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

Computation with Advice

Computation with advice is suggested as generalization of both computation with discrete advice and Type-2 Nondeterminism. Several embodiments of the generic concept are discussed, and the close connection to Weihrauch reducibility is pointed out. As a novel concept, computability with random advice is studied; which corresponds to correct solutions being guessable with positive probability. In the framework of computation with advice, it is possible to define computational complexity for certain concepts of hypercomputation. Finally, some examples are given which illuminate the interplay of uniform and non-uniform techniques in order to investigate both computability with advice and the Weihrauch lattice

Closed Choice and a Uniform Low Basis Theorem

We study closed choice principles for different spaces. Given information about what does not constitute a solution, closed choice determines a solution. We show that with closed choice one can characterize several models of hypercomputation in a uniform framework using Weihrauch reducibility. The classes of functions which are reducible to closed choice of the singleton space, of the natural numbers, of Cantor space and of Baire space correspond to the class of computable functions, of functions computable with finitely many mind changes, of weakly computable functions and of effectively Borel measurable functions, respectively. We also prove that all these classes correspond to classes of non-deterministically computable functions with the respective spaces as advice spaces. Moreover, we prove that closed choice on Euclidean space can be considered as "locally compact choice" and it is obtained as product of closed choice on the natural numbers and on Cantor space. We also prove a Quotient Theorem for compact choice which shows that single-valued functions can be "divided" by compact choice in a certain sense. Another result is the Independent Choice Theorem, which provides a uniform proof that many choice principles are closed under composition. Finally, we also study the related class of low computable functions, which contains the class of weakly computable functions as well as the class of functions computable with finitely many mind changes. As one main result we prove a uniform version of the Low Basis Theorem that states that closed choice on Cantor space (and the Euclidean space) is low computable. We close with some related observations on the Turing jump operation and its initial topology

The descriptive theory of represented spaces

This is a survey on the ongoing development of a descriptive theory of represented spaces, which is intended as an extension of both classical and effective descriptive set theory to deal with both sets and functions between represented spaces. Most material is from work-in-progress, and thus there may be a stronger focus on projects involving the author than an objective survey would merit.Comment: survey of work-in-progres

Real Computation with Least Discrete Advice: A Complexity Theory of Nonuniform Computability

It is folklore particularly in numerical and computer sciences that, instead of solving some general problem $f:Ato B$, additional structural information about the input $xin A$ (that is any kind of promise that $x$ belongs to a certain subset $A\u27subseteq A$) should be taken advantage of. Some examples from real number computation show that such discrete advice can even make the difference between computability and uncomputability. We turn this into a both topological and combinatorial complexity theory of information, investigating for several practical problem show much advice is necessary and sufficient to render them computable. Specifically, finding a nontrivial solution to a homogeneous linear equation $Acdotvec x=0$ for a given singular real $ntimes n$-matrix $A$ is possible when knowing $rank(A)in{0,1,ldots,n-1}$; and we show this to be best possible. Similarly, diagonalizing (i.e. finding a basis of eigenvectors of) a given real symmetric $ntimes n$-matrix $A$ is possible when knowing the number of distinct eigenvalues: an integer between $1$ and $n$ (the latter corresponding to the nondegenerate case). And again we show that $n$--fold (i.e. roughly $log n$ bits of) additional information is indeed necessary in order to render this problem (continuous and) computable; whereas finding emph{some single} eigenvector of $A$ requires and suffices with $Theta(log n)$--fold advice

From Bolzano-Weierstra{\ss} to Arzel\`a-Ascoli

We show how one can obtain solutions to the Arzel\`a-Ascoli theorem using suitable applications of the Bolzano-Weierstra{\ss} principle. With this, we can apply the results from \cite{aK} and obtain a classification of the strength of instances of the Arzel\`a-Ascoli theorem and a variant of it. Let AA be the statement that each equicontinuous sequence of functions f_n: [0,1] --> [0,1] contains a subsequence that converges uniformly with the rate 2^-k and let AA_weak be the statement that each such sequence contains a subsequence which converges uniformly but possibly without any rate. We show that AA is instance-wise equivalent over RCA_0 to the Bolzano-Weierstra{\ss} principle BW and that AA_weak is instance-wise equivalent over WKL_0 to BW_weak, and thus to the strong cohesive principle StCOH. Moreover, we show that over RCA_0 the principles AA_weak, BW_weak + WKL and StCOH + WKL are equivalent