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    If the logical capture of NP is sensitive to how inputs a... — Carmelics
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    Challenges→NP is captured by the logic SO-exists (second-order existential logic)

    If the logical capture of NP is sensitive to how inputs are encoded as structures, then SO-exists does not characterize NP simpliciter but only NP-relative-to-a-chosen-representation.

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    Reasons For

    1 perspective
    Reason for
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    • 1.Computational complexity classes are defined relative to formal models; changing the model (e.g., Turing machine vs circuit) changes what problems fall into NP.
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    • 2.Graph encoding as adjacency matrix vs. adjacency list demonstrably affects which algorithms run in polynomial time for the same problem.
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    • 3.If a property depends on arbitrary representational choices, it describes the representation, not the underlying problem class itself.
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    Reasons Against

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    Reason against
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    • 1.NP is defined over decision problems with abstract input lengths, not concrete encodings; encoding sensitivity affects runtime, not membership.
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    • 2.Standard encoding conventions (binary, reasonable size bounds) are established in complexity theory precisely to abstract away representation arbitrariness.
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    • 3.If NP were representation-relative, no comparison of algorithms across papers or fields would be meaningful—yet computational practice assumes it is.
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    Key Terms

    NP (in computational complexity)(in logic and computer science)
    A category of computer science problems that are easy to check if an answer is correct, but potentially hard to solve from scratch—like solving a puzzle versus verifying a completed puzzle.
    SO-exists(in mathematical logic)
    A technical statement asserting the existence of something according to a particular logical framework (likely second-order logic, an advanced form of logical notation).
    Sensitive to(as used in philosophical analysis)
    Responsive to or dependent on; something is 'sensitive to' a factor if that factor changes or affects it.
    encoded as structures(in computer science and logic)
    Converted or represented in the form of organized patterns or systems.
    logical capture(in philosophy of logic)
    The idea of fully describing or defining something using the rules of logic.
    relative-to-a-chosen-representation(in philosophy of logic and epistemology)
    Dependent on which specific way you decide to describe or organize something, rather than being independent of how it's presented.
    simpliciter(as used in philosophy)
    A Latin phrase meaning 'simply' or 'in itself'—used to mean something is true in the most basic or straightforward sense.

    Connections

    2 topics

    Proof of definition segments1 linkedTruth & Knowledge1 linked

    Related

    Computational complexity classes are defined relative to formal models; changing...Graph encoding as adjacency matrix vs. adjacency list demonstrably affects which...

    Details

    Type
    claim
    Perspectives
    2 (1 for, 1 against)
    Edits
    1 edit
    If NP were representation-relative, no comparison of algorithms across papers or...
    If a property depends on arbitrary representational choices, it describes the re...
    +3 moreShow less
    NP is captured by the logic SO-exists (second-order existential logic)NP is defined over decision problems with abstract input lengths, not concrete e...Standard encoding conventions (binary, reasonable size bounds) are established i...