A lower bound on the oscilation complexity of context-free languages

The paper deals with the oscilation complexity OSC introduced in [W 79] for context-free languages. It is known that CF \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$CF \subseteq OSC\left( {log} \right)$$ \end{document} OSC(log), but it was an open question whether there exist context-free languages requiring logarithmic oscilation. We exhibit a language L ∈ CF for which a logarithmic lower bound on its oscilation complexity is established. As a consequence we can conclude that the Dyck language D2 requires logarithmic oscilation. Our proof method is a rather involved counting argument which exploits specific properties of L.