1,415 research outputs found

    Bounded time computation on metric spaces and Banach spaces

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    We extend the framework by Kawamura and Cook for investigating computational complexity for operators occurring in analysis. This model is based on second-order complexity theory for functions on the Baire space, which is lifted to metric spaces by means of representations. Time is measured in terms of the length of the input encodings and the required output precision. We propose the notions of a complete representation and of a regular representation. We show that complete representations ensure that any computable function has a time bound. Regular representations generalize Kawamura and Cook's more restrictive notion of a second-order representation, while still guaranteeing fast computability of the length of the encodings. Applying these notions, we investigate the relationship between purely metric properties of a metric space and the existence of a representation such that the metric is computable within bounded time. We show that a bound on the running time of the metric can be straightforwardly translated into size bounds of compact subsets of the metric space. Conversely, for compact spaces and for Banach spaces we construct a family of admissible, complete, regular representations that allow for fast computation of the metric and provide short encodings. Here it is necessary to trade the time bound off against the length of encodings

    Computational Problems in Metric Fixed Point Theory and their Weihrauch Degrees

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    We study the computational difficulty of the problem of finding fixed points of nonexpansive mappings in uniformly convex Banach spaces. We show that the fixed point sets of computable nonexpansive self-maps of a nonempty, computably weakly closed, convex and bounded subset of a computable real Hilbert space are precisely the nonempty, co-r.e. weakly closed, convex subsets of the domain. A uniform version of this result allows us to determine the Weihrauch degree of the Browder-Goehde-Kirk theorem in computable real Hilbert space: it is equivalent to a closed choice principle, which receives as input a closed, convex and bounded set via negative information in the weak topology and outputs a point in the set, represented in the strong topology. While in finite dimensional uniformly convex Banach spaces, computable nonexpansive mappings always have computable fixed points, on the unit ball in infinite-dimensional separable Hilbert space the Browder-Goehde-Kirk theorem becomes Weihrauch-equivalent to the limit operator, and on the Hilbert cube it is equivalent to Weak Koenig's Lemma. In particular, computable nonexpansive mappings may not have any computable fixed points in infinite dimension. We also study the computational difficulty of the problem of finding rates of convergence for a large class of fixed point iterations, which generalise both Halpern- and Mann-iterations, and prove that the problem of finding rates of convergence already on the unit interval is equivalent to the limit operator.Comment: 44 page

    Proof mining in metric fixed point theory and ergodic theory

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    In this survey we present some recent applications of proof mining to the fixed point theory of (asymptotically) nonexpansive mappings and to the metastability (in the sense of Terence Tao) of ergodic averages in uniformly convex Banach spaces.Comment: appeared as OWP 2009-05, Oberwolfach Preprints; 71 page

    Computability and analysis: the legacy of Alan Turing

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    We discuss the legacy of Alan Turing and his impact on computability and analysis.Comment: 49 page

    Ultraproducts and metastability

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    Given a convergence theorem in analysis, under very general conditions a model-theoretic compactness argument implies that there is a uniform bound on the rate of metastability. We illustrate with three examples from ergodic theory

    Revising Type-2 Computation and Degrees of Discontinuity

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    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
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