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**\(\Sigma_ 2SPACE(n)\) is closed under complement.**
*(English)*
Zbl 0654.68053

A number of results on the collapsing of complexity hierarchies have been obtained in recent years. This paper is concerned with the \(\Sigma_ k\)SPACE\((n)\)-hierarchy of languages accepted by linear space bounded alternating Turing machines. It is shown that the hierarchy collapses to \(\Sigma_ 2\)SPACE\((n)\) by proving that \(\Sigma_ k\)SPACE\((n)\supseteq \Pi_ k\)SPACE\((n).\)

However, since this paper was published, an improvement to this result has been obtained by Immerman, who has proved that \(N\)SPACE\((n)=\)co-\(N\)SPACE\((n)\). Since \(\Sigma_ 1\)SPACE\((n)=N\)SPACE\((n)\) we see that the hierarchy actually collapses to \(\Sigma_ 1\)SPACE\((n)=\Pi_ 1\)SPACE\((n)\). Details of this improved result, which has quite a short proof, can be found in the article by J. Hartmanis [The collapsing hierarchies - the structural complexity column, Bull. EATCS 33, 26-39 (1987)].

Although the author proves a much weaker result, it is interesting to note the use of census functions, (which tell us how many strings are shorter than a given integer). This technique has been used to prove a number of interesting results in complexity theory recently.

However, since this paper was published, an improvement to this result has been obtained by Immerman, who has proved that \(N\)SPACE\((n)=\)co-\(N\)SPACE\((n)\). Since \(\Sigma_ 1\)SPACE\((n)=N\)SPACE\((n)\) we see that the hierarchy actually collapses to \(\Sigma_ 1\)SPACE\((n)=\Pi_ 1\)SPACE\((n)\). Details of this improved result, which has quite a short proof, can be found in the article by J. Hartmanis [The collapsing hierarchies - the structural complexity column, Bull. EATCS 33, 26-39 (1987)].

Although the author proves a much weaker result, it is interesting to note the use of census functions, (which tell us how many strings are shorter than a given integer). This technique has been used to prove a number of interesting results in complexity theory recently.

Reviewer: Ph.W.Grant

### MSC:

68Q25 | Analysis of algorithms and problem complexity |

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

03D15 | Complexity of computation (including implicit computational complexity) |

68Q45 | Formal languages and automata |

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DOI

### References:

[1] | Chandra, A. K.; Kozen, D. C.; Stockmeyer, L. J., Alternation, J. Assoc. Comput. Mach., 28, 114-133 (1981) · Zbl 0473.68043 |

[2] | Hopcroft, J. E.; Ullman, J. D., Introduction to Automata Theory, Languages, and Computation (1979), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0196.01701 |

[3] | Kuroda, S. Y., Classes of languages and linear bounded automata, Inform. and Control, 7, 207-223 (1964) · Zbl 0199.04002 |

[4] | Mahaney, S., Sparse complete sets for NP: Solution of a conjecture of Berman and Hartmanis, J. Comput. System Sci., 25, 130-143 (1982) · Zbl 0493.68043 |

[5] | Stockmeyer, L. J., The polynomial-time hierarchy, Theoret. Comput. Sci., 3, 1-22 (1976) · Zbl 0353.02024 |

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