A Proof System for Tree Adjoining Grammars

Many TAG-based systems employ a particular tree adjoining grammar to generate the intended structures of the set of sentences they aim to describe. However, in most cases, the underlying set of elementary trees is more or less hand-made or maybe derived from a given tree data-base. We present a formal framework that allow to specify tree adjoining grammars by logical formulae. Based on this formalism we can check whether a given specification is TAG-consistent or whether a given TAG meets some particular properties. In addition, we sketch a method that generates a TAG from a given logical specification. As formal foundation, we employ a particular version of modal hybrid logic to specify the properties of T/D-trees. Such trees structurally combine a derived TAG-tree T and its associated derivation tree D. Finally, we sketch a labeled tableau calculus that constructs a set of tree automata representing the elementary trees of the specified TAG and a special tree automaton for the corresponding derivation trees. In literature, we find some approaches specifying TAGs, or more generally, mildly context-sensitive grammar formalisms, that gradually vary in their underlying framework. Commonly, either starts with a logical description of recognizable sets of trees (Thatcher and Wright, 1968). However, they differ in their method of leaving the context-free paradigm. The approach mentioned in (Morawietz and Mönnich, 2001) and (Michaelis, Mönnich and Morawietz, 2000) uses a ‘lifting’ function that encodes a TAG into a regular tree grammar. In (Rogers, 1999) (and related works) we find a logical description of TAGs that is based on a 3-dimensional view of trees. The important issue of this approach is to combine the derived TAG-tree and its derivation tree to a single 3-dimensional structure. Similarly, we also consider the derived TAG-tree and its derivation tree employ so-called T/D-trees. However we only associate the nodes of the derived tree with the corresponding node in the derivation tree. Consequently, all nodes of the same instance of an elementary tree refer to the same corresponding node in the derived tree. Therefore, we can specify structural properties of the derived TAG-tree and of the derivation tree at the same time. Using the links to the derivation tree, we can identify nodes in the TAG tree that belong to the same instance of some elementary tree. In contrast to the other approaches mentioned above which encode the TAG-tree into other kind of structures, we keep the original derived TAG tree as a structural unit. Consequently, we can directly access the nodes and the structural properties of the TAG tree without employing a particular projection function or any other special coding issues. In essence, our formalism employs modal hybrid logic that combines the simplicity of modal logic and the expressivity of classical logic. The use of so-called nominals in hybrid logic offer explicit references to certain tree nodes which is (directly) possible in modal approaches. We introduce the hybrid language HLTAG that specifies properties of the combined structure of derived TAG-trees and their derivation trees. Using this language we specify a number of TAG axioms which establish a notion of TAG-consistency. Further, we briefly illustrate a formalism that constructs a number of tree automata representing the underlying TAG for a given TAG-consistent HLTAG formula.