Propagating chain-free normal forms for EOL systems

We establish two types of normal forms for EOL systems. We first show that each ε-free EOL language can be generated by a propagating EOL system in which each derivation tree is chain-free. By this we mean that it contains at least one path from the root to the grandfather of a leaf in which each node has more than one son. We use this result to prove that each ε-free EOL language can be generated by a propagating EOL system in which each production has a right side of length at most two and which does not contain nonterminal chainproductions, i.e., productions A → B for nonterminals A and B. As applications of our results we give a simple proof for the decidability of the finiteness problem for EOL systems and solve an open problem concerning completeness of EOL forms