## James Rogers

## Capturing Linguistic Theories Model-Theoretically

**Arbeitspapiere des SFB 340, Bericht Nr. 72 (1996)**, 26pp.

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### Abstract

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.