On the Transformation of LL(k)-linear Grammars to LL(1)-linear

It is proved that every LL(k)-linear grammar can be transformed to an equivalent LL(1)-linear grammar. The transformation incurs a blow-up in the number of nonterminal symbols by a factor of \(m^{2k-O(1)}\), where m is the size of the alphabet. A close lower bound is established: for certain LL(k)-linear grammars with n nonterminal symbols, every equivalent LL(1)-linear grammar must have at least \(n \cdot (m-1)^{2k-O(\log k)}\) nonterminal symbols.