By Marco Kuhlmann

Since 2002, FoLLI has provided an annual prize for striking dissertations within the fields of good judgment, Language and data. This e-book is predicated at the PhD thesis of Marco Kuhlmann, joint winner of the E.W. Beth dissertation award in 2008. Kuhlmann’s thesis lays new theoretical foundations for the research of non-projective dependency grammars. those grammars have gotten more and more vital for methods to statistical parsing in computational linguistics that care for loose observe order and long-distance dependencies. the writer presents new formal instruments to outline and comprehend dependency grammars, offers new dependency language hierarchies with polynomial parsing algorithms, establishes the sensible importance of those hierarchies via corpus reviews, and hyperlinks his paintings to the phrase-structure grammar culture via an equivalence end result with tree-adjoining grammars. The paintings bridges the gaps among linguistics and theoretical machine technology, among theoretical and empirical techniques in computational linguistics, and among formerly disconnected strands of formal language research.

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**Example text**

Congruences on the same chain can be compared with respect to the coarseness of their quotients: given a set S and two partitions Π1 , Π2 of S, we say that Π1 is coarser than Π2 (and that Π2 is ﬁner than Π1 ), if for every class C2 ∈ Π2 , there is a class C1 ∈ Π1 such that C2 ⊆ C1 . The set of convex partitions of a given set together with the ‘coarser-than’ relation forms a complete lattice. As a consequence, there is a coarsest congruence on a given set. 2 (continued). 1a is the coarsest congruence on the set S1 in C6 (all other congruence relations on S1 have more equivalence classes), the topmost relation is the ﬁnest congruence on S1 .

Note that, by this deﬁnition, the set Ω(k) is exactly what we need in order to encode the set of all block-ordered trees with up to k lists per node, and therefore, the set of all segmented dependency structures with block-degree at most k. Speciﬁcally, the set Ω(1) is essentially identical to our previous deﬁnition of Ω for projective dependency structures. 3. 2 Algebraic Framework 43 0121 01, 1 0 1, 0 0 Fig. 6. 4 for node 1: for node 2: L = 12453 , L = 12453 , L = 12252 , L = ✷ 23 ✷ 3 , order [1] = 1252 ; order [2] = 23 ✷ 3 ; for node 3: for node 4: L = 12453 , L = 12453 , L = ✷✷ 4 ✷ 3 , L = ✷✷ 4 ✷✷ , order [3] = 4 ✷ 3 ; order [4] = 4 ; for node 5: L = 12453 , L = ✷✷✷ 5 ✷ , order [5] = 5 .

Let D = (V ; , ) be a dependency structure, and let ≡ be a congruence relation on D. The segmentation of D by ≡ is the structure D := (V ; , , R), where R is a new ternary relation on V deﬁned as follows: (u, v1 , v2 ) ∈ R :⇐⇒ v1 ≡ v2 ∧ ∀w ∈ [v1 , v2 ]. w ∈ u . The elements of the set V /≡ are called the segments of D . ✷ We write v1 ≡u v2 instead of (u, v1 , v2 ) ∈ R. 2. 5 shows how we visualize segmented dependency structures: we use boxes to group nodes that belong to the same segment; all other congruences ≡u are uniquely determined by this choice.