Tape Bounds for Some Subclasses of Deterministic Context-Free Languages

There are several interesting observations to be made concerning the tape complexity of context-free languages. An early result given by Lewis et al. (1965) is that every context-free language can be recognized by an off-line deterministic Turing machine of O((log n) 2) tape complexity. This is still the best result known. Sudborough (1975) shows that if all linear context-free languages can be recognized by off-line deterministic Turing machines of O(log n) tape complexity, then the nondeterministic and deterministic context-sensitive languages are identical. He also discusses a deterministic context-free language (abbreviated DCFL) which is log n tape complete for the family of DCFL's (Sudborough, 1976@ Some closure properties on the class of O(log n) tape complexity languages (Ritchie and Springsteel, 1972) and on the class of O(log n) tape complexity functions (Lind, 1974) are known. It is also known that the class of O(log n) tape complexity context-free languages is closed under the star operation if and only if the deterministic and nondeterministic O(log n) tape complexity classes are identical (Flajolet and Steyaert, 1974; Monien, 1975). These results focus attention on the class of O(log n) tape complexity languages. In particular, it is natural to ask whether large subclasses of the deterministic context-free languages are recognizable by off-line deterministic Turing machines of O(log n) tape complexity. The class of languages recognizable by deterministic one-counter automata (Valiant, 1973; Valiant and Paterson, 1975) is a trivial example of such a subclass. Ritchie and Springstael (1972) show that Dyck languages, standard languages, structured context-free languages, and bounded context-free languages are recognizable by deterministic two-way marking automata. Hence they are all in the class of deterministic O(log n) tape complexity languages (Ritchie and Springstael, 1972; Hartmanis, 1972). It is also known that any parenthesis language (Lynch, 1975; Mehlhorn, 1975), any two-sided Dyck