Most of the machinery employed in categorial grammars can be
translated into unification-based grammar formalisms without problem.
Complex categories, for instance, can be represented by feature
structures in which the attributes VAL, DIR, and ARG are
assigned values representing the value, directionality, and argument
of the complex category, respectively. Categorial rules, such as
application, can be encoded as a rule rewriting a *value* as a
*functor* and *argument*, with feature constraints expressing
the categorial interpretation of this rule.

The variable notation used by Hoeksema, however, cannot be translated
so easily, as in general there is no feature structure subsuming all
possible instantiations of a $-category. Therefore, we have chosen
for a different approach. Instead of assigning polymorphic types to
VR and PE verbs directly, we stipulate that these verbs
are assigned all categories that can be derived from some initial
category by means of applying *rightward disharmonic division*
zero or more times:

The initial category of *willen* is VP/VP, for instance,
and thus it is assigned the categories VP/VP, (NP\VP)/(NP\VP), etc. Note that
rightward disharmonic division associates *willen* with a set of
categories that is identical to all possible instantiations of the
polymorphic categories proposed by Hoeksema.

The reason for pointing out this equivalence is that it suggests a
method for incorporating polymorphism into unification-based
frameworks. The various categories of a VR verb can be defined
using a recursive, relational, constraint *division*. An example
of a lexical entry, as well as a first approximation of the definition
of *division* is presented in (9). Note that we use
definite clauses to implement lexical entries as well as constraints.
Symbols starting with a capital represent variables, matrices
represent feature structures, and VP represents the feature
structure encoding of the corresponding linguistic category.

The *division*-relation holds if its second argument can be derived by
applying the categorial rule (rightward disharmonic) division to the first
argument an arbitrary number of times.

A consequence of defining lexical entries using recursive constraints is that certain entries may be infinitely ambiguous. Carpenter [3] has shown that such grammars in general are not decidable. In [1] we argue that processing with the particular kind of grammar used here is possible if one interleaves the evaluation of recursive lexical constraints with the derivation of syntactic structure.

1998-09-29