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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 are reasonably similar in that the same semantic interpretation can be assigned to both.
Vijay-Shanker and Weir (1994)〔Vijay-Shanker, K. and Weir, David J. 1994. ''The Equivalence of Four Extensions of Context-Free Grammars''. Mathematical Systems Theory 27(6): 511-546.〕 demonstrates that Linear Indexed Grammars, Combinatory Categorial Grammars, Tree-adjoining Grammars, and Head Grammars are weakly equivalent formalisms, in that they all define the same string languages.
On the other hand, if the two grammars generate the same set of derivation trees (or more generally, the same set of abstract syntactic objects), then the two languages are strongly equivalent. Chomsky (1963)〔Chomsky, N. 1963. ''Formal properties of grammar''. In R. D. Luce, R. R. Bush and E. Galanter, editors, Handbook of Mathematical Psychology, volume II:323-418. John Wiley and Sons, Inc.〕 introduces the notion of strong equivalence, and argues that only strong equivalence is relevant when comparing grammar formalisms. Kornai and Pullum (1990)〔Kornai, A. and Pullum, G. K. 1990. ''The X-bar Theory of Phrase Structure''. Language, 66:24-50.〕 and Miller (1994)〔Miller, Philip H. 1999. ''Strong Generative Capacity''. CSLI publications.〕 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)〔(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. )〕 offers a proof of the strong equivalency of the LTAG and HPSG formalisms.
==References==
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