Recursively enumerable subsets of Rq in two computing models Blum-Shub-Smale machine and Turing machine

Ning Zhong
1998 Theoretical Computer Science  
In this paper we compare recursively enumerable subsets of R" in two computing models over real numbers: the Blum-Shub-Smale machine and the oracle Turing machine. We prove that any Turing RE open subset of RY is a BSS RE set, while a Turing RE closed set may not be a BSS RE set. As an application we show that the Julia set of any computable hyperbolic polynomial is decidable in the Turing computing model. @ 1998-Elsevier Science B.V. All rights reserved 1989 [2]. The work of Turing, Godel,
more » ... ch and others in the 30s forms the core of classical computation theory. Although much of the classical theory of computation deals with computing over the natural numbers, certain approaches have considered other underlying domains. One such approach is recursive analysis, which studies the computability of reals and continuous functions of real variables. The subject is a natural development of computability theory for functions from natural numbers to natural numbers, and has been well studied, e.g. [6, lo]. Let R be the set of real numbers. In recursive analysis the standard definition for "computable open sets" of Rq, now commonly known as recursively enumerable open sets, or RE open sets, goes as follows: an open set U of R4 is RE open if it is the union of a sequence of q-balls, {x : Ix -ail <r;}, where (7;) ' The author thanks Professor Marian B. Pour-El for introducing her to the area of recursive analysis 0304-3975/98/$19.00 @ 1998 -Elsevier Science B.V. All rights reserved PII s0304-3975(97)00008-x 80 N. Zhony I Theoretical Cornpuier S&nce I97 (1998) [79][80][81][82][83][84][85][86][87][88][89][90][91][92][93][94] and {aj] are computable sequences of real numbers and q-vectors of Rq, respectively [9] . This de~nition is in accordance with the notion of classical RE sets of the natural numbers, i.e. when the set N of the natural numbers is considered as a subspace of Rq, RE open sets of N are the same as the ordinary RE sets in the classical recursion theory. The computing model adopted here is oracle Turing machines where, roughly, an oracle Turing machine is a classical Turing machine equipped with an oracle which can supply rational approximations of real numbers on demand. On the other hand, Blum et al. [2] developed a model for computation over ordered rings in 1989. This model provides an interplay between algebra, analysis, scientific computation, and topology. In this computational model, a set Y c Rq is RE if it is the halting set of some machine over R. Y is said to be decidable (recursive) if Y and its complement are both RE sets. This theory, as the recursive analysis approach, also
doi:10.1016/s0304-3975(97)00008-x fatcat:v2vkfo77ere5hptrtcryohrzoq