A Declarative Characterization of Different Types of Multicomponent Tree Adjoining Grammars

Multicomponent Tree Adjoining Grammars (MCTAG) is a formalism that has been shown to be useful for many natural language applications. The definition of MCTAG however is problematic since it refers to the process of the derivation itself: a simultaneity constraint must be respected concerning the way the members of the elementary tree sets are added. This way of characterizing MCTAG does not allow to abstract away from the concrete order of derivation. In this paper, we propose an alternative definition of MCTAG that characterizes the trees in the tree language of an MCTAG via the properties of the derivation trees (in the underlying TAG) the MCTAG licences. This definition gives a better understanding of the formalism, it allows a more systematic comparison of different types of MCTAG, and, furthermore, it can be exploited for parsing. 1 TAG and MCTAG Tree Adjoining Grammar (TAG, [Joshi and Schabes, 1997]) is a tree-rewriting formalism. A TAG consists of a finite set of trees (elementary trees). The nodes of these trees are labelled with nonterminals and terminals (terminals only label leaf nodes). Starting from the elementary trees, larger trees are derived by substitution (replacing a leaf with a new tree) and adjunction (replacing an internal node with a new tree). In case of an adjunction, the tree being adjoined has exactly one leaf that is marked as the foot node (marked with an asterisk). Such a tree is called an auxiliary tree. When adjoining it to a node n, in the resulting tree, the subtree with root n from the old tree is attached to the foot node of the auxiliary tree. Non-auxiliary elementary trees are called initial trees. A derivation starts with an initial tree. In a final derived tree, all leaves must have terminal labels. For a sample derivation see Fig. 1. Definition 1 (Tree Adjoining Grammar) A Tree Adjoining Grammar (TAG) is a tuple G = 〈I, A,N, T 〉 with – N and T being disjoint finite sets, the nonterminals and terminals – I being a finite set of initial trees with nonterminals N and terminals T , and – A being a finite set of auxiliary trees with nonterminals N and terminals T . NP John S NP VP V laughs VP ADV VP∗ always derived S tree: NP VP John ADV VP always V laughs derivation tree: laugh 1 2 john always Fig. 1. TAG derivation for John always laughs Definition 2 (TAG derivation and tree language) Let G = 〈I, A,N, T 〉 be a TAG. Let γ and γ be finite trees. – γ ⇒ γ in G iff there is a node position p and a γ 0 that is either elementary or derived from some elemenentary tree such that γ = γ[p, γ 0]. 1 ∗ ⇒ is the reflexive transitive closure of ⇒. – The tree language of G is LT (G) := {γ | there is an α ∈ I such that α ∗ ⇒ γ and all leaves in γ have terminal labels}. TAG derivations are represented by derivation trees that record the history of how the elementary trees are put together. A derived tree is the result of carrying out the substitutions and adjunctions, i.e., the derivation tree describes uniquely the derived tree. Each edge in a derivation tree stands for an adjunction or a substitution. The edges are labelled with Gorn addresses. E.g., the derivation tree in Fig. 1 indicates that the elementary tree for John is substituted for the node at address 1 and always is adjoined at node address 2. Definition 3 (TAG derivation tree) Let G = 〈I, A,N, T 〉 be a TAG. Let γ be a tree derived as follows in G: γ = γ0[p1, γ1] . . . [pk, γk] where γ0 is an instance of an elementary tree and the substitutions/adjunctions of the γ1, . . . , γk are all the substitutions/adjunctions to γ0 that are performed to derive γ. Then the corresponding derivation tree has a root labelled with γ0 that has k daugthers. The edges from γ0 to these daughters are labelled with p1, . . . , pk, and the daughters are the derivation trees for the derivations of γ1, . . . , γk. A TAG extension that is useful for linguistic applications is multicomponent TAG (MCTAG, [Weir, 1988]). An MCTAG contains sets of elementary trees. In each derivation step, one of the tree sets is chosen and its trees are added simultaneously. Depending on the nodes to which the trees from the set attach, different kinds of MCTAGs are distinguished: if the nodes are required to be part of the same elementary tree, the MCTAG is tree-local, if they are required to be part of the same tree set, the grammar is set-local and otherwise it is non-local. 1 For trees γ, γ1, . . . , γn and pairwise different node positions p1, . . . , pn in γ, γ[p1, γ1] . . . [pn, γn] denotes the result of subsequently substituting/adjoining the γ1, . . . , γn to the nodes in γ with addresses p1, . . . , pn respectively. 2 The root address is ǫ, and the jth child of a node with address p has address pj.