A Survey of Space Complexity

@article{Michel1992ASO,
  title={A Survey of Space Complexity},
  author={Pascal Michel},
  journal={Theor. Comput. Sci.},
  year={1992},
  volume={101},
  pages={99-132}
}
  • Pascal Michel
  • Published 13 July 1992
  • Mathematics
  • Theor. Comput. Sci.
Separating the Lower Levels of the Sublogarithmic Space Hierarchy
TLDR
It is shown that for each S between loglogn and logn, Σ2SPACE(S) and Σ3SPACE (S) are not closed under complement, which implies the hierarchy Σ1SPACE(-1,2,3) ⊂Σ2 SPACE(-3,4) and the power of weak and strong sublogarithmic space bounded ATMs is compared.
The Sublogarithmic Alternating Space World
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The Separation of NP and PSPACE
TLDR
This paper shows that NP 6= PSPACE via the premise of NTIME[S(n)] ⊆ DSPACE[S (n)], and then by diagonalization over all polynomial-time nondeterministic Turing machines via universal nondetergetic Turing machine M0 running in O(n ) space for any k ∈ N1.
Some Accepting Powers of Three-Dimensional Synchronized Alternating Turing Machines
  • Ito
  • Business
  • 2007
This paper introduces a three-dimensional synchronized alternating Turing machine (3-SATM), and investigates fundamental properties of 3-SATM ′s whose input tapes are restricted to cubic ones. The
Some Accepting Powers of Three-Dimensional Synchronized Alternating Turing Machines
This paper introduces a three-dimensional synchronized alternating Turing machine (3-SATM), and investigates fundamental properties of 3-SATM 0 s whose input tapes are restricted to cubic ones. The
Guest Column: One-Tape Turing Machine Variants and Language Recognition
TLDR
Two restricted versions of one-tape Turing machines are presented and for d = 2 deterministic limited automata are equivalent to deterministic pushdown automata, namely they characterize deterministic context-free languages.
Restricted Turing Machines and Language Recognition
TLDR
In 1965 Hennie proved that one-tape deterministic Turing machines working in linear time are equivalent to finite automata, namely they characterize regular languages, and in 1967 Hibbard proved that for each integer \(d\ge 2\), one-Tape Turing machines that are allowed to rewrite each tape cell only in the first d visits are equivalents to pushdown automata.
Nondeterministic One-Tape Off-Line Turing Machines and Their Time Complexity
TLDR
This paper shows that the running time of each nondeterministic machine accepting a nonregular language must grow at least as n log n, and proves that under this measure, each accepting computation should exhibit a crossing sequence of length at least log log n.
Resolution of The Linear-Bounded Automata Question
This work resolve a longstanding open question in automata theory, i.e. the linear-bounded automata question ( shortly, LBA question), which can also be phrased succinctly in the language of
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References

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Some properties of a dual class defined by exponential lower bounds are shown, as a version of Lupanov theorem and a characterization in terms of oracle Turing machines.
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TLDR
If the time bounds defining two nondeterministic complexity classes are too close for separation by the two known techniques, then they are almost too close to separation by any relativizable technique, implying $\operatorname{NSPACE}(\log n) = \operatORName{DSPACE})$.
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TLDR
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It is shown that the lower bounds on tape complexity of [1] depend on neither the halting assumption nor determinism, and that below log n tape complexity there exists a dense hierarchy of complexity classes for two-way nondeterministic devices.
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