Revising Type-2 Computation and Degrees of Discontinuity

By the sometimes so-called MAIN THEOREM of Recursive Analysis, every computable real function is necessarily continuous. Weihrauch and Zheng (TCS'2000), Brattka (MLQ'2005), and Ziegler (ToCS'2006) have considered different relaxed notions of computability to cover also discontinuous functions. The present work compares and unifies these approaches. This is based on the concept of the JUMP of a representation: both a TTE-counterpart to the well known recursion-theoretic jump on Kleene's Arithmetical Hierarchy of hypercomputation: and a formalization of revising computation in the sense of Shoenfield. We also consider Markov and Banach/Mazur oracle-computation of discontinuous fu nctions and characterize the computational power of Type-2 nondeterminism to coincide with the first level of the Analytical Hierarchy.Comment: to appear in Proc. CCA'0

Levels of discontinuity, limit-computability, and jump operators

We develop a general theory of jump operators, which is intended to provide an abstraction of the notion of "limit-computability" on represented spaces. Jump operators also provide a framework with a strong categorical flavor for investigating degrees of discontinuity of functions and hierarchies of sets on represented spaces. We will provide a thorough investigation within this framework of a hierarchy of $\Delta^0_2$-measurable functions between arbitrary countably based $T_0$-spaces, which captures the notion of computing with ordinal mind-change bounds. Our abstract approach not only raises new questions but also sheds new light on previous results. For example, we introduce a notion of "higher order" descriptive set theoretical objects, we generalize a recent characterization of the computability theoretic notion of "lowness" in terms of adjoint functors, and we show that our framework encompasses ordinal quantifications of the non-constructiveness of Hilbert's finite basis theorem

Real Hypercomputation and Continuity

By the sometimes so-called 'Main Theorem' of Recursive Analysis, every computable real function is necessarily continuous. We wonder whether and which kinds of HYPERcomputation allow for the effective evaluation of also discontinuous f:R->R. More precisely the present work considers the following three super-Turing notions of real function computability: * relativized computation; specifically given oracle access to the Halting Problem 0' or its jump 0''; * encoding real input x and/or output y=f(x) in weaker ways also related to the Arithmetic Hierarchy; * non-deterministic computation. It turns out that any f:R->R computable in the first or second sense is still necessarily continuous whereas the third type of hypercomputation does provide the required power to evaluate for instance the discontinuous sign function.Comment: previous version (extended abstract) has appeared in pp.562-571 of "Proc. 1st Conference on Computability in Europe" (CiE'05), Springer LNCS vol.352

Comparing Representations for Function Spaces in Computable Analysis

This paper compares different representations (in the sense of computable analysis) of a number of function spaces that are of interest in analysis. In particular subspace representations inherited from a larger function space are compared to more natural representations for these spaces. The formal framework for the comparisons is provided by Weihrauch reducibility. The centrepiece of the paper considers several representations of the analytic functions on the unit disk and their mutual translations. All translations that are not already computable are shown to be Weihrauch equivalent to closed choice on the natural numbers. Subsequently some similar considerations are carried out for representations of polynomials. In this case in addition to closed choice the Weihrauch degree LPO∗ shows up as the difficulty of finding the degree or the zeros. As a final example, the smooth functions are contrasted with functions with bounded support and Schwartz functions. Here closed choice on the natural numbers and the lim degree appear.SCOPUS: ar.jinfo:eu-repo/semantics/publishe