It would be extremely helpful if someone can suggest foundational textbooks on Recursive Analysis (Computability over Reals) which explains connections between Computability and the Topological properties of continuous sets. I could not find any, except "Recursive Analysis" by R.L. Goodstein but I dont have any feedback on that book. I would also like to ask for suggestions regarding a book on Complexity over Real Numbers which again deals with the connection between the complexity and the topology of a set. The only book I found on the net was "Complexity and Real Computation" by Lenore Blum et al but I am not sure if that is the book which I am looking for. Thanks in advance.
There are some good books - 1. Computable Analysis - Pour-El and Richards (an older reference) 2. Computable Analysis - Weihrauch
There's also the Blum-Shub-Smale Model, which is the model explored in "Complexity and Real Computation".
The complexity theory of computability of reals is explored in 1. Computational Complexity of real functions - Ker-I Ko
A regular conference which explores this area is "Computability and Complexity in Analysis", held annually.
For computability, see
- Klaus Weihrauch, "Computable Analysis", 2010.
It also has a chpater on complexity.
See also PhD theses of Jens Blanc and Andrej Bauer.
Another interesting paper is
- Viggo Stoltenberg-Hansen and John Tucker, "Computability on Topological Spaces via Domain Representations", 1997.