Computability on computable metric spaces
Theoretical Computer Science
Theoretical Computer Science - Special issue on real numbers and computers
Computable analysis: an introduction
Computable analysis: an introduction
Complexity theory for operators in analysis
Proceedings of the forty-second ACM symposium on Theory of computing
Complexity Theory for Operators in Analysis
ACM Transactions on Computation Theory (TOCT)
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This paper shows effective fixed point theorems for computable contractions. Effective fixed point theorem for computable contractions over a computable metric space is easily shown. A function over a computable metric space is represented by a Type-1 function, and the fixed point of a contraction is given by iteration of such Type-1 function. If the contraction is computable, then its fixed point is also computable. If the support space is not computably separable, the method above is not available. The function space of an interval into real numbers is not computably separable with polynomial time computability. This paper show the fixed point theorem for such non-computably separable spaces. This theorem is proved with iteration of Type-2 functionals. As an example of that, this paper shows that Takagi function is a polynomial time computable function.