Strong optimal lower bounds for Turing machines that accept nonregular languages
In this paper, simultaneous lower bounds on space and input head reversals for deterministic, nondeterministic and alternating Turing machines accepting nonregular languages are studied.
Three notions of space complexity, namely strong, middle, and weak, are considered; moreover, another notion called accept, is introduced. For all cases we obtain tight lower bounds. In particular, we prove that while in the deterministic and nondeterministic case these bounds are “strongly” optimal—in the sense that we exhibit a nonregular language over a unary alphabet exactly fitting them—in the alternating case optimal lower bounds for tally languages turn out to be higher than those for arbitrary languages.
Unable to display preview. Download preview PDF.
- [Alb85]M. Alberts. Space complexity of alternating Turing machines. In Fundamentals of Computation Theory, Proceedings, Lecture Notes in Computer Science 199, pages 1–7. Springer Verlag, 1985.Google Scholar
- [AM75]H. Alt and K. Mehlhorn. A language over a one symbol alphabet requiring only O(log log n) space. SIGAGT news, 7:31–33, 1975.Google Scholar
- [BDG87]J. Balcázar, J. Díaz, and J. Gabarró. Structural Complexity I. EATCS Monographs on Theoretical Computer Science 11. Springer Verlag, 1987.Google Scholar
- [BMP94a]A. Bertoni, C. Mereghetti, and G. Pighizzini. On languages accepted with simultaneous complexity bounds and their ranking problem. In Mathematical Foundations of Computer Science 1994, Proceedings, Lecture Notes in Computer Science 841, pages 245–255. Springer Verlag, 1994.Google Scholar
- [BMP94b]A. Bertoni, C. Mereghetti, and G. Pighizzini. An optimal lower bound for nonregular languages. Information Processing Letters, 50:289–292, 1994. Corrigendum. ibid., 52:339, 1994.Google Scholar
- [DH94]C. Damm and M. Holzer. Inductive counting below LOGSPACE. In Mathematical Foundations of Computer Science 1994, Proceedings, Lecture Notes in Computer Science 841, pages 276–285. Springer Verlag, 1994.Google Scholar
- [Fre79]R. Freivalds. On time complexity of deterministic and nondeterministic Turing machines. Latvijskij Matematiceskij Eshegodnik, 23:158–165, 1979. (In Russian).Google Scholar
- [Gef91]V. Geffert. Nondeterministic computations in sublogarithmic space and space constructibility. SIAM J. Computing, 20:484–498, 1991.Google Scholar
- [Gef93]V. Geffert. Tally version of the Savitch and Immerman-Szelepcsényi theorems for sublogarithmic space. SIAM J. Computing, 22:102–113, 1993.Google Scholar
- [HU69]J. Hopcroft and J. Ullman. Some results on tape-bounded Turing machines. Journal of the ACM, 16:168–177, 1969.Google Scholar
- [HU79]J. Hopcroft and J. Ullman. Introduction to automata theory, languages, and computations. Addison-Wesley, Reading, MA, 1979.Google Scholar
- [Iwa93]K. Iwama. ASPACE(o(log log n)) is regular. SIAM J. Computing, 22:136–146, 1993.Google Scholar
- [LR94]M. LiŚekiewicz and R. Reischuk. The complexity world below logarithmic space. In Structure in Complexity Theory, Proceedings, pages 64–78, 1994.Google Scholar
- [SHL65]R. Stearns, J. Hartmanis, and P. Lewis. Hierarchies of memory limited computations. In IEEE Conf. Record on Switching Circuit Theory and Logical Design, pages 179–190, 1965.Google Scholar
- [Sip80]M. Sipser. Halting space-bounded computations. Theoretical Computer Science, 10:335–338, 1980.Google Scholar
- [Sud80]I. Sudborough. Efficient algorithms for path system problems and applications to alternating and time-space complexity classes. In Proc. 21st IEEE Symposium on Foundations of Computer Science, pages 62–73, 1980.Google Scholar
- [Sze88]A. Szepietowski. Remarks on languages acceptable in log log n space. Information Processing Letters, 27:201–203, 1988.Google Scholar
- [Sze94]A. Szepietowski. Turing Machines with Sublogarithmic Space. Lecture Notes in Computer Science 843. Springer Verlag, 1994.Google Scholar