Second Workshop on Treebanks and Linguistic
  Theories, Mathematical Modelling in
• Physics, Engineering and Cognitive Sciences,
  pages 189200, Växjö, Sweden, 2003.