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**A time-space tradeoff for language recognition.**
*(English)*
Zbl 0533.68047

This paper presents a language L and proves that if L is recognized in time T(n) and space S(n), then it must be \(T^ 2(n)S(n)>cn^ 3\) for some constant c. The underlying nondeterministic machine model allows an arbitrary (but constant) number k of heads on the (read-only) input tape. The storage can be organized arbitrarily, even random access is allowed. The multiple access to the input is an improvement over the previously known lower bounds of \(\Omega(n^ 2)\) for \(S\cdot T\) for the recognition of certain languages, because there only one head was allowed. The proof of the lower bound is based on a counting argument on the number of different configurations for a given time and space. From the main result it follows as a corollary that the class of languages recognizable by 2- way multihead deterministic finite automata is properly contained in the corresponding nondeterministic class, if T(n) grows more slowly than \((n^ 3/\log n)^{1/2}\) for infinitely many n.

Reviewer: H.Alt

### MSC:

68Q25 | Analysis of algorithms and problem complexity |

68Q45 | Formal languages and automata |

68Q05 | Models of computation (Turing machines, etc.) (MSC2010) |

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\textit{P. Dúriś} and \textit{Z. Galil}, Math. Syst. Theory 17, 3--12 (1984; Zbl 0533.68047)

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### References:

[1] | A. Cobham, The recognition problem for perfect squares,Proceedings 7th IEEE Symposium on SWAT, Berkeley, Calif., October 1966, 78–87. |

[2] | A. B. Borodin, M. J. Fischer, D. G. Kirkpatrick, N. A. Lynch, and M. Tompa, A time-space tradeoff for sorting on non-oblivious machines.Proceedings 20th IEEE Symposium on FOCS, San Juan, Puerto Rico, October 1979, 319–327. · Zbl 0462.68011 |

[3] | A. B. Borodin and S. A. Cook, A time-space tradeoff for sorting on a general sequential model of computation,Proceedings 12th ACM, STOC., Los Angeles, Calif., April 1980, 294–301. |

[4] | Z. Galil and J. Seiferas, Time-space-optimal string matching,J. Computer Sys. Sciences 26 (1983), 280–294. · Zbl 0509.68101 · doi:10.1016/0022-0000(83)90002-8 |

[5] | R. L. Rivest and A. C. C. Yao,k+1 heads are better thank, JACM 25, 2(1978), 337–340. · Zbl 0372.68011 · doi:10.1145/322063.322065 |

[6] | L. Janiga, Real-time computations of two-way multihead finite automata, inFundamentals of Computation Theory (FCT 79) (L. Budach ed.), Akademic Verlag, Berlin, 214–219. |

[7] | P. Dúriś and Z. Galil, Fooling a two way automaton or One pushdown store is better than one counter for two way machines,Theoretical Computer Science 21 (1982), 39–53. · Zbl 0486.68084 · doi:10.1016/0304-3975(82)90087-1 |

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