R70-3 On Stochastic Languages

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 contextfree 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 n3 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 n5 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 n4 operations. Again, just like for the n3 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 n2 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. These results add to our understanding of the computational complexity aspects of context-free languages, and further, they draw our attention to the difficult problem of establishing least-upper time bounds for the recognition of context-free languages by multitape as well as by one-tape Turing machines. J. HARTMANIS Dept. of Computer Science Cornell University Ithaca, N. Y. 14850