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    SPACE(s1(n)) is a proper subset of SPACE(s2(n)) when s2 g... — Carmelics
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    SPACE(s1(n)) is a proper subset of SPACE(s2(n)) when s2 grows sufficiently faster than s1

    Modality & Possibility
    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.s1(n) and s2(n) are space constructible
      ?

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    • 2.The limit of s1(n) / s2(n) as n approaches infinity equals 0
      ?

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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.The diagonal construction used to separate complexity classes requires that a machine can simulate another within bounded space, but this simulation adds a constant-factor overhead.
      ?

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    • 2.Hartmanis and Stearns' original hierarchy results show that below logarithmic space, the gap conditions for proper separation become insufficient to absorb simulation overhead, meaning the limit condition on s1/s2 alone does not guarantee separation in sub-log regimes.
      ?

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    • 3.Therefore the claim's generality is undermined: sufficiency of the ratio condition is domain-restricted, not universally valid across all space bounds.
      ?

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    Reason against 2 of 2
    ?
    • 1.Space constructibility presupposes a background computational model, yet the choice of model (Turing machine, RAM, etc.) affects which functions count as constructible.
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    • 2.If the constructibility criterion is model-relative, the subset relation established by the Space Hierarchy Theorem inherits that relativity and cannot ground a model-independent claim about 'proper' containment.
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    Topics

    Modality & PossibilityProof of definition segments

    Connections

    2 topics

    All sources support it1 linkedTruth & Knowledge1 linked

    Related

    Hartmanis and Stearns' original hierarchy results show that below logarithmic sp...If the constructibility criterion is model-relative, the subset relation establi...Space constructibility presupposes a background computational model, yet the cho...The diagonal construction used to separate complexity classes requires that a ma...
    +3 moreShow less
    The limit of s1(n) / s2(n) as n approaches infinity equals 0Therefore the claim's generality is undermined: sufficiency of the ratio conditi...s1(n) and s2(n) are space constructible

    Similar

    SPACE(s1(n)) is a proper subset of SPACE(s2(n)) when s2(n) grows suffi...100%NTIME(t1(n)) is a proper subset of NTIME(t2(n)) when t2(n) grows suffi...95%NTIME(t1(n)) is a proper subset of NTIME(t2(n)) when t2 grows sufficie...95%TIME(t1(n)) is a proper subset of TIME(t2(n)) when t2 grows sufficient...95%

    Source

    AI-extracted1/3 agreementValid
    SEP: computational-complexity
    View source passageHide passage
    On the other hand, if \(x \not\in X\), then all of \(N\)’s computations from \(C_0(x)\) are required to lead to rejecting states. Non-deterministic machines are sometimes described as making undetermined ‘choices’ among different possible successor configurations at various points during their computation. But what the foregoing definitions actually describe is a tree \(\mathcal{T}^N_{C_0}\) of all possible computation sequences starting from a given configuration \(C_0\) for a deterministic mac
    Extraction notes

    Validity: Extracted via Max plan + API grounding/validity checks

    Details

    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit