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    If problem identity is encoding-relative, polynomial-time... — Carmelics
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    Challenges→If any NP-complete problem has a polynomial time algorithm, then all problems in NP have polynomial time algorithms

    If problem identity is encoding-relative, polynomial-time reducibility ≤_P does not transitively preserve the property of being 'the same computational task'.

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    1 reason for
    1 reason against

    Reasons For

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    Reason for
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    • 1.Problem identity depends on how we represent inputs/outputs; different encodings can obscure or reveal task structure differently.
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    • 2.A ≤_P B and B ≤_P C doesn't guarantee A and C share the same computational essence if encoding choices differ at each reduction step.
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    • 3.Transitivity assumes sameness is preserved, but encoding-relative identity means 'same task' is observer-dependent, not intrinsic.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Polynomial-time reductions are defined to work for any standard encoding; encoding choice doesn't affect computational class membership.
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    • 2.If problem identity were truly encoding-relative, we couldn't meaningfully discuss NP-completeness, which assumes encoding-invariant identity.
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    • 3.Transitivity failure would require A ≤_P B ≤_P C but A ≰_P C under same encoding—no standard example demonstrates this.
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    Related

    A ≤_P B and B ≤_P C doesn't guarantee A and C share the same computational essen...If any NP-complete problem has a polynomial time algorithm, then all problems in...If problem identity were truly encoding-relative, we couldn't meaningfully discu...Polynomial-time reductions are defined to work for any standard encoding; encodi...
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    Problem identity depends on how we represent inputs/outputs; different encodings...Transitivity assumes sameness is preserved, but encoding-relative identity means...Transitivity failure would require A ≤_P B ≤_P C but A ≰_P C under same encoding...

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