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On Primitive Recursive Algorithms And The Greatest Common Divisor Function
, 2003
"... We establish linear lower bounds for the complexity of nontrivial, primitive recursive algorithms from piecewise linear given functions. The main corollary is that logtime algorithms for the greatest common divisor from such givens (such as Stein's) cannot be matched in e#ciency by primitive recurs ..."
Abstract

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We establish linear lower bounds for the complexity of nontrivial, primitive recursive algorithms from piecewise linear given functions. The main corollary is that logtime algorithms for the greatest common divisor from such givens (such as Stein's) cannot be matched in e#ciency by primitive recursive algorithms from the same given functions. The question is left open for the Euclidean algorithm, which assumes the remainder function. In 1991, Colson [3] proved a remarkable theorem about the limitations of primitive recursive algorithms, which has the following consequence: Colson's Corollary. If a primitive recursive derivation of min(x, y) is expressed faithfully in a programming language, then one of the two computations min(1, 1000) and min(1000, 1) will take at least 1000 steps.