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Computing the noncomputable
 Contemporary Physics
"... We explore in the framework of Quantum Computation the notion of computability, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm that exploits the quantum adiabatic which is equivalent to the Turing halting problem and known to be mathematically non ..."
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Cited by 32 (7 self)
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We explore in the framework of Quantum Computation the notion of computability, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm that exploits the quantum adiabatic which is equivalent to the Turing halting problem and known to be mathematically
Noncomputable Julia sets
 Journ. Amer. Math. Soc
"... Polynomial Julia sets have emerged as the most studied examples of fractal sets generated by a dynamical system. Apart from the beautiful mathematics, one of the reasons for their popularity is the beauty of the computergenerated images of such sets. The algorithms used to draw these pictures vary; ..."
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Cited by 36 (10 self)
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of quadratic polynomials. Under the definition we use, a set is computable, if, roughly speaking, its image can be generated by a computer with an arbitrary precision. Under this notion of computability we show: Main Theorem. There exists a parameter value c ∈ C such that the Julia set of
The Similarity Metric
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 2003
"... A new class of distances appropriate for measuring similarity relations between sequences, say one type of similarity per distance, is studied. We propose a new "normalized information distance", based on the noncomputable notion of Kolmogorov complexity, and show that it is in this class ..."
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Cited by 280 (34 self)
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A new class of distances appropriate for measuring similarity relations between sequences, say one type of similarity per distance, is studied. We propose a new "normalized information distance", based on the noncomputable notion of Kolmogorov complexity, and show that it is in this class
NONCOMPUTABLE CONDITIONAL DISTRIBUTIONS
, 2011
"... We study the computability of conditional probability, a fundamental notion in probability theory and Bayesian statistics. In the elementary discrete setting, a ratio of probabilities defines conditional probability. In more general settings, conditional probability is defined axiomatically, and t ..."
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Cited by 11 (5 self)
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We study the computability of conditional probability, a fundamental notion in probability theory and Bayesian statistics. In the elementary discrete setting, a ratio of probabilities defines conditional probability. In more general settings, conditional probability is defined axiomatically
Noncomputable Spectral Sets
, 2007
"... iii For my Mama, whose *minimal index is computable (because it’s finite). ..."
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Cited by 6 (4 self)
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iii For my Mama, whose *minimal index is computable (because it’s finite).
Noncomputability of consciousness
"... With the great success in simulating many intelligent behaviors using computing devices, there has been an ongoing debate whether all conscious activities are computational processes. In this paper, the answer to this question is shown to be no. A certain phenomenon of consciousness is demonstrated ..."
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Cited by 1 (1 self)
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to be fully represented as a computational process using a quantum computer. Based on the computability criterion discussed with Turing machines, the model constructed is shown to necessarily involve a noncomputable element. The concept that this is solely a quantum effect and does not work for a classical
Clustering by compression
 IEEE Transactions on Information Theory
, 2005
"... Abstract—We present a new method for clustering based on compression. The method does not use subjectspecific features or background knowledge, and works as follows: First, we determine a parameterfree, universal, similarity distance, the normalized compression distance or NCD, computed from the l ..."
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Cited by 297 (25 self)
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developed by one of the authors, is provably optimal. However, the optimality comes at the price of using the noncomputable notion of Kolmogorovcomplexity. We propose axioms to capture the realworld setting, and show that the NCD approximates optimality. To extract a hierarchy of clusters from the distance matrix
Computability, noncomputability, and hyperbolic systems
 Appl. Math. Comput
, 2012
"... In this paper we study the computability of the stable and unstable manifolds of a hyperbolic equilibrium point. These manifolds are the essential feature which characterizes a hyperbolic system, having many applications in physical sciences and other fields. We show that (i) locally these manifolds ..."
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Cited by 1 (1 self)
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In this paper we study the computability of the stable and unstable manifolds of a hyperbolic equilibrium point. These manifolds are the essential feature which characterizes a hyperbolic system, having many applications in physical sciences and other fields. We show that (i) locally these manifolds can be computed, but (ii) globally they cannot, since their degree of computational unsolvability lies on the second level of the Borel hierarchy. We also show that Smale’s horseshoe, the first example of a hyperbolic invariant set which is neither an equilibrium point nor a periodic orbit, is computable. 1
EXISTENCE OF NONCOMPUTABLE FUNCTIONS
"... N: natural numbers ℘(N): set of all subsets of natural numbers N → N: functions from N to N Progs: programs in C++ (or Java) ..."
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N: natural numbers ℘(N): set of all subsets of natural numbers N → N: functions from N to N Progs: programs in C++ (or Java)
Noncomputability, unpredictability, and financial markets
, 2012
"... One of the most significant achievements from theoretical computer science was to show that there are noncomputable problems, which cannot be solved through algorithms. Although the formulation of such problems is mathematical, they often can be interpreted as problems derived from other fields, li ..."
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Cited by 1 (0 self)
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One of the most significant achievements from theoretical computer science was to show that there are noncomputable problems, which cannot be solved through algorithms. Although the formulation of such problems is mathematical, they often can be interpreted as problems derived from other fields
Results 1  10
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