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    Home/Original/inverse
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    Inverse View

    It is not the case that Feasibility of computation is preserved when a function is defined by limited recursion on notation

    ?Set your confidence on the premises below to see your aggregate.

    Reasons For

    2 perspectives
    Reason for 1 of 2
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    • 1.The bound on recursive steps being logarithmic does not bound the size of intermediate values computed during those steps.
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    • 2.If intermediate values grow polynomially or worse, the total bit-complexity of the computation may escape feasibility even with few recursive steps.
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    • 3.Cobham's original framework requires bounding both the number of steps and the bitwidth of values; limited recursion on notation alone addresses only the former.
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    Reason for 2 of 2
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    • 1.Feasibility is a semantic, resource-sensitive notion tied to actual physical computation, not merely a structural property of definitional schemes.
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    • 2.A function class closed under limited recursion on notation can still contain functions whose constants or basis functions encode infeasible computations in their descriptions.
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    • 3.As Slot and van Emde Boas argued, the choice of machine model and encoding can shift the boundary of feasibility independently of the recursion-theoretic characterization.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.The basis functions F_0 are feasibly computable
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    • 2.Feasibility is preserved under composition of functions
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    • 3.Limited recursion on notation recurses on the binary length of y — proportional to log_2(y) — rather than on the value of y itself as in ordinary primitive recursion, so the number of recursive steps is bounded by the bit-length of the input
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