The universal generation problem for LFG grammars is the problem of determining whether a given grammar derives any terminal string with a given f-structure. It is known that this problem is decidable for acyclic f-structures. In this brief note, we show that for those f-structures the problem is nonetheless intractable. This holds even for grammars that are off-line parsable.

The formalism for Lexical-Functional Grammar (LFG) was introduced in the 1980s as one of the first constraint-based grammatical formalisms for natural language. It has led to substantial contributions to the linguistic literature and to the construction of large-scale descriptions of particular languages. Investigations of its mathematical properties have shown that, without further restrictions, the recognition, emptiness, and generation problems are undecidable, and that they are intractable in the worst case even with commonly applied restrictions. However, grammars of real languages appear not to invoke the full expressive power of the formalism, as indicated by the fact that algorithms and implementations for recognition and generation have been developed that run—even for broad-coverage grammars—in typically polynomial time. This article formalizes some restrictions on the notation and its interpretation that are compatible with conventions and principles that have been implicit or informally stated in linguistic theory. We show that LFG grammars that respect these restrictions, while still suitable for the description of natural languages, are equivalent to linear context-free rewriting systems and allow for tractable computation.

This article describes an approach to Lexical-Functional Grammar (LFG) generation that is based on the fact that the set of strings that an LFG grammar relates to a particular acyclic f-structure is a context-free language. We present an algorithm that produces for an arbitrary LFG grammar and an arbitrary acyclic input f-structure a context-free grammar describing exactly the set of strings that the given LFG grammar associates with that f-structure. The individual sentences are then available through a standard context-free generator operating on that grammar. The context-free grammar is constructed by specializing the context-free backbone of the LFG grammar for the given f-structure and serves as a compact representation of all generation results that the LFG grammar assigns to the input. This approach extends to other grammatical formalisms with explicit context-free backbones, such as PATR, and also to formalisms that permit a context-free skeleton to be extracted from richer specifications. It provides a general mathematical framework for understanding and improving the operation of a family of chart-based generation algorithms.