Left-Fitting Translations

One rather natural method for defining translations is by specifying a pair of grammars generating the translation. If for each leftmost derivation d in the input grammar generating an input word x the "corresponding" derivation d' in the output grammar generates an output word x', we call this grammar pair left-fitting. This concept is motivated by the usual parsing algorithms yielding leftmost derivations and by the fact that the left-fitting translations are more powerful than the syntax-directed translations. It is shown that it is decidable whether or not a given pair of context-free grammars is left-fitting or not essentially using the fact that the set of left derivation trees of a context-free grammar is semilinear. By means of certain structure properties of left-fitting translations, it is shown that they form a proper hierarchy in their so called intercalation number.