Continuous Turing Machine: Real Function Computability and Iteration Issues

Contemporary computer theory is governed by the discretization of continuous problems. Classical Turing machines (TMs) are originally built to solve computation and computability problems, which main feature is discreteness. However, even some simple numerical calculations problems, e. g., iterations in R-n, generate difficulties to be described or solved by constructing a TM. This paper explores the computability of continuous problems by proposing a class of continuous Turing machines (CTMs) that are an extension of TMs. CTMs can be applied to the standard for the precision of algorithms. First, computable real numbers are precisely defined by CTMs and their computations are regarded as the running of the CTMs. CTMs introduce the coded recursive descriptions, machine states, and operations with the characters of computer instructions in essence compared with usual computable continuous models. Hence, they can precisely present continuous computations with the form of processes. Second, the concepts of CTM computable and CTM handleable are proposed. Moreover, the basic concepts on approximation theory such as convergency, metric space, and fixed-point in R-n are defined in a new space CTMRn. Finally, an iterative algorithm is shown by constructing a CTM to solve linear equations.