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114
On the complexity of real functions
, 2005
"... We establish a new connection between the two most common traditions in the theory of real computation, the BlumShubSmale model and the Computable Analysis approach. We then use the connection to develop a notion of computability and complexity of functions over the reals that can be viewed as an ..."
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Cited by 15 (5 self)
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We establish a new connection between the two most common traditions in the theory of real computation, the BlumShubSmale model and the Computable Analysis approach. We then use the connection to develop a notion of computability and complexity of functions over the reals that can be viewed as an extension of both models. We argue that this notion is very natural when one tries to determine just how “difficult ” a certain function is for a very rich class of functions. 1
Graph Homomorphisms with Complex Values: A Dichotomy Theorem
"... Graph homomorphism problem has been studied intensively. Given an m × m symmetric matrix A, the graph homomorphism function is defined as ZA(G) = Aξ(u),ξ(v), ξ:V →[m] (u,v)∈E where G = (V, E) is any undirected graph. The function ZA(G) can encode many interesting graph properties, including counting ..."
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Cited by 15 (8 self)
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Graph homomorphism problem has been studied intensively. Given an m × m symmetric matrix A, the graph homomorphism function is defined as ZA(G) = Aξ(u),ξ(v), ξ:V →[m] (u,v)∈E where G = (V, E) is any undirected graph. The function ZA(G) can encode many interesting graph properties, including counting vertex covers and kcolorings. We study the computational complexity of ZA(G) for arbitrary complex valued symmetric matrices A. Building on work by Dyer and Greenhill [6], Bulatov and Grohe [2], and especially the recent beautiful work by Goldberg,
Computability, noncomputability and undecidability of maximal intervals of IVPs
 Trans. Amer. Math. Soc
"... Abstract. Let (α, β) ⊆ R denote the maximal interval of existence of solution for the initialvalue problem { dx = f(t, x) dt x(t0) = x0, where E is an open subset of R m+1, f is continuous in E and (t0, x0) ∈ E. We show that, under the natural definition of computability from the point of view o ..."
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Cited by 15 (14 self)
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Abstract. Let (α, β) ⊆ R denote the maximal interval of existence of solution for the initialvalue problem { dx = f(t, x) dt x(t0) = x0, where E is an open subset of R m+1, f is continuous in E and (t0, x0) ∈ E. We show that, under the natural definition of computability from the point of view of applications, there exist initialvalue problems with computable f and (t0, x0) whose maximal interval of existence (α, β) is noncomputable. The fact that f may be taken to be analytic shows that this is not a lack of regularity phenomenon. Moreover, we get upper bounds for the “degree of noncomputability” by showing that (α, β) is r.e. (recursively enumerable) open under very mild hypotheses. We also show that the problem of determining whether the maximal interval is bounded or unbounded is in general undecidable. 1.
Filled Julia sets with empty interior are computable. eprint, math.DS/0410580
"... Abstract. We show that if a polynomial filled Julia set has empty interior, then it is computable. 1. ..."
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Cited by 14 (8 self)
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Abstract. We show that if a polynomial filled Julia set has empty interior, then it is computable. 1.
Computational complexity of twodimensional regions
 SIAM J. Comput
, 1995
"... The computational complexity of bounded sets of the twodimensional plane is studied in the discrete computational model. We introduce four notions of polynomialtime computable sets in R 2 and study their relationship. The computational complexity of the winding number problem, the membership probl ..."
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Cited by 12 (3 self)
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The computational complexity of bounded sets of the twodimensional plane is studied in the discrete computational model. We introduce four notions of polynomialtime computable sets in R 2 and study their relationship. The computational complexity of the winding number problem, the membership problem, the distance problem and the area problem is characterized by the relations between discrete complexity classes of the NP theory. 1
Efficiently Approximable RealValued Functions
 Electronic Colloquium on Computational Complexity
, 2000
"... We consider a class, denoted APP, of realvalued functions f : f0; 1g n ! [0; 1] such that f can be approximated, to within any ffl ? 0, by a probabilistic Turing machine running in time poly(n; 1=ffl). We argue that APP can be viewed as a generalization of BPP, and show that APP contains a nat ..."
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Cited by 12 (2 self)
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We consider a class, denoted APP, of realvalued functions f : f0; 1g n ! [0; 1] such that f can be approximated, to within any ffl ? 0, by a probabilistic Turing machine running in time poly(n; 1=ffl). We argue that APP can be viewed as a generalization of BPP, and show that APP contains a natural complete problem: computing the acceptance probability of a given Boolean circuit; in contrast, no complete problems are known for BPP. We observe that all known complexitytheoretic assumptions under which BPP is easy (i.e., can be efficiently derandomized) imply that APP is easy; on the other hand, we show that BPP may be easy while APP is not, by constructing an appropriate oracle. 1 Introduction The complexity class BPP is traditionally considered a class of languages that can be efficiently decided with the help of randomness. While it does contain some natural problems, the "semantic" nature of its definition (on every input, a BPP machine must have either at least 3=4 or at...
Computational Complexity of Euclidean Sets: Hyperbolic Julia Sets are PolyTime Computable
, 2004
"... We investigate different definitions of the computability and complexity of sets in R k, and establish new connections between these definitions. This allows us to connect the computability of real functions and real sets in a new way. We show that equivalence of some of the definitions corresponds ..."
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Cited by 12 (9 self)
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We investigate different definitions of the computability and complexity of sets in R k, and establish new connections between these definitions. This allows us to connect the computability of real functions and real sets in a new way. We show that equivalence of some of the definitions corresponds to equivalence between famous complexity classes. The model we use is mostly consistent with [Wei00]. We apply the concepts developed to show that hyperbolic Julia sets are polynomial time computable. This result is a significant generalization of the result in [RW03], where polynomial time computability has been shown for a restricted type of hyperbolic Julia sets. ii Acknowledgements First of all, I would like to thank my graduate supervisor, Stephen Cook. Our weekly meetings not only allowed me to complete this thesis, but also gave me a much broader and deeper understanding of the entire field of theoretical computer science. Working with him has made my learning process a pleasant one.
Complexity and Real Computation: A Manifesto
 International Journal of Bifurcation and Chaos
, 1995
"... . Finding a natural meeting ground between the highly developed complexity theory of computer science with its historical roots in logic and the discrete mathematics of the integers and the traditional domain of real computation, the more eclectic less foundational field of numerical analysis ..."
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Cited by 11 (0 self)
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. Finding a natural meeting ground between the highly developed complexity theory of computer science with its historical roots in logic and the discrete mathematics of the integers and the traditional domain of real computation, the more eclectic less foundational field of numerical analysis with its rich history and longstanding traditions in the continuous mathematics of analysis presents a compelling challenge. Here we illustrate the issues and pose our perspective toward resolution. This article is essentially the introduction of a book with the same title (to be published by Springer) to appear shortly. Webster: A public declaration of intentions, motives, or views. k Partially supported by NSF grants. y International Computer Science Institute, 1947 Center St., Berkeley, CA 94704, U.S.A., lblum@icsi.berkeley.edu. Partially supported by the LettsVillard Chair at Mills College. z Universitat Pompeu Fabra, Balmes 132, Barcelona 08008, SPAIN, cucker@upf.es. P...
Computability of probability measures and MartinLöf randomness over metric spaces
 Information and Computation
"... In this paper we investigate algorithmic randomness on more general spaces than the Cantor space, namely computable metric spaces. To do this, we first develop a unified framework allowing computations with probability measures. We show that any computable metric space with a computable probability ..."
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Cited by 11 (5 self)
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In this paper we investigate algorithmic randomness on more general spaces than the Cantor space, namely computable metric spaces. To do this, we first develop a unified framework allowing computations with probability measures. We show that any computable metric space with a computable probability measure is isomorphic to the Cantor space in a computable and measuretheoretic sense. We show that any computable metric space admits a universal uniform randomness test (without further assumption). 1