R70-1 A Note on Computing Time for the Recognition of Context- Free Languages by a Single-Tape Turing Machine

Abstract

Although considerable progress has been made in the understanding of the more "algebraic" properties of context-free languages, the quantitative aspects of the recognition and parsing of context- free languages are still far from well understood. It was shown by Kasami and Younger that for every context-free language there exists a multitape Turing machine which recognizes this language and uses no more than n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> operations to process input words of length n. For one-tape Turing machines Hartmanis showed, using the Younger algorithm, that every context-free language can be recognized in n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5</sup> operations. This result has now been improved in the paper under review. It is shown, by a careful implementation of the Torii–Kasami–Ozaki algorithm on a one-tape Turing machine, that every context-free language can be recognized in n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> operations. Again, just like for the n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> operation multitape result, it is not known how good this time bound is. On the other hand, the authors have been able to show in this paper that for linear context-free languages n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> is the least-upper time bound for their recognition on one-tape Turing machines. This is done by showing that every linear language is n2-recognizable, and by recalling that there are linear languages whose recognition requires n2 operations on a one-tape machine.