The Axiomatic Derivation of Absolute Lower Bounds

Abstract

The ancient Euclidean algorithm computes the greatest common divisor gcd(m, n) of two natural numbers from (or relative to) the remainder operation rem, which is assumed as primitive; it requires no more than 2 log(min(m, n)) applications of the remainder operation to compute gcd(m, n) (for m, n ges 2), and it is not known to be optimal: Conjecture: for every algorithm a which computes on Nopf from rem the greatest common divisor function, there is a constant r > 0 such that for infinitely many pairs a ges b ges 1, calpha(a, b) ges rlog2(a), where calpha(m,n) counts the number of calls to "the remainder oracle" required by a for the computation of gcd(m, n). The conjecture claims a logarithmic lower bound for all algorithms which compute gcd(m, n) from the remainder operation, not just those expressed by a specific class of computation models. In this lecture the author develops an approach to the theory of algorithms in the style of abstract model theory which makes it possible to make precise and (on occasion) prove the existence of non-trivial, absolute lower bounds for a wide variety of problems and specified primitives, including many of the results in the bibliography.