Robert E. Wall

Parsing with Categorial Grammar in Predictive Normal Form

Steedman (1985, 1987), Dowty (1987), Moortgat (1988), Morrill (1988), and others have proposed that Categorial Grammar, a theory of syntax in which grammatical categories are viewed as functional types, be generalized in order to analyze “noncanonical” natural language constructions such as whextraction and nonconstituent conjunction. A consequence of these augmentations is an explosion of semantically equivalent derivations admitted by the grammar, a problem we have characterized as spurious ambiguity from the parsing perspective (Wittenburg, 1986). In Wittenburg (1987), it was suggested that the offending rules of these grammars could take an alternate predictive form that would eliminate the problem of spurious ambiguity. This approach, consisting of compiling grammars into forms more suitable for parsing, is within the tradition of discovering normal forms for phrase structure grammars, and thus our title. Our approach stands in contrast to those which are attempting to address the spurious ambiguity problem in Categorial Grammars through the parsing algorithm itself rather than through the grammar (see Gardent & Bes, 1989; Pareschi & Steedman, 1987) and also to those addressing the problem by proof-theoretic means in the Lambek calculus tradition (Bouma, 1989; Hepple & Morrill, 1989; Koenig, 1989; Lambek, 1958; Moortgat, 1986, 1988). We follow the line of Steedman (1985, 1987), Dowty (1987), and various strains of Categorial Unification Grammar (Karttunen, 1986; Uszkoreit, 1986; Wittenburg, 1986; Zeevat, Klein & Calder, 1987) in that we assume a finite number of combinatory rules and study the behavior of parsers that apply these rewrite rules in roughly the phrase-structure parsing tradition.

Basic Concepts of Algebra

An algebra A is a set A together with one or more operations f i . We may represent an algebra by writing

Pushdown Automata, Context Free Grammars and Languages

{\rm{ }}A = \left\langle {A,{\rm{ }}{f_1},{\rm{ }}{f_2}{\rm{ }}...{\rm{, }}{f_n}} \right\rangle

Relations and Functions

(9-1)

Languages Between Context Free and Context Sensitive

A set is an abstract collection of distinct objects which are called the members or elements of that set. Objects of quite different nature can be members of a set, e.g. the set of red objects may contain cars, blood-cells, or painted representations. Members of a set may be concrete, like cars, blood-cells or physical sounds, or they may be abstractions of some sort, like the number two, or the English phoneme /p/, or a sentence of Chinese. In fact, we may arbitrarily collect objects into a set even though they share no property other than being a member of that set. The subject matter of set theory and hence of Part A of this book is what can be said about such sets disregarding the actual nature of their members.

Formal Systems, Axiomatization, and Model Theory

We turn next to a class of automata which are more powerful than the finite automata in the sense that they accept a larger class of languages. These are the pushdown automata (pda’s).

Boolean and Heyting Algebras

We have seen that a pushdown automaton can carry out computations which are beyond the capability of a finite automaton, which is perhaps the simplest sort of machine able to accept an infinite set of strings. At the other end of the scale of computational power is the Turing machine (after the English mathematician A. M. Turing, who devised them), which can carry out any set of operations which could reasonably be called a computation.

Basic Concepts of Logic and Formal Systems

Recall that there is no order imposed on the members of a set. We can, however, use ordinary sets to define an ordered pair, written 〈a, b〉 for example, in which a is considered the first member and b is the second member of the pair. The definition is as follows: