On ϵ productions for terminals in E0L forms

When working with E0L forms, equality of languages and of language families has traditionally been considered modulo the empty word ? to avoid trivial but cumbersome distinctions of cases. Despite this fact, the role of ?-productions is often significant in L form theory particular, it has previously been shown that vomplete forms must have ?-productions We strengthen this result by proving that vomplete forms must even have ?-productions for nonterminals. Based on this theorem we establish that there are E0L forms with the property that any form equivalent E0L form must generate the empty word. Thus, despite the fact that we define equality of languages modulo ?, we are still unable to ignore the presence of the empty word in languages when studying L forms. The proof of our main theorem is based on a number of lemmas which allow to show that certain E0L forms are not form equivalent. In view of the difficulty of proving or disproving form equivalence of arbitrary E0L forms (conjectured to be undecidable) these lemmas may well be of independent interest.