Restricted Turing machines and language recognition

In 1965 Hennie proved that one-tape deterministic Turing machines working in linear time are equivalent to finite automata, namely they characterize regular languages. This result has been improved in different directions, by obtaining optimal lower bounds for the time that one-tape deterministic and nondeterministic Turing machines need to recognize nonregular languages. On the other hand, in 1964 Kuroda showed that one-tape Turing machines that are not allowed to use any extra space, besides the part of the tape which initially contains the input, namely linear bounded automata, recognize exactly contextsensitive languages. In 1967 Hibbard proved that for each integer d ≥ 2, one-tape Turing machines that are allowed to rewrite each tape cell only in the first d visits are equivalent to pushdown automata. This gives a characterization of the class of context-free languages in terms of restricted Turing machines. We discuss these and other related models, by presenting an overview of some fundamental results related to them. Descriptional complexity aspects are also considered.