James Rogers

Capturing Linguistic Theories Model-Theoretically

Arbeitspapiere des SFB 340, Bericht Nr. 72 (1996), 26pp.
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Over the last ten or fifteen years there has been a shift, in generative linguistics, away from formalisms based on a procedural interpretation of grammars towards constraint-based formalisms---formalisms that define languages by specifying a set of constraints that characterize the set of well-formed structures analyzing the strings in the language. A natural extension of this trend is to define this set of structures model-theoretically---to define it as the set of mathematical structures that satisfy some set of logical axioms. In this paper we sketch L2KP, a monadic second-order framework for axiomatizing theories of syntax. We look, in particular at two components of Generalized Phrase-Structure Grammar (GPSG)---Feature Specification Defaults (FSDs) and the Exhaustive Constant Partial Ordering property (ECPO)---that illustrate the strengths of L2KP as a framework.

FSDs have been cited as a component of GPSG which cannot be captured declaratively. We show that the apparent dynamic aspect of these defaults is a consequence of the difficulty of capturing them within the formal framework in which GPSG was originally defined. The monadic second-order quantification of L2KP allows FSDs to be defined in a much more direct way, clarifying their nature significantly.

ECPO, despite being a fundamental principle of GPSG, has been ignored by existing model-theoretic treatments of GPSG. We argue that this is a consequence of the fact that these treatments focus on axiomatizing the set of trees licensed by a given GPSG grammar, while ECPO, being a universal principle of natural language, is a characteristic of the class of sets of trees that analyze natural languages. That is, it is not a property of trees, but is, rather, a property of sets of trees. L2KP, again principally because it supports second-order quantification, allows us to capture, in a limited way, universal principles like ECPO.

Finally, we note that, while L2KP is powerful enough to capture principles like FSDs and ECPO in a natural way, its formal expressive power is well-defined. The class of finite trees that are definable in L2KP are a very slight generalization of the class of sets of trees definable in the phrase-structure formalism underlying GPSG.