Feasible real random access machines

Abstract

We present a modified real RAM model which is equipped with the usual discrete and real-valued arithmetic operations and with a finite precision test <k which allows comparisons of real numbers only up to a variable uncertainty 1/k+1 Furthermore our feasible RAM has an extended semantics which allows approximative computations. Using a logarithmic complexity measure we prove that all functions computable on a feasible RAM in time O(t) can be computed on a Turing machine in time O(t2 · log(t) · log log (t)). Vice versa all functions computable on a Turing machine in time O(t) are computable on a feasible RAM in time O(t). Thus our real RAM model does not only express exactly the computational power of Turing machines on real numbers (in the sense of Grzegorczyk), but it also yields a high-level tool for realistic time complexity estimations on real numbers.