Two iteration theorems for the LL(k) languages

AbstractThe structure of derivation trees over an LL(k) grammar is explored and a property of these trees obtained which is shown to characterize the LL(k) grammars. This characterization, called the LL(k) Left Part Theorem, makes it possible to establish a pair of iteration theorems for the LL(k) languages. These theorems provide a general and powerful method of showing that a language is not LL(k) when that is the case. They thus provide for the first time a flexible tool with which to explore the structure of the LL(k) languages and with which to discriminate between the LL(k) and LR(k) language classes.Examples are given of LR(k) languages which, for various reasons, fail to be LL(k). Easy and rigorous proofs to this effect are given using our LL(k) iteration theorems. In particular, it is proven that the dangling-ELSE construct allowed in PL/I and Pascal cannot be generated by any LL(k) grammar. We also give a new and straightforward proof based on the LL(k) Left Part Theorem that every LL(k) grammar is LR(k)