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30
On the complexity of numerical analysis
 IN PROC. 21ST ANN. IEEE CONF. ON COMPUTATIONAL COMPLEXITY (CCC ’06
, 2006
"... We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis: • The BlumShubSmale model of computation over the reals. • A problem we call the “Generic Task of Numerical Computation, ” which captures an aspect of doing numerical computation ..."
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Cited by 48 (7 self)
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We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis: • The BlumShubSmale model of computation over the reals. • A problem we call the “Generic Task of Numerical Computation, ” which captures an aspect of doing numerical computation in floating point, similar to the “long exponent model ” that has been studied in the numerical computing community. We show that both of these approaches hinge on the question of understanding the complexity of the following problem, which we call PosSLP: Given a divisionfree straightline program producing an integer N, decide whether N> 0. • In the BlumShubSmale model, polynomial time computation over the reals (on discrete inputs) is polynomialtime equivalent to PosSLP, when there are only algebraic constants. We conjecture that using transcendental constants provides no additional power, beyond nonuniform reductions to PosSLP, and we present some preliminary results supporting this conjecture. • The Generic Task of Numerical Computation is also polynomialtime equivalent to PosSLP. We prove that PosSLP lies in the counting hierarchy. Combining this with work of Tiwari, we obtain that the Euclidean Traveling Salesman Problem lies in the counting hierarchy – the previous best upper bound for this important problem (in terms of classical complexity classes) being PSPACE. In the course of developing the context for our results on arithmetic circuits, we present some new observations on the complexity of ACIT: the Arithmetic Circuit Identity Testing problem. In particular, we show that if n! is not ultimately easy, then ACIT has subexponential complexity.
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.
Computing equilibria: A computational complexity perspective
, 2009
"... Computational complexity is the subfield of computer science that rigorously studies the intrinsic difficulty of computational problems. This survey explains how complexity theory defines “hard problems”; applies these concepts to several equilibrium computation problems; and discusses implications ..."
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Cited by 7 (2 self)
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Computational complexity is the subfield of computer science that rigorously studies the intrinsic difficulty of computational problems. This survey explains how complexity theory defines “hard problems”; applies these concepts to several equilibrium computation problems; and discusses implications for computation, games, and behavior. We assume
NONCOMPUTABLE CONDITIONAL DISTRIBUTIONS
"... Abstract. 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 axiomaticall ..."
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Cited by 7 (3 self)
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Abstract. 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 the search for more constructive definitions is the subject of a rich literature in probability theory and statistics. However, we show that in general one cannot compute conditional probabilities. Specifically, we construct a pair of computable random variables (X, Y) in the unit interval whose conditional distribution P[YX] encodes the halting problem. Nevertheless, probabilistic inference has proven remarkably successful in practice, even in infinitedimensional continuous settings. We prove several results giving general conditions under which conditional distributions are computable. In the discrete or dominated setting, under suitable computability hypotheses, conditional distributions are computable. Likewise, conditioning is a computable operation in the presence of certain additional structure, such as independent absolutely continuous noise.
Points on computable curves
 In Proceedings of the FortySeventh Annual IEEE Symposium on Foundations of Computer Science
, 1999
"... The “analyst’s traveling salesman theorem ” of geometric measure theory characterizes those subsets of Euclidean space that are contained in curves of finite length. This result, proven for the plane by Jones (1990) and extended to higherdimensional Euclidean spaces by Okikiolu (1992), says that a ..."
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Cited by 5 (3 self)
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The “analyst’s traveling salesman theorem ” of geometric measure theory characterizes those subsets of Euclidean space that are contained in curves of finite length. This result, proven for the plane by Jones (1990) and extended to higherdimensional Euclidean spaces by Okikiolu (1992), says that a bounded set K is contained in some curve of finite length if and only if a certain “square beta sum”, involving the “width of K ” in each element of an infinite system of overlapping “tiles” of descending size, is finite. In this paper we characterize those points of Euclidean space that lie on computable curves of finite length. We do this by formulating and proving a computable extension of the analyst’s traveling salesman theorem. Our extension, the computable analyst’s traveling salesman theorem, says that a point in Euclidean space lies on some computable curve of finite length if and only if it is “permitted ” by some computable “Jones constriction”. A Jones constriction here is an explicit assignment of a rational cylinder to each of the abovementioned tiles in such a way that, when the radius of the cylinder corresponding to a tile is used in place of the “width of K ” in each tile, the square beta sum is finite. A point is permitted by a Jones constriction if it is
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.
Computability of Julia Sets
, 2008
"... In this paper we settle most of the open questions on algorithmic computability of Julia sets. In particular, we present an algorithm for constructing quadratics whose Julia sets are uncomputable. We also show that a filled Julia set of a polynomial is always computable. ..."
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Cited by 4 (0 self)
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In this paper we settle most of the open questions on algorithmic computability of Julia sets. In particular, we present an algorithm for constructing quadratics whose Julia sets are uncomputable. We also show that a filled Julia set of a polynomial is always computable.
ON THE COMPUTABILITY OF CONDITIONAL PROBABILITY
"... Abstract. We study the problem of computing conditional probabilities, a fundamental operation in statistics and machine learning. In the elementary discrete setting, conditional probability is defined axiomatically and the search for more constructive definitions is the subject of a rich literature ..."
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Cited by 3 (3 self)
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Abstract. We study the problem of computing conditional probabilities, a fundamental operation in statistics and machine learning. In the elementary discrete setting, conditional probability is defined axiomatically and the search for more constructive definitions is the subject of a rich literature in probability theory and statistics. In the discrete or dominated setting, under suitable computability hypotheses, conditional probabilities are computable. However, we show that in general one cannot compute conditional probabilities. We do this by constructing a pair of computable random variables in the unit interval whose conditional distribution encodes the halting problem at almost every point. We show that this result is tight, in the sense that given an oracle for the halting problem, one can compute this conditional distribution. On the other hand, we show that conditioning in abstract settings is computable in the presence of certain additional structure, such as independent absolutely continuous noise. 1.
A NOTE ON UNIVERSALITY IN MULTIDIMENSIONAL SYMBOLIC DYNAMICS
, 901
"... there is a universal system, i.e. one that factors onto every other effective system. In particular, for d ≥ 3 there exist ddimensional shifts of finite type which are universal for 1dimensional subactions of SFTs. On the other hand, we show that there is no universal effective Zdsystem for d ≥ 2 ..."
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Cited by 3 (1 self)
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there is a universal system, i.e. one that factors onto every other effective system. In particular, for d ≥ 3 there exist ddimensional shifts of finite type which are universal for 1dimensional subactions of SFTs. On the other hand, we show that there is no universal effective Zdsystem for d ≥ 2, and in particular SFTs cannot be universal for subactions of rank ≥ 2. As a consequence, a decrease in entropy and Medvedev degree and periodic data are not sufficient for a factor map to exists between SFTs. We also discuss dynamics of cellular automata on their limit sets and show that (except for the unavoidable presence of a periodic point) they can model a large class of physical systems. 1.
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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Cited by 2 (0 self)
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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