Chapter 7 Some Observations on a “ Graphical ” Model-Theoretical Approach and Generative Models

Algebraic and logic based model-theoretic approaches to syntax have been extensively studied, revealing interesting and linguistically relevant properties of natural language formalisms. Büchi (1960) and Elgot (1961), for example, have examined the connections between monadic second order (MSO) logic and regular string and tree languages, and Doner (1970) and Thatcher and Wright (1968) have examined the relationship between MSO and regular tree languages (recognizable sets). Morawietz and Mönnich (2001) have provided a model-theoretic account of Tree-Adjoining Grammars (TAG). Rogers (2003) has investigated a model-theoretic characterization of TAGs in terms of MSO on three dimensional trees. Though related to word-word dependencies, the formalisms mentioned above are primarily phrase structure based. Dependency structures, which rely directly on word-word dependencies, have not been studied modeltheoretically until very recently. Bodirsky et al. (2005), Kuhlmann and Möhl (2006), and Kuhlmann and Nivre (2006) initiated a research program based on the properties of the dependency structures rather than in terms of some grammar formalism that generates such structures. Because of this emphasis on the graphs themselves independent of a derivational system, we refer to their approach as a “graphical” model-theoretic approach to syntactic structure. The formal notation used in Kuhlmann and Möhl’s (2007) recent work on characterizing dependency languages further facilitates viewing each of their dependency structures as a model of a set of well-formedness conditions. On the empirical front, Kuhlmann and Nivre (2006) investigate the adequacy of a number of formal properties of these graphical models for characterizing the data in two large scale dependency tree banks, the Prague Dependency Treebank (Hajič et al., 2001) and Danish Dependency Treebank (Kromann, 2003). Their results identify three properties that successfully describe more than 99% of the structures in these treebanks. Bodirsky et al. (2005) further show that two of these three properties ( well-nestedness and a gap degree≤ 1) are exactly the right properties for characterizing graphical models that correspond to derivation s in Lexicalized Tree Adjoining Grammar (LTAG). 1 That is, on the theoretical front, their work can be seen as a graphical model-based perspective of the generative approach taken by TAG. This paper reports some further explorations of the graphical model-theoretic account mentioned above to multi-component extensions of TAG (MC-TAG) and also comments on how the graphical model-theoretic approach and the TAG approach inform each other both when they converge and diverge. In Section 7.2, we provide a brief introduction to TAG,graph drawings(the graph models of interest), and the work of Bodirsky et al. (2005) and Möhl (2006) that relates the two. We also touch on how the convergence of the two approaches suggests that TAG provides an answer to the question of what the source of a certain type of discontinuity (non-projectivity) is. In Section 7.3, we review examples from Kuhlman (p.c.) and Kuhlmann and Möhl (2006) that show that their graph model is not extendible to tree-local MC-TAG. Because MC-TAGs allow only additional derivations, not additional derived phrase struc tures or strings (that is, they are weakly equivalent to TAGs), this divergence is somewhat surprising. Based on the success of the graph-model approach with respect to characterizing the Czech and Dutch structures in the previously mentioned treebanks, one is tempted to conclude that multi-component extensions are unnecessarily expressive. However, multi-component extensions have proven useful for characterizing a wide range of linguistic phenomena such as extraposition, anaphoric binding, quantifier scope ambiguities, that are difficult for basic TAG. In Section 7.4, we consider MC-TAGs in use. In particular, we examine the cases where MC-TAG extensions have been argued to be necessary for linguistically satisfying accounts and find that these MC-TAGs do in fact have graph models that satisfy the constraints identified by Bodirsky et al. (2005) due to certain properties