Observations on complete sets between linear time and polynomial time

Hemmerling, Armin

doi:10.1016/j.ic.2010.11.027

Cited by 2 publications

(16 citation statements)

References 15 publications

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“…Thus, for any oracle A satisfying LIN A =NLIN A = DTIME A (n), as it was given by Proposition 15 in [10], it follows that both ¬ H A (n) and ¬ G A (n) hold, as well as ¬ H A (f ) and ¬ G A (f ) for any time-constructible bound f . Hence G(f ) and H(f ) do not relativize if they hold.…”

Section: Definition 4 (Subhomogeneity) a Functionmentioning

confidence: 72%

“…The hardness of this problem was emphasized by showing that H(n) does not relativize. We want to recall this result from [10], since it marks the starting point of the present investigations. Moreover, we shall come back to relativizations at the end of this paper.…”

Section: Basic Notions and Results Around The Hierarchy Propertymentioning

confidence: 99%

“…• , • denotes an injective pairing function computable in linear time, with lineartime computable inverses (projections). It is not hard to show that V is NTIME(n)-complete with respect to linear-time reductions, see [10].…”

Section: Basic Notions and Results Around The Hierarchy Propertymentioning

confidence: 99%

“…So far, complete sets have mainly been considered for complexity classes defined by regular sets of bounds, as for P, NP, LIN, NLIN etc. The notion of regular set of bounds was introduced in [10]. It ensures a certain robustness of the related complexity classes and useful properties of complete sets of them.…”

Section: Complete Sets For Subhomogeneous Boundsmentioning

confidence: 99%

“…Moreover, it has to be noticed that fact (1) relativizes. It can easily be proved that V A is NTIME A (n)-complete with respect to linear-time reductions, see [10].…”

Section: ⊓ ⊔mentioning

confidence: 99%

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On the Tape-Number Problem for Deterministic Time Classes

Hemmerling¹

2013

Preprint

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For any time bound f , let H(f ) denote the hierarchy conjecture which means that the restriction of the numbers of work tapes of deterministic Turing machines to b ∈ N generates an infinite hierarchy of proper subclasses DTIME b (f ) ⊂ DTIME(f ). We show that H(f ) implies separations of deterministic from nondeterministic time classes. H(f ) follows from the gap property, G(f ), which says that there is a time-constructible f2 such that f ∈ o(f2) and DTIME(f ) = DTIME(f2). G(f ) implies further separations. All these relationships relativize.

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Section: Definition 4 (Subhomogeneity) a Functionmentioning

confidence: 72%

Section: Basic Notions and Results Around The Hierarchy Propertymentioning

confidence: 99%

Section: Basic Notions and Results Around The Hierarchy Propertymentioning

confidence: 99%

Section: Complete Sets For Subhomogeneous Boundsmentioning

confidence: 99%

“…Moreover, it has to be noticed that fact (1) relativizes. It can easily be proved that V A is NTIME A (n)-complete with respect to linear-time reductions, see [10].…”

Section: ⊓ ⊔mentioning

confidence: 99%

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On the Tape-Number Problem for Deterministic Time Classes

Hemmerling¹

2013

Preprint

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On Regular Sets of Bounds and Determinism versus Nondeterminism

Hemmerling

2013

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This paper illustrates the richness of the concept of regular sets of time bounds and demonstrates its application to problems of computational complexity. There is a universe of bounds whose regular subsets allow to represent several time complexity classes of common interest and are linearly ordered with respect to the confinality relation which implies the inclusion between the corresponding complexity classes. By means of classical results of complexity theory, the separation of determinism from nondeterminism is possible for a variety of sets of bounds below n • log * (n).The system of all regular bound sets ordered by confinality allows the order-isomorphic embedding of, e.g., the ordered set of real numbers or the Cantor discontinuum.

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Observations on complete sets between linear time and polynomial time

Cited by 2 publications

References 15 publications

On the Tape-Number Problem for Deterministic Time Classes

On the Tape-Number Problem for Deterministic Time Classes

On Regular Sets of Bounds and Determinism versus Nondeterminism

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