Equivalence (formal languages)

In formal language theory, weak equivalence of two grammars means they generate the same set of strings, i.e. that the formal language they generate is the same. In compiler theory the notion is distinguished from strong (or structural) equivalence, which additionally means that the two parse trees[clarification needed] are reasonably similar in that the same semantic interpretation can be assigned to both.[1]

Vijay-Shanker and Weir (1994)[2] demonstrates that Linear Indexed Grammars, Combinatory Categorial Grammars, Tree-adjoining Grammars, and Head Grammars are weakly equivalent formalisms,[clarification needed] in that they all define the same string languages.

On the other hand, if two grammars generate the same set of derivation trees (or more generally, the same set of abstract syntactic objects), then the two grammars are strongly equivalent. Chomsky (1963)[3] introduces the notion of strong equivalence, and argues that only strong equivalence is relevant when comparing grammar formalisms. Kornai and Pullum (1990)[4] and Miller (1994)[5] offer a refined notion of strong equivalence that allows for isomorphic relationships between the syntactic analyses given by different formalisms. Yoshinaga, Miyao, and Tsujii (2002)[6] offers a proof that for any LTAG formalism, there is a strongly equivalent HPSG one.

  1. ^ Stefano Crespi Reghizzi (2009). Formal Languages and Compilation. Springer. p. 57. ISBN 978-1-84882-049-4.
  2. ^ Vijay-Shanker, K. and Weir, David J. (1994). "The Equivalence of Four Extensions of Context-Free Grammars". Mathematical Systems Theory. 27 (6): 511–546. doi:10.1007/BF01191624. S2CID 12336597.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  3. ^ Cite error: The named reference Chomsky.1963 was invoked but never defined (see the help page).
  4. ^ Kornai, A. and Pullum, G. K. 1990. The X-bar Theory of Phrase Structure. Language, 66:24-50.
  5. ^ Miller, Philip H. 1999. Strong Generative Capacity. CSLI publications.
  6. ^ "Yoshinaga, N., Miyao Y., and Tsujii, J. 2002. A formal proof of strong equivalence for a grammar conversion from LTAG to HPSG-style. In the Proceedings of the TAG+6 Workshop:187-192. Venice, Italy" (PDF). Archived from the original (PDF) on 2011-08-28. Retrieved 2012-02-05.