Effective Fixed Point Theorem over a Non-Computably Separable Metric Space

Abstract

This paper shows effective fixed point theorems for computable contractions. Effective fixed point theorem for computable constractions 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.

Acknowledgements

The author would like to thank Yasugi Mariko, Kamo Hiroyasu, and anonymous referees for discussions and comments. This work was presented at the Forth Workshop on Computability and Complexity in Analysis. The author also thanks the participants of the workshop for the comments.