Computational Pitfalls in Tractable Grammatical Formalisms
Marten Trautwein
Abstract:
In this dissertation we apply complexity theory to various grammatical
formalisms. We presuppose that the reader has basic knowledge of the formal
aspects of linguistics and the formal language theory. Familiarity with the
rudiments of complexity theory is required to fully understand the material
that that we present.
The merit of applying complexity theory to grammatical formalisms is that
the complexity analyses provide different kinds of information. The
complexity analyses do not only show whether a problem is difficult, but
also why it is difficult and possibly even how to restrict the problem in
order to make it less difficult. The effect of the complexity analyses are
twofold. On the one hand, the analyses provide statements concerning the
development of the theory of the grammatical formalisms. On the other hand,
the analyses express general statements concerning implementations of these
grammatical formalisms. In this dissertation the main emphasis lies on the
development of theories.
Chapter 1 motivates our approach to the complexity analyses of grammatical
formalisms. This approach is based on the descriptions of grammatical
formalisms as they appear in actual practice.
Chapter 2 studies "restricted attribute-value grammars" (R-AVGs). We
introduce a liberal variation on the well-known "off-line parsability
constraint," the "honest parsability constraint." We prove that the R-AVGs
that satisfy this liberal constraint generate precisely all sets in the
complexity class NP. This provides a direct link between weak generative
capacity and complexity.
In Chapter 3 we introduce "Categorial Unification Grammar" (CUG) and
analyze its complexity. We prove that the recognition problem of CUG is
NP-complete. By means of this complexity result we settle the weak
generative capacity of CUG. We show that "Functional Unification Grammar"
(FUG), "Head-driven Phrase Structure Grammar" (HPSG), and "Lexical
Functional Grammar" (LFG) can simulate CUG (Chapters 4,5, and 6).
Furthermore, the recognition problems of these grammatical formalisms are
shown to be NP-complete. The weak generative capacity of these grammatical
formalisms follows from these simulations and complexity results.
Chapter 7 handles "Functional Grammar" (FG). We introduce FG by means of
many examples that clarify the relevant parts. Four processes are
distinguished within the generation process of FG: the "fund formation
process," the "clause structure formation process," the "form specification
process," and the "order specification process." These four processes are
considered one afterthe other. We formulate each process as a decision
problem, and determine its complexity.
Finally, Chapter 8 contains our overall conclusions and suggests directions
for further research.