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.
![\enumsentence{
$
\begin{array}[t]{l}
\mbox{\it lex}(\con{willen},\var{Sign}) \; ...
...\att{val } \var{X}\\
\att{dir $/$} \\
\att{arg } \var{Y}
}).
\end{array}$}](img9.png)
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.