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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

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2 / 25 Dependency analysis This isa sentence The depedency analysis of a sentence consists of two relations: a tree ( dependencies among words) a total order ( word order) formal model: drawings A drawing is a relational structure (V;S, ), where (V;S) forms a tree and (V; ) is a total order. Definition

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3 / 25 The task two relaxations of projectivity: gap degree well-nestedness constraint language for well-nested drawings saturation algorithm for enumeration definition of TAG drawing TAG’ness = well-nestedness + gap 1 structural properties of drawings description language Tree Adjoining Grammar (TAG)

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4 / 25 Part I Structural properties of drawings

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5 / 25 Some terminology for any node v of a drawing (V;S, ) we define: yield(v) := S*v cover(v) = Convex-Hull(S*v) 13452 yield(5)={2,4,5} cover(5)={2,3,4,5} Example A drawing is projective, iff the yield of each node equals its cover. Definition

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6 / 25 Example 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. 13452 67 node 1 has two gaps: {3} and {5,6} its gap degree is two 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

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7 / 25 Contributions related to gaps Algorithm to compute the gap degree O(n*g) (n= number of nodes; g=gap degree ; g<n) A drawing has at most O(n²) gaps and at most O(n*log(n)) different gaps 1247356 1247356 worst case examples O(n*log(n))O(n²)

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8 / 25 Well-nestedness A drawing is well-nested if is satisfies the constraint: All disjoint subtrees are non-interleaving. Two subtrees T 1, T 2 interleave, if there are nodes l 1, r 1 T 1 and l 2, r 2 T 2 such that l 1 l 2 r 1 r 2. l1l1 l2l2 r1r1 r2r2 interleaving subtrees l1l1 l2l2 r1r1 r2r2 non-interleaving subtrees l1l1 r1r1 l2l2 r2r2 l1l1 l2l2 r1r1 r2r2 Definition Well-nestedness is a second kind of relaxation of projectivity Orthogonal to the gap degree projective well-nested

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9 / 25 Contributions related to well-nesterness A well-nested drawing has at most O(n) different gaps 1247356 1247356 worst case examples O(n*log(n))O(n²) bound for well-nested drawings: O(n) not well-nested

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10 / 25 Contributions related to well-nestedness 123 4123 123 4123 123 123 projective planarwell-nested 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

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11 / 25 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: ?

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12 / 25 Contributions related to well-nestedness abcdefgh a c e f gap forest of node b: Example a c e f b

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13 / 25 Part II A description language for drawings

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14 / 25 Description language projective drawings are describable as tree structure + local order This isa sentence is subjobj This: subj sentence: obj det a: det for well-nested drawings local order is not sufficient: goal: find local description for order in well-nested drawings This isa sentence a a

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15 / 25 Description language description of order in our approach: extended form of gap forests (contains some additional nodes and edge labels) Example cadb e c a d b e b c self 1 2 ed 22 11 each well-nested drawing has a unique description order is described locally for each node

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16 / 25 Description language underspecified description of gap forests with constraint language 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)... Example c a d b e constraints: self is in the first gap of c b before c......

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17 / 25 Part III Tree Adjoining Grammar

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18 / 25 Tree Adjoining Grammar (TAG) TAG derivation combines elementary trees into derived tree like VB NP1 NP2 * Dan NP2 S VB does what NP1 like VB NP1 NP2 VB Dan S does what derivation tree records the combining operations lexicalised TAG: each elementary tree corresponds to one word does like Dan what derived tree elementary trees derivation tree

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19 / 25 TAG drawings consist of derivation tree + leaf-order of derived tree structural characterisation: TAG drawings = wellnested drawings with gap degree ≤ 1 Theorem: like VB NP1 NP2 VB Dan S does what derived tree does like Dan what derivation tree what doesDan like drawing +

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20 / 25 finally a technical detail: Reducing well-nestedness to a cycle-test

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21 / 25 b c a d c a d b ■ tree edges ■ gap edges Two types of edges: tree edges and gap edges Theorem: A drawing is well-nested if and only if its gap graph is acyclic b c a d c a d b ■ tree edges ■ gap edges x x The gap graph of a drawing

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22 / 25 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. l1l2r1r2 If the drawing is not well-nested, there exist two disjoint subtrees with interleaving nodes: l1r1l2r2 drawing with interleaving subtreesgap graph with cycle The gap graph of a drawing

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23 / 25 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 y1 x1 y2 x2 yn xn... If x1 and x2 are not disjoint either x1 dominates x2or x2 dominates x1 y1 x1 y2 x2 yn xn... y1 x1 y2 x2 yn xn... The gap graph of a drawing

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24 / 25 Assume that the drawing is well- nested. Then the path implies C(x1) C(x2) y1 x1 y2 x2 yn xn... y1x2 x1 y1x2 x1 The gap graph of a drawing Proof: II. If the gap graph contains a cycle, the drawing is not well-nested. C(x1) C(x2) ... C(xn) C(x1) C(x1) C(x1)

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25 / 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

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26 / 25 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 constraints. In Proceedings of the 19th Annual Symposium on Theoretical Aspects of Computer Science (STACS 2002), 2002. Mike Daniels and W. Detmar Meurers. Improving the efficiency of parsing with discontinuous constituents. In Shuly Wintner, editor, Proceedings of NLULP’02: The 7th International Workshop on Natural Language Understanding and Logic Program- ming, number 92 in Datalogiske Skrifter, pages 49–68, Copenhagen, 2002. Roskilde Universitetscenter. Denys Duchier and Joachim Niehren. Dominance constraints with set operators. In Proceedings of the First International Conference on Computational Logic (CL2000), volume 1861 of Lecture Notes in Computer Science, pages 326–341. Springer, July 2000.

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27 / 25 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 Kuboˇn. 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 159–174. 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 189–200, Växjö, Sweden, 2003.

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