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14
Semicomputability of the Fréchet distance between surfaces
 In Proc. 21st European Workshop on Computational Geometry
, 2005
"... The Fréchet distance is a distance measure for parameterized curves or surfaces. Using a discrete approximation, we show that for triangulated surfaces it is upper semicomputable, i.e., there is a nonhalting Turing machine which produces a monotone decreasing sequence of rationals converging to th ..."
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Cited by 9 (0 self)
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The Fréchet distance is a distance measure for parameterized curves or surfaces. Using a discrete approximation, we show that for triangulated surfaces it is upper semicomputable, i.e., there is a nonhalting Turing machine which produces a monotone decreasing sequence of rationals converging to the result. It follows that the decision problem, whether the Fréchet distance of two given surfaces lies below some specified value, is recursively enumerable. 1
The emperor’s new recursiveness: The epigraph of the exponential function in two models of computability
 In Masami Ito and Teruo Imaoka, editors, Words, Languages & Combinatorics III
, 2003
"... In his book “The Emperor’s New Mind ” Roger Penrose implicitly defines some criteria which should be met by a reasonable notion of recursiveness for subsets of Euclidean space. We discuss two such notions with regard to Penrose’s criteria: one originated from computable analysis, and the one introdu ..."
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Cited by 7 (0 self)
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In his book “The Emperor’s New Mind ” Roger Penrose implicitly defines some criteria which should be met by a reasonable notion of recursiveness for subsets of Euclidean space. We discuss two such notions with regard to Penrose’s criteria: one originated from computable analysis, and the one introduced by Blum, Shub and Smale. 1
Some Properties of the Effectively Uniform Topological Space
 Computability and Complexity in Analysis, LNCS 2064
, 2000
"... Our main objective is to investigate algorithmic features latent in some discontinuous functions in terms of uniform topological spaces. We develop the theory of computability structure and computable functions on a uniform topological space, and will apply the results to some real functions which a ..."
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Cited by 6 (3 self)
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Our main objective is to investigate algorithmic features latent in some discontinuous functions in terms of uniform topological spaces. We develop the theory of computability structure and computable functions on a uniform topological space, and will apply the results to some real functions which are discontinuous in the Euclidean space. Keywords: Effectively uniform topological space; Computability structure; Computable function; Binary tree structure, Cylinder function, Amalgamated space of reals, Function with jump points 1 Introduction We have recently been speculating on computability problems of some discontinuous real functions such as the Gauian function ([?]), Rademacher functions ([?]) and Walsh functions ([?]). These are functions either on the whole real line or on the interval [0; 1], jumping at integer points or at some binary rationals, and assuming computable values. Throughout this article, knowledge of the notion of computability of a real number or a sequence of...
A note on Rademacher functions and computability
, 2001
"... We will speculate on some computational properties of the system of Rademacher functions fOE n g. The nth Rademacher function OE n is a step function on the interval [0; 1), jumping at finitely many dyadic rationals of size 1 2 n and assuming values f1; 01g alternatingly. Key Words Rademacher fun ..."
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Cited by 6 (1 self)
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We will speculate on some computational properties of the system of Rademacher functions fOE n g. The nth Rademacher function OE n is a step function on the interval [0; 1), jumping at finitely many dyadic rationals of size 1 2 n and assuming values f1; 01g alternatingly. Key Words Rademacher functions, computability problems of discontinuous function, L p [0; 1]space, computability structure, 6 0 1 law of excluded middle, limiting recursion 1 Introduction In [?], PourEl and Richards proposed to treat computational aspects of some discontinuous functions by regarding them as points in some appropriate function spaces. It will then be of general interest to find examples of discontinuous functions which can be regarded as computable in PourEl and Richards approach. We are working on the integer part function [x] in [?] in this respect. It is not difficult to claim that it is a computable point in a function space. It is also important to find out what sort of a principle,...
Computable de Finetti measures
, 2009
"... We prove a uniformly computable version of de Finetti’s theorem on exchangeable sequences of real random variables. As a consequence, exchangeable stochastic processes in probabilistic functional programming languages can be automatically rewritten as procedures that do not modify nonlocal state. A ..."
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Cited by 4 (1 self)
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We prove a uniformly computable version of de Finetti’s theorem on exchangeable sequences of real random variables. As a consequence, exchangeable stochastic processes in probabilistic functional programming languages can be automatically rewritten as procedures that do not modify nonlocal state. Along the way, we prove that a distribution on the unit interval is computable if and only if its moments are uniformly computable.
Can we compute the similarity between surfaces
 In preparation
"... A suitable measure for the similarity of shapes represented by parameterized curves or surfaces is the Fréchet distance. Whereas efficient algorithms are known for computing the Fréchet distance of polygonal curves, the same problem for triangulated surfaces is NPhard. Furthermore, it remained open ..."
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A suitable measure for the similarity of shapes represented by parameterized curves or surfaces is the Fréchet distance. Whereas efficient algorithms are known for computing the Fréchet distance of polygonal curves, the same problem for triangulated surfaces is NPhard. Furthermore, it remained open whether it is computable at all. Using a discrete approximation we show that it is upper semicomputable, i.e., there is a nonhalting Turing machine which produces a monotone decreasing sequence of rationals converging to the Fréchet distance. It follows that the decision problem, whether the Fréchet distance of two given surfaces lies below a specified value, is recursively enumerable. Furthermore, we show that a relaxed version of the Fréchet distance, the weak Fréchet distance can be computed in polynomial time. For this, we give a computable characterization of the weak Fréchet distance in a geometric data structure called the free space diagram. 1
Singular Coverings and NonUniform Notions of Closed Set Computability
, 2007
"... Abstract. The empty set of course contains no computable point. On the other hand, surprising results due to Zaslavskiĭ, Tseĭtin, Kreisel, and Lacombe assert the existence of nonempty cor.e. closed sets devoid of computable points: sets which are even ‘large ’ in the sense of positive Lebesgue mea ..."
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Abstract. The empty set of course contains no computable point. On the other hand, surprising results due to Zaslavskiĭ, Tseĭtin, Kreisel, and Lacombe assert the existence of nonempty cor.e. closed sets devoid of computable points: sets which are even ‘large ’ in the sense of positive Lebesgue measure. We observe that a certain size is in fact necessary: every nonempty cor.e. closed real set without computable points has continuum cardinality. This initiates a comparison of different notions of computability for closed real subsets nonuniformly like, e.g., for sets of fixed cardinality or sets containing a (not necessarily effectively findable) computable point. By relativization we obtain a bounded recursive rational sequence of which every accumulation point is not even computable with support of a Halting oracle. Finally the question is treated whether compact sets have cor.e. closed connected components; and every starshaped cor.e. closed set is asserted to contain a computable point. 1
Computability of discontinuous functions Functional space approach  A gerenal report at Dagstuhl Seminar
"... Contents 1 Our standpoint We are to discuss how to view notions of computability for discontinuous functions. We confine ourselves to realvalued functions from some spaces. Our standpoint in studying computability problems in mathematics is doing mathematics. That is, we would like to talk abou ..."
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Contents 1 Our standpoint We are to discuss how to view notions of computability for discontinuous functions. We confine ourselves to realvalued functions from some spaces. Our standpoint in studying computability problems in mathematics is doing mathematics. That is, we would like to talk about computable functions and other mathematical objects just as one talks about continuous functions, integrable functions, etc. In any naive notion of computability of a function (on a compact set), uniform continuity is inherent. On the other hand, one often approximates discontinuous functions, for instance in numerical computations and drawing graphs. Very often such approximations are successful and satisfactory. It is therefore meaningful and important to speculate on computability of discontinuous functions. According to our standpoint, it is a mathematical investigation to formulate computability o
On the Computability of Rectifiable Simple Curve ⋆ (Extended Abstract)
"... Abstract. In mathematics curves are defined as the images of continuous real functions defined on closed intervals and these continuous functions are called parameterizations of the corresponding curves. If only simple curves of finite lengths are considered, then parameterizations can be restricted ..."
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Abstract. In mathematics curves are defined as the images of continuous real functions defined on closed intervals and these continuous functions are called parameterizations of the corresponding curves. If only simple curves of finite lengths are considered, then parameterizations can be restricted to the injective continuous functions or even to the continuous lengthnormalized parameterizations. In addition, a plane curve can also be considered as a connected onedimensional compact subset of points. By corresponding effectivizations, we will introduce in this paper four versions of computable curves and show that they are all different. More interestingly, we show also that four classes of computable curves cover even different sets of points.
Real Computation with Least Discrete Advice: A Complexity Theory of Nonuniform Computability
, 2009
"... It is folklore particularly in numerical and computer sciences that, instead of solving some general problem f: A → B, additional structural information about the input x ∈ A (that is any kind of promise that x belongs to a certain subset A ′ ⊆ A) should be taken advantage of. Some examples from ..."
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It is folklore particularly in numerical and computer sciences that, instead of solving some general problem f: A → B, additional structural information about the input x ∈ A (that is any kind of promise that x belongs to a certain subset A ′ ⊆ 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 problems how much advice is necessary and sufficient to render them computable. Specifically, finding a nontrivial solution to a homogeneous linear equation A · x = 0 for a given singular real n × nmatrix A is possible when knowing rank(A) ∈ {0, 1,..., n−1}; and we show this to be best possible. Similarly, diagonalizing (i.e. finding a basis of eigenvectors of) a given real symmetric n × nmatrix 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 some single eigenvector of A requires and suffices with Θ(log n)–fold advice.