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

    All sources support itProof of definition segments
    ?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 functions with s2(n) >= s1(n) >= n
      ?

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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 space hierarchy theorem's proof relies on a diagonalization argument that presupposes a universal TM can simulate any TM with bounded overhead.
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    • 2.The simulation overhead assumption embeds a non-trivial empirical claim about machine architecture that is not derivable from pure mathematical definitions of SPACE classes.
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    • 3.If the overhead of universal simulation is not tightly bounded, the strict separation between SPACE(s1(n)) and SPACE(s2(n)) may not hold for all constructible function pairs satisfying the ratio condition.
      ?

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    Reason against 2 of 2
    ?
    • 1.Space constructibility is a non-trivial precondition: not all mathematically well-defined functions s(n) >= n are space constructible in the required sense.
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    • 2.The claim's scope is implicitly restricted to a proper subset of function pairs, making the theorem's generality philosophically overstated when presented without that qualification.
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    Topics

    Proof of definition segmentsAll sources support it

    Related

    If the overhead of universal simulation is not tightly bounded, the strict separ...Space constructibility is a non-trivial precondition: not all mathematically wel...The claim's scope is implicitly restricted to a proper subset of function pairs,...The limit of s1(n) / s2(n) as n approaches infinity equals 0
    +3 moreShow less
    The simulation overhead assumption embeds a non-trivial empirical claim about ma...The space hierarchy theorem's proof relies on a diagonalization argument that pr...s1(n) and s2(n) are space constructible functions with s2(n) >= s1(n) >= n

    Similar

    SPACE(s1(n)) is a proper subset of SPACE(s2(n)) when s2 grows sufficie...100%NTIME(t1(n)) is a proper subset of NTIME(t2(n)) when t2(n) grows suffi...96%TIME(t1(n)) is a proper subset of TIME(t2(n)) when t2(n) grows suffici...95%NTIME(t1(n)) is a proper subset of NTIME(t2(n)) when t2 grows sufficie...95%

    Source

    AI-extracted1/3 agreementValid
    SEP: computational-complexity
    View source passageHide passage
    2 Complexity classes and the hierarchy theorems Recall that a complexity class is a set of languages all of which can be decided within a given time or space complexity bound \(t(n)\) or \(s(n)\) with respect to a fixed model of computation. g. non-recursive ones) it is standard to restrict attention to complexity classes defined when \(t(n)\) and \(s(n)\) are time or space constructible. e. a string of \(n\) 1s) halts after exactly \(t(n)\) steps. Similarly, \(s(n)\) is said to be space constru
    Extraction notes

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

    Details

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