On the Syntactic Structures of Unrestricted Grammars I. Generative Grammars and Phrase Structure Grammars

Formal definitions for the syntactic structures of unrestricted grammars are given. The traditional forms for grammar productions give rise to “generative grammars” with “derivation structures” (where productions have the form α → β), and “phrase structure grammars” with “phrase structures” (where productions have the form A → B/μ-ν), two distinct notions of grammar and syntactic structure which become indistinguishable in the context free case, where the structures are trees. Parallel theories are developed for both kinds of grammar and structure. We formalize the notion of structural equivalence for derivations, extended to unrestricted grammars, and we prove that two derivations are structurally equivalent if and only if they have the same syntactic structure. Structural equivalence is an equivalence relation over the derivations of a grammar, and we give a simpler proof of a theorem by Griffiths that each equivalence class contains a rightmost derivation. We also give a proof for the uniqueness of the rightmost derivation, following a study of some of the properties of syntactic structures. Next, we investigate the relationship between derivation structures and phrase structures and show that the two concepts are nonisomorphic. There is a natural correspondence between generative productions and phrase structure productions, and, by extension, between the two kinds of grammars and between their derivations. But we show that the correspondence does not necessarily preserve structural equivalence, in either direction. However, if the correspondence from the productions of a phrase structure grammar to the productions of a generative grammar is a bijection, then structural equivalence on the generative derivations refines the image under the correspondence of structural equivalence on the phrase structure derivations.