Complexity and real computation
Complexity and real computation
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During the last few decades two traditions of computing have grown and grown further apart.Firstly, there is the tradition of discrete computation. It has its roots in work in mathematical logic at the truth of the century involving the decidability of the arithmetic. Abstract computational devices such as the Turing machine and computability concepts such as recursive function are legacies of this tradition.Secondly, there is the long-standing tradition of algorithmics results in algebra and analysis which we will refer to as the numerical tradition. Algorithms such as Gaussian elimination or Newton's method and negative results such as Galois' theorem on the non-solvability by radicals of polynomial equations of degree at least 5 are legacies of this tradition.The arrival of the digital computer set a stage in which both traditions could meet. The need of feasibility in practice for computable functions brought along the need for a complexity theory. But while the discrete tradition was very successful at building such a theory, at the end of the 80's there was little akin to complexity theory within the numerical tradition.This difference in the theoretical foundations for both traditions is apparent in the 1989 SIAM's John von Neumann Lecture given by Steve Smale [4]. The following table is taken from there (below, “Scientific computation” corresponds to the numerical tradition and “Computer Science” to the discrete one). Models of computations are abstract devices designed to model computations in the real world and set a formal framework in which the power and limitations of computation can be rigorously proved. Real world computations, however, depend on too many parameters and the consideration of all of them would make the abstract model too cumbersome and therefore useless. A key question is how does one select a set of features to be considered in the formal model.In this talk we shall try to motivate one such choice (with two main variants) for a model of computation over the real numbers. Then, we shall survey some advances towards the laying of foundations within the numerical tradition as suggested in [4].Some of the work laying these foundations can be traced back some decades, for instance, to the work of Turing [5]. Some other is more recent (e.g. the foundational paper [2]). Two appropriate references for our talk are [1, 3].