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Additional info for Context-free grammars: Covers, normal forms, and parsing
3. COVERS AND NORMAL FORMS: AN INTRODUCTION In this monograph, from now on, unless stated otherwise, we assume that whenever we define a cover homomorphism g = <~,~> between grammars G = (N,Z,P,S) and G' = (N',I',P',S') then E = E' and ~ is the identity homomorphism. Rence we only consider homomorphism ~. Both in Gray and Harrison  and in Aho and Ullman  results and remarks are presented on the existence and nonexistence of certain covers of grammars with grammars in some normal form. Some of these remarks are not correct.
PROPERTIES OF COVERS It is useful to put forward a few general properties of grammar covers. The most frequently used property will be the transitivity of covers, that is, if grammar G 2 covers grammar G 1 and grammar G 3 covers grammar G2, then grammar G 3 covers G I. |. Proof. If G3[f/g]G 2 and G2[g/h]G 1 then G3[f/h]G I. Trivial. For future applications we generalize the idea of transitivity. 1. Let G and G' be CFG's such that G'[f/h]G for some parse relations fG' and h G and a cover homomorphism ~.
Therefore, if we have a proper NLR grammar G, we may first eliminate the single productions (cf. 1 to obtain a grammar G' in GNF such that a faithful and external full grammar functor H : S(G') ÷ S(G) can be defined. Notice that the condition mentioned before A l g o r i t h m 5 . can also be handled functorially. For example, for the given example the functor H should satisfy H(Ha) = a, H ( A ÷ ~Ha~) = A ÷ ~a~ and H(H a + a) = id a . This can be generalized in an obvious way, and clearly, such a functor is faithful and externally full.