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    The supporting argument's transitivity premise holds for ... — Carmelics
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    Challenges→The existence of a polynomial time algorithm for any single NP-complete problem would entail the existence of polynomial time algorithms for all problems in NP.

    The supporting argument's transitivity premise holds for decision problems under standard Turing reductions, but Ladner's theorem demonstrates that if P≠NP there exist problems in NP neither in P nor NP-complete, complicating the universality of the entailment.

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    Key Terms

    Ladner's theorem(as used in computational complexity theory)
    A mathematical result proved by Richard Ladner showing that (assuming a certain unsolved problem is true) there are computational problems that aren't easy but also aren't the hardest kind of hard. It's named after its discoverer.
    NP
    The class of problems for which membership can be verified efficiently once an appropriate certificate is provided.
    NP-complete(Complexity theory classification of hardest problems within NP.)
    A problem is NP-complete if it is in NP and every problem in NP is polynomial-time reducible to it, making it among the most difficult problems in NP.
    P≠NP(as used in computer science and mathematics)
    A famous unsolved question asking whether problems that are quick to check are also quick to solve. Most computer scientists believe they're different, but no one has proven it yet.

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    Turing reduction(as used in computer science and computational theory)
    A way of comparing how hard two problems are by asking: if you could instantly solve one problem, could you use that to solve another? It's named after Alan Turing, a pioneering computer scientist.
    Universality of entailment(as used in logic)
    The idea that a logical rule works everywhere and always produces reliable conclusions without exceptions.
    transitivity(Applied to the temporal relation 'earlier than' on a set of worlds W)
    A property of a relation R such that if wRv and vRu, then wRu

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    The existence of a polynomial time algorithm for any single NP-complete problem ...

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