Abstract
Given a continuous functional \(f \colon X \to Y\) and y ∈ Y, we wish to compute x ∈ X such that f(x) = y, if such an x exists. We show that if x is unique and X and Y are subspaces of Kleene–Kreisel spaces of continuous functionals with X exhaustible, then x is computable uniformly in f, y and the exhaustion functional \(\forall_X \colon 2^X \to 2\). We also establish a version of the above for computational metric spaces X and Y, where is X computationally complete and has an exhaustible set of Kleene–Kreisel representatives. Examples of interest include functionals defined on compact spaces X of analytic functions.
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References
Normann, D.: Recursion on the countable functionals. Lec. Not. Math., vol. 811. Springer, Heidelberg (1980)
Escardó, M.: Exhaustible sets in higher-type computation. Log. Methods Comput. Sci. 4(3), 3:3, 37 (2008)
Weihrauch, K.: Computable analysis. Springer, Heidelberg (2000)
Bauer, A.: A relationship between equilogical spaces and type two effectivity. MLQ Math. Log. Q. 48(suppl. 1), 1–15 (2002)
Bishop, E., Bridges, D.: Constructive Analysis. Springer, Berlin (1985)
Simpson, A.: Lazy functional algorithms for exact real functionals. In: Brim, L., Gruska, J., Zlatuška, J. (eds.) MFCS 1998. LNCS, vol. 1450, pp. 323–342. Springer, Heidelberg (1998)
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Escardó, M. (2009). Computability of Continuous Solutions of Higher-Type Equations. In: Ambos-Spies, K., Löwe, B., Merkle, W. (eds) Mathematical Theory and Computational Practice. CiE 2009. Lecture Notes in Computer Science, vol 5635. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03073-4_20
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DOI: https://doi.org/10.1007/978-3-642-03073-4_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03072-7
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