Abstract
Having presented two non-equivalent formalisms of generative grammar,1 each with its own fully developed hierarchy of language and complexity classes, we return in this chapter to the formal properties of natural language. Section 12.1 compares the complexity of the LA- and PS-grammatical language classes. Section 12.2 describes the inclusion relations between language classes in the PS- and the LA-hierarchy, respectively. Section 12.3 defines a context-free language which is a C3-language. Section 12.4 describes the orthogonal relation between the context-free languages and the classes of C-languages. Section 12.5 investigates ambiguity in natural language, and concludes that the natural languages are in the class of C1-LAGs, thus parsing in linear time.
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References
Namely PS-grammar in Chapters 8 and 9, and LA-grammar in Chapters 10 and 11.
Hardest Context-Free Language, S. Greibach 1973.
Hierarchy lemma, Hoperoft and Ullman 1979, p. 228.
See Hoperoft and Ullman 1979, p. 228, Theorem 9.8. The A-LAGs generate the recursive languages while the B-LAGs generate the context-sensitive languages, as shown in 11.1 and 11. 2.
Cf. Hoperoft and Ullman 1979, p. 324ff.
Assuming that the proper inclusion of C2 C C3 could be shown directly (for example, by means of a pumping lemma for C2-languages), then this would imply P C NP only if it can be shown that C2 = P, which is improbable.
Not only is a PSPACE-complete problem not likely to be in P, it is also not likely to be in NP. Hence the property whose existence is PSPACE-complete probably cannot even be verified in polynomial time using a polynomial length `guess.’
Because L.. is a deterministic context-free language, it can be parsed in linear time in PS-grammar. Cf. B. Stubert 1993, p. 71, Lemma 5. 1.
See also M. Harrison 1978, p. 219 ff., in the same vein.
A.K. Joshi et al. 1975.
Hoperoft and Ullman 1979, p. 389 f. T. Hayashi 1973 proved a pumping lemma for index languages.
First proposed in CoL, p. 219–232 and 239–247.
Syntactically, the prepositional phrases are categorized as (adv and pnm) using the multicat notation (see Section 15.2).
At least on the syntactic level.
The structural properties assigned by CoNSyx to natural language syntax may and should be tested empirically in the continuing analysis of various different languages. Possible counterexamples to 12.5.7 would be constructions of natural language with +recursive syntactic ambiguities which do not allow an alternative treatment based on semantic doubling. Given the mechanism of natural communication within the SLIM theory of language, it seems unlikely that such constructions will be found.
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© 2001 Springer-Verlag Berlin Heidelberg
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Hausser, R. (2001). LA- and PS-hierarchies in Comparison. In: Foundations of Computational Linguistics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04337-0_13
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DOI: https://doi.org/10.1007/978-3-662-04337-0_13
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