A domain-theoretic approach to computability on the real line
Theoretical Computer Science - Special issue on real numbers and computers
An abstract data type for real numbers
Theoretical Computer Science
Concrete models of computation for topological algebras
Theoretical Computer Science - Special issue on computability and complexity in analysis
Computable analysis: an introduction
Computable analysis: an introduction
Topological properties of real number representations
Theoretical Computer Science
Real number computation through gray code embedding
Theoretical Computer Science
Compact metric spaces as minimal-limit sets in domains of bottomed sequences
Mathematical Structures in Computer Science
Electronic Notes in Theoretical Computer Science (ENTCS)
Streams with a bottom in functional languages
ESOP'05 Proceedings of the 14th European conference on Programming Languages and Systems
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When a topological space X can be embedded into the space 驴;驴,n驴 of n驴-sequences of 驴, then we can define the corresponding computational notion over X because a machine with n + 1 heads on each tape can input/output sequences in 驴;驴,n驴. This means that the least number n such that X can be topologically embedded into 驴;驴,n驴 serves as a degree of complexity of the space. We prove that this number, which we call the computational dimension of the space, is equal to the topological dimension for separable metric spaces. First, we show that the weak inductive dimension of 驴;驴,n驴 is n, and thus the computational dimension is at least as large as the weak inductive dimension for all spaces. Then, we show that the N枚beling's universal n-dimensional space can be embedded into 驴;驴,n驴 and thus the computational dimension is at most as large as the weak inductive dimension for separable metric spaces. As a corollary, the 2-dimensional Euclidean space R2 can be embedded in {0, 1}驴,2驴 but not in 驴;驴,1驴 for any character set 驴, and infinite dimensional spaces like the set of closed/open/compact subsets of Rm and the set of continuous functions from Rl to Rm can be embedded in 驴;驴驴 but not in 驴;驴,n驴 for any n.