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Computing over the reals: Foundations for scientific computing
 Notices of the AMS
"... We give a detailed treatment of the “bitmodel ” of computability and complexity of real functions and subsets of R n, and argue that this is a good way to formalize many problems of scientific computation. In Section 1 we also discuss the alternative BlumShubSmale model. In the final section we d ..."
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Cited by 32 (3 self)
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We give a detailed treatment of the “bitmodel ” of computability and complexity of real functions and subsets of R n, and argue that this is a good way to formalize many problems of scientific computation. In Section 1 we also discuss the alternative BlumShubSmale model. In the final section we discuss the issue of whether physical systems could defeat the ChurchTuring Thesis. 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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Abstract. We show that if a polynomial filled Julia set has empty interior, then it is computable. 1.
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 14 (13 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.
On computational complexity of Siegel Julia sets
 Commun. Math. Physics
"... Abstract. It has been previously shown by two of the authors that some polynomial Julia sets are algorithmically impossible to draw with arbitrary magnification. On the other hand, for a large class of examples the problem of drawing a picture has polynomial complexity. In this paper we demonstrate ..."
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Cited by 10 (4 self)
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Abstract. It has been previously shown by two of the authors that some polynomial Julia sets are algorithmically impossible to draw with arbitrary magnification. On the other hand, for a large class of examples the problem of drawing a picture has polynomial complexity. In this paper we demonstrate the existence of computable quadratic Julia sets whose computational complexity is arbitrarily high. 1. Foreword Let us informally say that a compact set in the plane is computable if one can program a computer to draw a picture of this set on the screen, with an arbitrary desired magnification. It was recently shown by the second and third authors, that some Julia sets are not computable [BY]. This in itself is quite surprising to dynamicists – Julia sets are among the “most drawn ” objects in contemporary mathematics, and numerous algorithms exist to produce their pictures. In the cases when one has not been able to produce informative pictures (the dynamically pathological cases, like maps with a Cremer or a highly Liouville Siegel point) the feeling had been that this was due to the immense computational resources required by the known algorithms.
COMPUTATIONAL UNSOLVABILITY OF DOMAINS OF ATTRACTION OF NONLINEAR SYSTEMS
, 2009
"... Let S be the domain of attraction of a computable and asymptotically stable hyperbolic equilibrium point of the nonlinear system ˙x = f(x). We show that the problem of determining S is computationally unsolvable. We also present an upper bound of the degree of unsolvability of this problem. ..."
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Cited by 6 (6 self)
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Let S be the domain of attraction of a computable and asymptotically stable hyperbolic equilibrium point of the nonlinear system ˙x = f(x). We show that the problem of determining S is computationally unsolvable. We also present an upper bound of the degree of unsolvability of this problem.
Constructing NonComputable Julia Sets
 Proc. of STOC 2007
"... While most polynomial Julia sets are computable, it has been recently shown [12] that there exist noncomputable Julia sets. The proof was nonconstructive, and indeed there were doubts as to whether specific examples of parameters with noncomputable Julia sets could be constructed. It was also unk ..."
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Cited by 4 (0 self)
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While most polynomial Julia sets are computable, it has been recently shown [12] that there exist noncomputable Julia sets. The proof was nonconstructive, and indeed there were doubts as to whether specific examples of parameters with noncomputable Julia sets could be constructed. It was also unknown whether the noncomputability proof can be extended to the filled Julia sets. In this paper we give an answer to both of these questions, which were the main open problems concerning the computability of polynomial Julia sets. We show how to construct a specific polynomial with a noncomputable Julia set. In fact, in the case of Julia sets of quadratic polynomials we give a precise characterization of Julia sets with computable parameters. Moreover, assuming a widely believed conjecture in Complex Dynamics, we give a polytime algorithm for computing a number c such that the Julia set J z 2 +cz is noncomputable. In contrast with these results, we show that the filled Julia set of a polynomial is always computable.
Computable Symbolic Dynamics
, 2008
"... We investigate computable subshifts and the connection with effective symbolic dynamics. It is shown that a decidable Π 0 1 class P is a subshift if and only if there is a computable function F mapping 2 N to 2 N such that P is the set of itineraries of elements of 2 N. Π 0 1 subshifts are construct ..."
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We investigate computable subshifts and the connection with effective symbolic dynamics. It is shown that a decidable Π 0 1 class P is a subshift if and only if there is a computable function F mapping 2 N to 2 N such that P is the set of itineraries of elements of 2 N. Π 0 1 subshifts are constructed in 2 N and in 2 Z which have no computable elements. We also consider the symbolic dynamics of maps on the unit interval. 1
Computability of Countable Subshifts in One Dimension ⋆
, 2011
"... We investigate the computability of countable subshifts in one dimension, and their members. Subshifts of CantorBendixson rank two contain only eventually periodic elements. Any rank two subshift in 2 Z is is decidable. Subshifts of rank three may contain members of arbitrary Turing degree. In cont ..."
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We investigate the computability of countable subshifts in one dimension, and their members. Subshifts of CantorBendixson rank two contain only eventually periodic elements. Any rank two subshift in 2 Z is is decidable. Subshifts of rank three may contain members of arbitrary Turing degree. In contrast, effectively closed (Π 0 1) subshifts of rank three contain only computable elements, but Π 0 1 subshifts of rank four may contain members of arbitrary ∆ 0 2 degree. There is no subshift of rank ω + 1.
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
Randomness, Computation and Mathematics
"... Abstract. This article examines some of the recent advances in our understanding of algorithmic randomness. It also discusses connections with various areas of mathematics, computer science and other areas of science. Some questions and speculations will be discussed. 1 ..."
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Abstract. This article examines some of the recent advances in our understanding of algorithmic randomness. It also discusses connections with various areas of mathematics, computer science and other areas of science. Some questions and speculations will be discussed. 1