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Continuous Turing Machine: Real Function Computability and Iteration Issues |
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PP: 2405-2416 |
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Author(s) |
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Xiaoliang Chen,
Wen Song,
Zexia Huang,
Mingwei Tang,
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Abstract |
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| 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 Rn, 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 Rn are defined in a new space CTMRn . Finally, an iterative algorithm is shown by constructing a CTM to solve linear
equations. |
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